Algebraic

Real
Analysis
PETER FREYD
pjf @ upenn.edu

J
J

Special thanks to Mike Barr and Don von Osdol for editorial assistance and
to the Executive Director of the FMS Foundation for making it all possible.

An earlier draft appeared in TAC (2008–07–02). This is the current revision (2019–06–25).

The earlier version can be found at http://www.tac.mta.ca/tac/ or by anonymous

ftp from ftp://ftp.tac.mta.ca/pub/tac/html/volumes/20/10/20-10.{dvi,fps,pdf}
The current revision is available at

http://www.math.upenn.edu/~pjf/analysis.pdf
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The Houdini Diagram

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 ALG
GEBRA
AIC REA
A
ALAANALYSIS

Frontispiece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
1. Diversion: The Proximate Origins, or: Coalgebraic Real Analysis . . . . . ??
2. The Equational Theory of Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
3. The Initial Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
4. Lattice Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
5. Diversion: Lukasiewicz vs. Girard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
6. Diversion: The Final Interval Coalgebra as a Scale . . . . . . . . . . . . . . . . . . . ??
7. Congruences, or: >–Faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
8. The Linear Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
9. Lipschitz Extensions and I-Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
10. Simple Scales and the Existence of Standard Models . . . . . . . . . . . . . . . . . ??
11. A Few Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
12. Non–Semi-Simple Scales and the Richter Scale . . . . . . . . . . . . . . . . . . . . . . . ??
13. A Construction of the Reals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
14. The Enveloping D-module. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
15. The Semi-Simplicity of Free Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
16. Diversion: Harmonic Scales and Differentiation . . . . . . . . . . . . . . . . . . . . . . . ??
17. Subintervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
18. Extreme Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
19. Diversion: Chromatic Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
20. The Representation Theorem for Free Scales . . . . . . . . . . . . . . . . . . . . . . . . . ??
21. Finitely Presented Scales, or: How Brouwer Made Topology Algebraic ??
22. Complete Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
23. Scales vs. Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
24. Injective Scales, or: Order-Complete Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
25. Diversion: Finitely Presented Chromatic Scales . . . . . . . . . . . . . . . . . . . . . . . ??
26. Appendix: Lattice-Ordered Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . ??
27. Appendix: Computational Complexity Issues . . . . . . . . . . . . . . . . . . . . . . . . . ??
28. Appendix: Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
29. Appendix: Continuously vs Discretely Ordered Wedges . . . . . . . . . . . . . . . ??
30. Appendix: Signed-Binary Expansions:
The Contrapuntal Procedure and Dedekind Sutures. . . . . . ??
31. Appendix: Dedekind Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
32. Appendix: The Peneproximate Origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
33. Addendum: A Few Latex Macros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
34. Addendum: Heyting Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
35. Addendum: Wilson Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
36. Addendum: Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
37. Addendum: Extending the Reach of the Theorem on Standard Models ??
38. Addendum: Boolean-Algebra Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
39. Addendum: On the Definition of I-Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
40. Addendum: Proofs for Section 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
41. Addendum: Extreme Points, Faces and Convex Sets . . . . . . . . . . . . . . . . . . ??
42. Addendum: Scale Spectra Are Compact Normal . . . . . . . . . . . . . . . . . . . . . . ??
43. Addendum: Stream Docking and Other Signed-Binary Automata . . . . . ??
44. Addendum: The Rimsky Scale and Dedekind Incisions. . . . . . . . . . . . . . . . ??
45. Addendum: Lebesgue Integration and Measure, Rethought . . . . . . . . . . . ??
46. Addendum: A Few Subscorings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

2
 ALG
GEBRA
AIC REA
A
ALAANALYSIS

0. Introduction
The title is wishful thinking; there ought to be a subject that deserves the name “algebraic
real analysis.”
Herein is a possible beginning.
For reasons that can easily be considered abstruse we were led to the belief that the closed
interval—not the entire real line—is the basic structure of interest. Before describing those
abstruse reasons, a theorem:
Let G be a compact group and I the closed interval. (We will not say which closed interval;
to do so would define it as a part of the reals, belying the view of the closed interval as the
fundamental structure.) Let C(G) be the set of continuous maps from G to I. We wish to
view this as an algebraic structure, where the word “algebra” is in the very general sense,
something described by operations and equations. In the case at hand, the only operators
that will be considered right now are the constants, “top” and “bottom,” denoted > and ⊥,
and the binary operation of “midpointing,” denoted x||y. (There are axioms that will define
the notion of “closed midpoint algebra” but since the theorem is about specific examples
they’re not now needed.) C(G) inherits this algebraic structure in the usual way (f |g, for
example, is the map that sends σ ∈ G to (f σ) | (gσ) ∈ I). We use the group structure on
G to define an action of G on C(G), thus obtaining a representation of G on the group of
automorphisms of the closed midpoint algebra. (Fortunately no knowledge of the axioms is
necessary for the definition of automorphism—or even homomorphism.) Let (C(G), I) be the
set of closed–midpoint-algebra homomorphisms from C(G) to I. Again we obtain an action
of G.
0.1 Theorem: There is a unique G–fixed-point in (C(G), I)
There is an equivalent way of stating this:
0.2 Theorem: There is a unique G-invariant homomorphism from the closed midpoint
algebra C(G) to I.
This theorem is mostly von Neumann’s: the unique G-invariant R homomorphism is integra-
tion, that is, it is the map that sends f : G → I to f dσ. But it is not entirely
von Neumann’s: we have just characterized integration on compact groups without a
single mention of semiquations or limits (see Section 40, p??–??, for proofs). The only non-
algebraic notion that appeared was at the very beginning in the definition of C(G) as the set of
continuous maps (in Section 11, p??–??, we will obtain a totally algebraic characterization).
The fact that we are stating this theorem for I and not the reals, R, is critical. Consider
the special case when G is the one-element group; the theorem says that the identity map on I
is the only midpoint-preserving endomorphism that fixes > and ⊥ (we said that the theorem
is mostly von Neumann’s; this part is not, see Section 4, p??–??). It actually suffices to
assume that the endomorphism fixes any two points (Lemma 13.1, p??) but with the axiom
ℵ
of choice and a standard rational Hamel-basis argument we can find 22 0 counterexamples
for this assertion if I is replaced with R (even when the number of designated fixed-points is
not just two but countably infinite).
We do not need a group structure or even von Neumann to make the point. Consider this
remarkably simple characterization of definite integration of continuous maps from interval
to interval.

3
 ALG
GEBRA
AIC REA
A
ALAANALYSIS

Z Z
> dx = > ⊥ dx = ⊥

R
Z Z Z
f (x) | g(x) dx = f (x) dx | g(x) dx

R
Z Z Z
f (x) dx = f (⊥|x) dx | f (>|x) dx [1]

R
No semiquations. No limits. The first three equations say just that integration is a homo-
morphism of closed midpoint algebras. The fourth equation says that the mean value of a
function on I is the midpoint of the two mean-values of the function on the two halves of I.
The fourth equation—as any numerical or theoretical computer scientist will tell you—
is a “fixed-point characterization.” When Church proved that his and Gödel’s notion of
computability were coextensive he used the fact that all computation can be reduced to
finding fixed-points. (The word “point” here is traditional but misleading. The fixed-point
under consideration here is, as it was for Church, an operator on functions—rather far
removed from the public notion of point.)
If we seek a fixed-point of an operator the first thing to try, of course, is to iterate the
operator on some starting point and to hope that the iterations converge. So let
Z
f (x) dx
R

denote an “initial approximation” operator, to wit, an arbitrary operator from C(I) to I that
satisfies the first three equations. Define a sequence of operators, iteratively, as follows:
Z Z Z
f (x) dx = f (⊥|x) dx f (>|x) dx
R

n+1 n n

where each new operator is to be considered an improvement of the previous. (One should
verify that we automatically maintain the first three equations in each iteration.) Thus the
phrase “fixed-point” here turns out to mean an operator so good that it can not be improved.
(What is being asserted is that there is a unique such operator.) Wonderfully enough: no
matter what closed midpoint homomorphism is chosen as the initial approximation, the values
of these operators are guaranteed to converge.
If we take the initial approximation to be evaluation on ⊥, that is, if we take
Z
f (x) dx = f (⊥)
R

then what we are saying turns out to be only that “left Riemann sums” work for integration. If
we use f (>) for the initial approximation we obtain “right Riemann sums.” For the “trapezoid
rule” use the midpoint of these two initial operators, f (⊥)|f (>). For “Simpson’s rule” use
1
6
f (⊥) + 13 f (⊥|>) + 16 f (>). [2]
[1 ] There was an appendix for Latex macros, but the powers that be deemed such to be beneath the dignity of this journal.
It appears here as Section 33 (p??–??).
[ 2 ] If this work’s title is to be taken seriously we will be obliged to give an algebraic description of the limits used in the previous
Q
paragraph. Here’s one way: let IN = N I denote the closed midpoint algebra of all sequences in I. The first step is to identify
sequences that agree almost everywhere to obtain the quotient algebra IN → A . The latter will be shown—for entirely algebraic
reasons—to have a closed midpoint homomorphism to I (Theorem 10.4, p??) and we could use any such homomorphism to
define the sequential limits appearing in the previous paragraph. There is, of course, an obvious objection: we would be assigning

4
 ALG
GEBRA
AIC REA
A
ALAANALYSIS

1. Diversion: The Proximate Origins, or: Coalgebraic Real Analysis

The point of departure for this approach to analysis is the use of the closed interval as the
fundamental structure; the reals are constructed therefrom. A pause to describe how I was
prompted to explore such a view.
The community of theoretical computer scientists (in the European sense of the phrase)
had found something called “initial-algebra” definitions of data types to be of great use. Such
definitions typically tell one how inductive programs—and then recursive programs—are to be
defined and executed.[3] It then became apparent that some types required a dual approach:
something called “final-coalgebra definitions.”[4] Such can tell one how “co-inductive” and
“co-recursive” programs are to be defined and executed. (One must really resist here the
temptation to say “co-defined” and “co-executed.”)
Thus began a search for a final-coalgebra definition of that ancient data type, the reals.
There is, actually, always a trivial answer to such a question: every object is automatically
the final coalgebra of the functor constantly equal to that object. What was being sought was
not just a functor with a final coalgebra isomorphic to the object in question but a functor
that supplies its final coalgebra with the structure of interest. In 1999 an answer was found
not for the reals but for the closed interval.[5] (To this date, no one has found a functor whose
final coalgebra is usefully the reals.)
Consider, then, the category whose objects are sets with two distinguished points, denoted
> and ⊥ and whose maps are the functions that preserve > and ⊥. Given a pair of objects, X
and Y, we define their ordered wedge, denoted X ∨Y , to be the result of identifying the
top of X with the bottom of Y. [6] This construction can clearly be extended to the maps to
obtain the “ordered-wedge functor.” The closed interval can be defined as the final coalgebra
of the functor that sends X to X ∨X. Let me explain.
First (borrowing from the topologists’ construction of the ordinary wedge), X∨Y is taken
as the subset of pairs, hx, yi, in the product X × Y that satisfy the disjunctive condition:
x = > or y = ⊥.[7] A map, then, from X to X ∨X may be construed as a pair of self-maps,
∧ ∨ ∨ ∧
whose values are denoted x and x , such that for all x either x = > or x = ⊥. The final
coalgebra we seek is the terminal object in the category whose objects are these structures.
limits to all sequences not just convergent ones; worse, the homomorphism would be not at all unique. Remarkably enough
we can turn this inside out: an element in IN is convergent iff it is in the joint equalizer of all homomorphisms of the form
IN → A → I. Put another way, lim an = b iff h({an }) = b whenever h : IN → I is a closed midpoint homomorphism that
respects almost-everywhere equivalence. (See Section 11, p??–??, for an approach to this definition of limits that does not require
the axiom of choice.) This approach can be easily modified to supply limits of I-valued functions at points in arbitrary topological
spaces. More interesting: it can be used to define derivatives. Let F be the set of all functions f from the standard interval
[−1, +1] to itself such that |f x| ≤ |x|. We will regard F as a closed midpoint algebra where the identity function is taken as >
and the negation map as ⊥. Now identify functions that name the same germ at 0 (that is, that agree on some neighborhood of
0) to obtain a quotient algebra F → A. The joint equalizer of all homomorphisms of the form F → A → [−1, +1] is precisely
the set of functions differentiable at 0; the common values delivered by all such homomorphisms are the derivatives of those
functions. That is, f 0 (0) = b iff H(f ) = b whenever H : F → [−1, +1] is a closed midpoint homomorphism that depends
only on the germs at 0 of its arguments. (Again, see Section 11, p??–??.) And, yes, for each n there’s a similar construction
for nth derivatives.
[ 3 ] Let me brag here: I won a prize for one of a series of papers on this subject: Recursive types reduced to inductive

types 5th Annual IEEE Symposium on Logic in Computer Science [lics] (Philadelphia, PA. 1990), p408–507 IEEE
Comput. Soc. Press, Los Alamitos, CA, 1990. The series culminated with Remarks on algebraically compact categories,
Applications of categories in computer science (Durham, 1991), p95–106. London Math. Soc. Lecture Note Ser., 177,
Cambridge Univ. Press, Cambridge. 1992.
[4 ] We don’t actually need the general definitions, but for the record let T be an arbitrary endofunctor. X → TX
A T -coalgebra is just a map X → T X. A map between T -coalgebras is illustrated by the commutative dia- f ↓ ↓ Tf
gram on the right. If the resulting category of T -coalgebras has a final object, it’s called a final T -coalgebra. Y → TY
[ 5 ] First announced in a note I posted on 22 December, 1999, http://www.mta.ca/~cat-dist/1999/99-12.
[ 6 ] The word “wedge” and its notation are borrowed from algebraic topology where X ∨ Y is the result of joining the (single)

base-point of X to that of Y.
[ 7 ] Yes, X ∨ Y it a pushout: given f : X → Z and g : Y → Z such that f > = g ⊥ define X ∨ Y → Z by sending hx, y i to

f x if y = ⊥ else gy.

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To be formal, begin with the category whose objects are quintuples hX, ⊥, >, ∧, ∨i where
⊥, > ∈ X, and ∧, ∨ signify self-maps on X. The maps of the category are the functions that
preserve the two constants and the two self-maps. Then cut down to the full subcategory of
objects that satisfy the conditions:
∧ ∨
> = > =>
∧ ∨
= ⊥ =⊥
⊥
∧ ∨
∀x [ x = ⊥ or x = > ]
[8]
⊥ 6= >
We will call such a structure an interval coalgebra.[9]
I said that we will eventually construct the reals from I. But if one already has the reals
then one may chose ⊥ < > and define a coalgebra structure on [⊥, >] as
∨
x = min(2x−⊥, >)
∧
x = max(2x−>, ⊥)
Note that each of the two self-maps evenly expands a half interval to fill the entire
interval—one the bottom half the other the top half. We will call them zoom operators.
(By convention we will not say “zoom in” or “zoom out.” All zooming herein is expansive,
not contractive.)
The general definition of “final coalgebra” reduces—in this case—to the characterization
of such a closed interval, I, as the terminal object in this category.[10]
The general notion of “co-induction” reduces—in this case—to the fact that given any
f
quintuple hX, ⊥, >, ∧, ∨i satisfying the above-displayed conditions there is a unique X → I
∨ ∧
such that f (⊥) = ⊥, f (>) = >, f (x) = (f x)∨ and f (x) = (f x)∧. If I is taken as the unit
interval, that is, if ⊥ is taken as 0 and > as 1, then in the classical setting (and if one pays
no attention to computational feasibility) a quick and dirty construction of f is to define the
binary expansion of f (x) ∈ I by iterating (forever) the following procedure:
∨ ∧
If x = > then emit “1” and replace x with x
∨
else emit “0” and replace x with x.

[ 8 ] I did not say “with a pair of distinguished points” above. What I said was “with two distinguished points.” (Yes, I’m one

of those who object to the phrase “these two things are the same.”)
[ 9 ] The modal operations 3 for possibly and 2 for necessarily have received many formalizations but it is safe to say that no

one allows simultaneously both 3Φ 6= T and 2Φ 6= : “less than completely possible but somewhat necessary.” (The coalgebra
T

condition can be viewed here as a much weakened excluded middle: when the pair of unary operations are trivialized—that is,
each taken to be the identity operation—then 3Φ = T or 2Φ =
T
becomes just standard excluded middle.)
are fixed points for 3 and 2 then we have an example of an interval coalgebra
T
If we assume, for the moment, that T and
∧ ∨
where 2Φ = Φ and 3Φ = Φ . The finality of I yields what may be considered truth values for sentences (e.g., the truth value
of ⊥|> translates to “entirely possible but totally unnecessary” and a truth value greater than >|(>|⊥) means “necessarily
entirely possible”).
The fixed-point conditions are not, in fact, appropriate—true does not imply necessarily true nor does possibly false imply
false—but, fortunately, they’re not needed: an easy corollary of the finality of I says that it suffices to assume the disjointness
T
of the orbits of T and under the action of the two operators. If we work in a context in which the modal operations are
monotonic (that is, when Φ implies Ψ it is the case that 2Φ implies 2Ψ and 3Φ implies 3Ψ) it suffices to assume that
2Φ implies Φ, that Φ implies 3Φ and that 2n T never implies 3n . If this last condition has never previously been stated
T

it’s only because no one ever thought of needing it.

The same treatment of modal operators holds when 3 is interpreted as tenable/conceivable/allowed/foreseeable and 2 as
certain/known/required/expected. (Note that 3 and 2 are “De Morgan duals,” that is, 2Φ = ¬3(¬Φ) and 3Φ = ¬2(¬Φ).)
This topic will be much better discussed in the intuitionistic foundations considered in Section 29 (see [??], p??).
[ 10 ] If the case with ⊥ = > were allowed then the terminal object would be just the one-point set. (In some sense, then, the

separation of > and ⊥ requires no less than an entire continuum.)

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Note that the symmetry on I is forced by its finality: if > and ⊥ are interchanged and if
>- and ⊥-zooming are interchanged the definition of interval coalgebra is maintained, hence
there is a unique map from hI, ⊥, >, ∧, ∨i to hI, >, ⊥, ∨, ∧i that effects those interchanges and
it is necessarily an involution. It is the symmetry being sought.
The ≤ = relation on I may be defined as the most inclusive binary relation preserved by
∧ and ∨ that avoids > ≤
= ⊥. We will delay the (more difficult) proof that the characterization
yields a construction of the midpoint operator that figures so prominently in the opening
(and throughout this work).
The assertion that the final coalgebra may be taken as the unit interval [11] needs a full
proof—actually several proofs depending on the extent of constructive meaning one desires
in his notion of the closed interval (see Sections 29–31, p??–??). But we move now from the
coalgebraic theory with its disjunctive condition to an algebraic theory in the usual purely
equational sense.

2. The Equational Theory of Scales

The theory of scales is given by:

a constant top
denoted >;
a unary operation dotting
.
whose values are denoted x ;
a unary operation top-zooming
∧
whose values are denoted x and;
a binary operation midpointing
whose values are denoted x|y.
Define:
the constant bottom
.
by ⊥ = > ;
the constant center
by = ⊥|> and;
the unary operation bottom-zooming
.
∨ ∧.
by x = x.

We’ll use both | and | for midpointing and—when convenient—denote (x|y)∧ as

∧ ∨ ∧ ∨
x | y or xd | y and (x|y)∨ as x | y.
y Indeed, we will often treat | and | as binary
operations.[12]

[ 11 ] It is also the final coalgebra of any finite iteration of ordered wedges. If we take the n-fold ordered wedge, X ∨ X ∨ · · · ∨ X,

as the set of n-tuples of the form hx0 , x1 , . . . , xn−1 i such that either xi = > or xi+1 = ⊥ for i = 0, 1, . . . , n−2
then the coalgebra structure is an n-tuple of functions z0 , z1 , . . . , zn−1 such that for each x ∈ X and each ` i = 0, 1, . . . , ´n−2
either zi (x) = > or zi+1 (x) = ⊥. The coalgebra structure on the unit interval is given by zi (x) = max 0, min(nx−i , 1) for
i = 0, 1, . . . , n−1. Given x ∈ X obtain the base-n expansion for its corresponding element in the unit interval by iterating
(forever) the following procedure:

Let i = 0;
While zi (x) = > and i < n−1 replace i with i+1;
Emit “ i ” and replace x with zi (x).

[ 12 ] Indeed—in my naive innocence, much to my surprise—they are associative binary operations. See [??], p??.

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The six axioms (7 equations) that define the theory of scales:

idempotent:
x|x = x
[13]
commutative:
x|y = y|x
[14][15]
medial:
(v|w) | (x|y) = (v|x) | (w|y)
[16]
constant:
.
x |x =
[17]
unital: ∧ ∨
>| x = x = ⊥| x
the scale identity:
d∧ ∧d∨
∨
xdy = x y x y
The standard model is the closed real interval I of all real numbers from −1 through +1.
More generally, let D be the ring of dyadic rationals (those with denominator a power of 2).
In D-modules (or as we will pronounce them, “dy-modules”) with total orderings we may
choose elements ⊥ < >, and define a scale as the set of all elements from ⊥ through > with
. ∧ ∨
x|y = (x + y)/2, x = ⊥ + > − x, x = max(2x −>, ⊥) (hence x = min(2x −⊥, >)). [18] The
standard interval in D, that is, the interval from −1 through +1, will be shown in the next
section to be isomorphic to the initial scale (the scale freely generated by its constants). It
will be denoted I, the standard D-interval (pronounced “dy-interval”).[19]
The verification of all but the last of the defining equations on the standard interval is
entirely routine. It will take a while before the scale identity reveals its secrets: how it first
became known; how it can be best viewed; why it is true for the standard models.[20]
[ 13 ] This axiom can be replaced with a single instance: ⊥|> = >|⊥. See [??] on p??.
[ 14 ] Sometimes “middle-two interchange.”
[ 15 ] The medial law has a geometric interpretation: it says that the midpoints of a cycle of four edges on a tetrahedron are

the vertices of a parallelogram (“the medial parallelogram”). That is, view four points A, B, C, D in general position in Euclidean
space of dimension three (or more). Consider the closed path from A to B to D to C back to A and note that the four successive
midpoints A|B, B|D, D|C, C|A appear in the medial law (A|B) | (C|D) = (A|C) | (B|D). This equation says, among other things,
that the two line segments, the one from A|B to C|D and the one from A|C to B|D, having a point in common, are coplanar,
forcing the four midpoints, A|B, B|D, D|C, C|A to be coplanar. The medial law says, further, that these two coplanar line
segments have their midpoints in common, And that says—indeed, is equivalent with—A|B, B|D, D|C, C|A being the vertices
of a parallelogram. (A traditional proof is obtainable from the observation that the two line segments, the one from A|B to A|C
and the other from D|B to D|C, are both parallel to—and half the length of—the line segment from B to C.)
[ 16 ] A technically simpler equation is the two-variable u . .
|u = v |v.
.
[ 17 ] The commutative axiom can be removed entirely if the first (left) unital law is replaced with ⊥ ∧| x = x. See Section 28

(p??–??).
[ 18 ] In Section 14 (p??–??) we will see that every scale has a faithful representation into a product of scales that arise in this

way.
[ 19 ] The free scale on one generator will be shown in Section 20 (p??–??), to be isomorphic to the scale of continuous piecewise

affine functions (usually called piecewise linear) from I to I where each affine piece is given using dyadic rationals. We will
give a generalization of the notion of piecewise affine so that the result generalizes: a free scale on n generators is isomorphic
to the scale of all functions from In to I that are continuous piecewise affine with each piece given by dyadic rationals. The
result further generalizes: essentially for every finitely presented scale there is a finite simplicial complex such that the scale is
isomorphic to the scale of continuous piecewise affine maps with dyadic coefficients from the complex to I . Their full subcategory
can then be described in a piecewise affine manner. See Sections 21–22 (p??–??).
[ 20 ] As forbidding as the scale identity appears, this writer, at least, finds comfort in the fact that the Jacobi identity for Lie

algebras looks at first sight no less forbidding. Indeed, the scale identity has 2 variables and the standard Jacobi identity 3 (with
each variable appearing three times in each); the scale identity has 1 binary operation and it appears 4 times, the Jacobi has 2
binary operations, one of which appears twice and the other 6 times. (Its more efficient—and meaningful—form is a bit simpler:
[[x, y], z] = [x, [y, z]] − [y, [x, z]].) By these counts even the high-school distributivity law is worse than the scale identity (it has
3 variables that appear a total of 7 times and 2 binary operations that appear a total of five times). Yeah, the scale identity has
a total of 11 operations but 7 of them are just unary.

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An ad hoc verification of the scale identity on the standard model may be obtained by
noting first that:
 
∧ −1 if x ≤
=0 ∨ 2x+1 if x ≤=0
x= x=
2x−1 if x ≥ =0 +1 if x ≥
=0

The scale identity then separates into four cases depending on the signatures of the two
variables. When both are positive the two sides of the identity quickly reduce to x + y − 1
and, when both are negative, to −1. In the mixed case (because of commutativity we know
in advance that the two mixed cases are equivalent) where x is positive and y is negative, the
x+y
left side is, of course, ([ \ The entire verification
) and the right side reduces to (−1) (x+y).
2
d = (−1) z∧ which is, in turn, quickly verified by considering the two
is now reduced to z/2
possible signatures of z. [21]
For a fixed element a we will use a| to denote the contraction at a, the unary operation
that sends x to a|x.
We will use freely:
2.1 Lemma: self-distributivity:
a|(x|y) = (a|x) | (a|y)

an immediate consequence of idempotence and the medial law: a|(x|y) = (a|a)|(x|y) =

(a|x)|(a|y). This is equivalent, of course, with a contraction being a midpoint
homomorphism.[22]
Define a/, the dilatation at a, by:
.\ ∨ [ 23 ]
a/ x = (a |⊥) | x

2.2 Lemma: Dilatation undoes contraction:

a/ (a|x) = x
because using the medial, constant, self-distributive and both unital laws:[24] a / (a|x) =
. \ ∨ . \ ∨ \ ∨ ∨
\
(a |⊥) | (a|x) = (a |a) | (⊥|x) = (⊥|>) | (⊥|x) = ⊥ | (>|x) = >
d|x = x. [25][26][27]
[ 21 ] In Section 15 (p??–??) we will show that the defining equations for scales are complete, that is, any new equation involving

only the operators under discussion is either a consequence of the given equations or is inconsistent with them. Put another way:
any equation not a consequence of these axioms fails in every non-degenerate scale. In particular, it fails in the initial scale. Put
still another way: any equation true for any non-degenerate scale is true in all scales. A consequence is that the equational theory
is decidable. In Section 27 (p??–??) it will be shown that the verification of an equation, though decidable, is NP-complete.
It may be noted that the previously stated faithful representation of free scales as scales of functions makes the equational
completeness clear: if an equation on n variables fails anywhere it fails in the free scale on n generators; but if the two sides
of the failed equation are not equal when represented as functions from the n-cube we may apply the evaluation operator at a
dyadic-rational point where the functions disagree to obtain two distinct points in the initial scale, I. Because the evaluation
operator is a homomorphism of scales we thus obtain a counterexample in I. Hence the set of equations that hold in all scales is
the same as the set of equations that hold in I, necessarily a complete equational theory. (Alas, the proof of the faithfulness of
the representation in question requires the equational completeness.)
[ 22 ] It’s worth finding the high-school–geometry proof for self-distributivity in the case that a, x and y are points in R.2
[ 23 ] For one way of finding this formula for dilatation see [??] (p??).
[ 24 ] See Section 46 (p??–??) for a subscoring of the following equations.

[ 25 ] ∧
The zooming operations may be viewed as special cases of dilatations. One can easily verify that > / x = x and in [??]
∨
(p??) we’ll verify that ⊥ / x = x.
[ 26 ] For those looking for a Mal0cev operator, txyz, note that y/ (x|z) is exactly that (tabb = a = tbba). On that subject see

New Entry 2.137 at http://www.math.upenn.edu/~pjf/amplifications.pdf

[ 27 ] Existential problem.

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We immediately obtain:
2.3 Lemma: the cancellation law:

If a|x = a|y then x = y.

Two important equations for dotting:

2.4 Lemma: the involutory law:

:
x = x

2.5 Lemma: dot-distributivity:

. . .
(u | v) = u | v

. : : . .
Both can be quickly verified using commutativity and cancellation: x | x = x | x = x |x
. . . . . . . . .
and (u|v)|(u|v) = (u|v) |(u|v) = u |u = (u |u) |(u |u) = (u |u) |(v |v) = (u| u)|(v | v) =
. .
(u|v)|(u | v). [28]
Given a term txy . . . z involving >, ⊥, midpointing, dotting, >- and ⊥-zooming, the
dual term is. the result of fully applying the distributivity and involutory laws
. .
to (t x · · · z). It has the effect of swapping ∧ with ∨ and > with ⊥. [29] If we replace
both sides of an equation with their dual terms we obtain the dual equation.
We have already seen one pair of dual equations, to wit, the unital laws. (That’s not
counting a whole bunch of self-dual equations.) The dual equation of the scale identity is:
∨ ∧∨∨ ∨∨∧
u | v = (u | v) || (u | v)
Note that we have not yet allowed dilatations in the terms to be dualized.[30]
As a direct consequence of the idempotent and unital laws we have that > is a fixed-point
∧ ∨
for >-zooming: > = > d|> = >. Dually, ⊥ = ⊥. > is a also a fixed-point for ⊥-zooming using
the unital law, scale identity (for the first time), idempotent, commutative and unital laws:
∨
d∧ ∧d∨ ∨d d∨ ∨ ∨ ∨ ∧
>=> d|> = > | > | > | > = > |> | >| > = > | > = >. Dually ⊥ = ⊥. That is:
2.6 Lemma: Both > and ⊥ are fixed-points for both ∧ and ∨.

[ 28 ] To see how the axiom ⊥|> = >|⊥ suffices for commutativity, first note that commutativity was not used to
. .
obtain the left cancellation law (a|x = a|y implies x = y). One consequence is that x = v implies x = v (use left can-
. . ∨
cellation on« x|x = x|v). Besides being monic, dotting is epic because the second unital law, ⊥ | x = x, when written in full says
.”∧ . .”∧
„“
. “
(⊥|x) = x, hence for all x there is v such that v = x (to wit, (⊥|x) ). Hence dotting is an invertible operation.
. .
If y|x = then y = x because if we let z be such that z = y then y|x = y|z and we use cancellation to obtain x = z
. . . . . .
and, hence, y = x . A consequence is (u|v) = u | v because it suffices to show (u | v)|(u|v) = which follows easily using the
. :
medial, constant and idempotence laws. Another consequence: when y = z and x =z . we obtain z =z .
. . : .
Quite enough to establish that the center is central: x| = (x|x)|(x|x) = (x|x)|(x|x) = (x | x)|x = |x. Finally, |(x|y) =
( |x)|( |y) = ( |x)|(y| ) = ( |y)|(x| ) = ( |y)|( |x) = |(y|x) and left cancellation yields x|y = y|x. See Section 46
(p??–??) for subscorings.
[ 29 ] Or: replace each operator with its horizontal mirror image.

[ 30 ]
In time we will be able to do so. We will show (in [??], p??) that dilatations are self-dual just as are midpointing and
„“
. ”∨ «∧ „“ . ”∧«∨
dotting. That is, we will show (a |⊥)|x = (a |>)|x .

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In our second direct use of the scale identity we replace its second variable with > and use
the unital law to obtain:[31]
2.7 Lemma: the law of compensation:
∨ ∧
x = x || x
d∧ ∧d∨ ∨d ∧d ∨ ∧
∨
because x = x|
d > = x | > | x | > = x |> | x |> = x | x .

A consequence:
2.8 Lemma: the absorbing laws:
∧ ∨
x| ⊥ = ⊥ and x| > = >
∨ ∧
because we can use the law of compensation and then cancellation on x|⊥ = (x | ⊥) | (x | ⊥) =
∧
x|(x | ⊥). [32]
2.9 Lemma: A scale is trivial iff > = ⊥.
Because if > = ⊥ then x = x|
d > = x|
d ⊥ = ⊥ for all x.
The unital laws (or, for that matter, the absorbing laws) easily yield:
2.10 Lemma:
∧ ∨
= ⊥ and = >
. .
The center is the only self-dual element, = . (If x = x then apply x| to both sides to
.
obtain x| x = x|x, that is, = x.)
If the second variable is replaced with in the scale identity (this is its third direct
∧ ∨
[33]
use ), the equations = ⊥ and = > yield a special case of the (not correct-in-general)
distributive laws for >- and ⊥-zooming:
2.11 Lemma: the central distributivity laws:
∧ ∧ ∨ ∨
x| = x |⊥ and x| = x |>

∨d ∧ ∧d ∨ ∨
d ∧d ∧
d =x
because x| | | x | = x |⊥ | x |> = ⊥| x . [34]
We will need:
2.12 Lemma:
.
x = y iff x| y =
. . .
because if x |y = we can use cancellation on x | y = x | x.
A consequence is what’s called “swap-and-dot:” given w|x = v|z swap-and-dot any pair
. .
of terms from opposite sides to obtain equations such as w| v = x|z. (From w|x = v|z infer
. . . . . . . .
= (w|x)|(v|z) = (w|x)|(v | z) = (w| v)|(x| z) = (w| v)|(x |z).)
Note that the commutative and medial laws say that (w|x)|(y|z) is invariant under all 24
∨ ∧
permutations of the variables (as, of course, are (w|x) | (y|z) and (w|x) | (y|z)).
31 ] ∨
[ 32
∨ ∨
>= > can now be viewed as a special case of the absorbing law: > = > | > = >.
[ 33 ] This is the last direct use of the scale identity until Section 4 (p??–??).
“ . ”∨∧ “ ”∨∧ “ ∨ ”∧ ∨
[ 34 ] As promised, we can now easily prove ⊥ / x = (⊥ |⊥)|x = |x = >| x = x.

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3. The Initial Scale

3.1 Theorem: The standard D-interval I (the standard interval of dyadic rationals) is
isomorphic to the initial scale and it is simple.
(Recall that for any equational theory “simple” means having exactly two quotient structures,
the identity and the trivial.) When coupled with the previous observation that ⊥ 6= > in all
non-trivial scales we thus obtain:
3.2 Theorem: I appears uniquely as a subscale of every non-trivial scale.
The proof is on the computational side as, apparently, it must be. It turns out that not all
of the axioms are needed for the proof and that leads to another theorem of interest in which—
for technical reasons—we define the theory of minor scales to be the result of removing the
scale identity but adding the absorbing laws (either one, by itself, would suffice).[35]
3.3 Theorem: The theory of minor scales has a unique equational completion (and using
Theorem 15.1 (p??), that complete theory is the theory of scales).
There are several ways of restating this fact: equations consistent with the theory are
consistent with each other; an equation is true for all scales iff it holds for any non-trivial
minor scale; an equation is true for all scales iff it is consistent with the theory of minor
scales; using the completeness of the theory of scales, every equation is either inconsistent
with the theory of minor scales or is a consequence of the scale identity.[36]
The proof is obtained by showing that the initial minor scale is I and is simple. (Thus
every consistent extension of the theory of minor scales, having a non-degenerate model, must
hold for every subalgebra, hence must hold for the initial model. The complete equational
theory of the initial model thus includes all equations consistent with the theory of minor
scales.)[37]
[ 35 ] “Technical reasons” means other than the existence of interesting examples. For an example of a minor scale that is not a

scale see Section 28 (p??–??).

[ 36 ] The same relationship holds between the theory of lattices and the theory of distributive lattices, and between the theories

of Heyting and Boolean algebras. A less well-known example: for any prime p , the unique equationally consistent extension of
the theory of characteristic-p unital rings is the theory of characteristic-p unital rings satisfying the further equation xp = x.
(This almost remains true when the unit is dropped: given a maximal consistent extension of the theory of “rngs” there is a
prime p such that the theory is either the theory of characteristic-p rngs that satisfy the same equation as above (xp = x), or
the theory of elementary p-groups with trivial multiplication (xy = 0).)
A telling pair of examples: the equational theory of lattice-ordered groups and the equational theory of lattice-ordered rings.
In each case the unique maximal consistent equational extension is the set of equations that hold for the integers. The first
case is decidable (and all one needs to add to obtain a complete set of axioms is the commutativity of the group operation—see
Section 28, p??–??). The second case is undecidable: the non-solvability of any Diophantine equation, P = 0, is equivalent to the
consistency of the equation 1 ∧ P 2 = 1. (Conversely, one may show that the consistency of any equation is equivalent to the non-
solvability of some Diophantine equation, for instance, by showing that the solvability of an equation involving lattice operations is
always equivalent to the solvability of an equation with one fewer lattice operation: given a term S in the theory of lattice-ordered
commutative rings replace S with S 2, if necessary, to insure that it has no negative values and let P  Q be an inner-most
instance of a lattice operation—that is, one in which P and Q are ordinary polynomials; let T be the result of replacing P  Q
in S with a fresh variable z; let a, b, c, d be four more fresh variables; let e denote the value of 1  (−1); then the solvability of
˜2 ˆ ˜2
S = 0 in Z is equivalent to the solvability of T + P + Q + e(a2 + b2 + c2 + d2 ) − 2z + (a2 + b2 + c2 + d2 )2 − (P − Q)2 = 0.)
ˆ
[ 37 ] We can not only drop axioms but structure: I is the free midpoint algebra on two generators (>,⊥) and the free

symmetric midpoint algebra on one generator where we understand the first three scale equations (idempotent, commutative
and medial) to define midpoint algebras and the first four (add the constant law) together with the involutory and distributive
laws for dotting to define symmetric midpoint algebras. In the opening section I talked about closed midpoint algebras with
reference to the structure embodied by top, bottom and midpointing, with the remark that the axioms were not needed in the
material of that section. Let me now legislate that the axioms are the first three scale equations together with the non-equational
Horn sentence of cancellation for midpointing. Since I is such, it will perforce be the case that I is the initial closed midpoint
algebra. (The set {⊥, >} is a two-element midpoint algebra but not a closed midpoint algebra when we take ⊥|> = ⊥ and
.
it is a symmetric midpoint algebra when we take >= ⊥.) For a symmetric closed midpoint algebra add dotting and
the constant law (the involutory and distributive laws are consequences of cancellation). I is also the initial symmetric closed
midpoint algebra.
It should be noted, however, that there are closed midpoint algebras, even symmetric closed midpoint algebras, that extend
the notion of midpoint. Choose an odd number of evenly spaced points on a circle and define the midpoint of any two of
.
them to be the unique equidistant point in the collection. Chose any two points for > and ⊥ and define x to be the unique
.
element such that x |x = ⊥|>. If one chooses ⊥, ⊥|>, > to be adjacent then the induced map from I is guaranteed to be

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We first construct the initial minor scale via a “canonical form” theorem and show that
it is simple. Define a term in the signature of scales to be of “grade −1” if it is either > or
⊥, of “grade 0” if it is , and for positive n a term is of “grade n” if it is either >|A or ⊥|A
where A is of grade n−1. We need to show that the elements named by graded terms form a
subscale. Closure under the unary operations—dotting, >- and ⊥-zooming—is straightforward
(but note that the absorbing laws are needed). For closure under midpointing we need an
inductive proof. We consider A|B where A is of grade a and B is of grade b. Because of
commutativity we may assume that a ≤ = b. The induction is first on a. The case a = −1
presents no difficulties. For a = 0 we must consider two sub-cases, to wit when b = 0 and
when b > 0. If b = 0 then A|B = . When b > 0 we may, without loss of generality, assume
that B = >|B 0 for B 0 of grade b−1. But then A|B = |(>|B 0 ) = (>|⊥) | (>|B 0 ) = >|(⊥|B 0 )
which is of grade b+1. If a > 0 there are, officially, four sub-cases to consider, but, without
loss of generality, we may assume that A = >|A0 and either B = >|B 0 or B = ⊥|B 0 where
A0 is of grade a − 1 and B 0 is of grade b − 1. In the homogeneous sub-case we have that
A|B = (>|A0 ) | (>|B 0 ) = >|(A0 |B 0 ) and by inductive hypothesis we know that A0 |B 0 is
named by a graded term, hence so is A|B. In the heterogeneous sub-case we have that
A|B = (>|A0 ) | (⊥|B 0 ) = (>|⊥) | (A0 |B 0 ) = |(A0 |B 0 ) and by inductive hypothesis we know
that A0 |B 0 is named by a graded term and we then finish by invoking again the case a = 0.
The simplicity of the initial scale—and the uniqueness of graded terms—also requires
induction. Suppose that A and B are distinct graded terms and that ≡ is a congruence such
that A ≡ B. Again we may assume that a ≤ = b. In the case a = −1 we may assume without
loss of generality that A = >. The sub-case b = −1 is, of course, the prototypical case (B = ⊥
∧ ∧
else A = B). For b = 0 we infer from > ≡ that > ≡ , hence > ≡ ⊥, returning to the
sub-case b = 0. For b > 0 we must consider the two sub-cases, B = >|B 0 and B = ⊥|B 0 where
∧
B 0 is of grade b−1. From > ≡ >|B 0 we may infer > ≡ >d |B 0, hence > ≡ B 0 and thus reduce
∧
to the earlier sub-case b−1. From > ≡ ⊥|B 0 we may infer > ≡ ⊥d
|B 0 immediately reducing
to the sub-case b = −1. For the case a = 0 we know from a ≤ = b and A 6= B that b > 0
and we may assume without loss of generality that B = >|B where B 0 is of grade b−1. But
0
∧ ∧
≡ B then says that ≡ B , hence > ≡ B 0 and we reduce to the case a = −1. For the case
a > 0 we again come down to two sub-cases. In the homogeneous sub-case A = >|A0 and
B = >|B 0 we infer from A ≡ B that >d |B 0 hence that A0 ≡ B 0. Since A 6= B we have
|A0 ≡ >d
that A0 6= B 0 and we reduce to the case a−1. Finally, in the heterogeneous sub-case A = >|A0
∨ ∨
and B = ⊥|B 0 we infer from A ≡ B that > | A0 ≡ ⊥ | B 0 hence that > ≡ B 0. Since the grade of
B 0 is positive such reduces to the case a = −1.
When we know that every non-trivial scale contains a minimal scale isomorphic to the
initial scale, then perforce we know that there is, up to isomorphism, only one non-trivial
minimal scale. Hence, to see that the initial scale is isomorphic to I it suffices to show that
I is without proper subscales, or—to put it more constructively—that every element in I
.
can be accounted for starting with >. Switching to D-notation, we have 1 = >, −1 => and
0 = >|⊥. We know that all other elements are of the form ±(2n+1)2-(m+1) where n and m
are natural numbers and 2n+1 < 2m+1 . Inductively (on m):
±(2n+1)2-(m+1) = ±n2-m | ±(n+1)2-m [ 38 ]

onto. I thus has an infinite number of closed midpoint quotients and it is far from simple (the simple algebras are precisely
the cyclic examples of prime order).
[ 38 ] Using, of course, that n+1 ≤ 2m (because, necessarily, 2n+2 ≤ 2m+1 ).

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4. Lattice Structure
The most primitive way of defining the natural partial order on a scale is to define u ≤
= v iff
there is an element w such that u|> = v|w. From this definition it is clear that any map that
preserves midpointing and > must preserve order (which together with von Neumann is quite
enough to prove the opening assertion of this work: see Section 40, p??–??).[39]
But in the presence of zooming we may remove the existential. First note that
∨
∃w z = >|w iff z=>
∨ ∨ ∨
because if z = >|w then the absorbing law says z = >|w = >. Conversely, if z = > then we
∧ ∨ ∧ ∧
may take w = z (the law of compensation gives us z = z | z = >| z).
If we use swap-and-dot and the involutory law to rewrite the existential condition for u ≤
=v
.
as ∃w [u |v = >|w] we are led to define a new binary operation to be denoted by borrowing
.∨
from J-Y Girard,[40] u −◦ v = u |v and we see by the absorbing law that
= v iff u −◦ v =
u≤ >.

We make this our official definition (u −◦ v may be read as “the extent to which u is less
than v” where > is taken as “true” [41] ) A neat way to encapsulate this material is with:
4.1 Lemma: the law of balance:
u | (u −◦ v) = v | (v −◦ u)

(One can see at once that u −◦ v = > implies that u|> = v|w is solvable.) To prove the law
. ∧. ∨.
of balance note that the law of compensation yields u| v= (u | v)|(u | v) and a swap-and-dot
.
∧. ∨. .∨
yields u|(u | v) = v |(u | v); the left side rewrites as u|(u | v) = u|(u −◦ v) and the right as
.∨
v |(v | u) = v |(v −◦ u).
We verify that ≤ = is a partial order as follows:
.∨ ∨
Reflexivity is immediate: x −◦ x = x | x = = >.
For antisymmetry, given x −◦ y = > = y −◦ x just apply the unital law to both sides of
the law of balance.
∨ ∨ ∨
Transitivity is not so immediate. We will need that u = > = v implies u | v = > (true
∨ ∨ ∨ ∨ ∧ ∨ ∨ ∧
because, using the law of compensation, u = > = v says u | v = (u| u) | (v | v) =
∧ ∨ ∧ ∨ ∧ ∧ . ∨ .
(> | u) | (> | v) = > | (u | v) = >). Hence if u −◦ v and v −◦ w are both > then so is (u|v) | (v|w).
∨ .
But this last term is equal (using the commutative, medial and constant laws) to |(u |w)
.∨
which by the central distributivity law is >|(u | w). Hence if u ≤ = v and v ≤= w we have that
>|(u −◦ w) = > and when both sides are >-zoomed we obtain u −◦ w = >.
Covariance of z| also follows from central distributivity:
.∨ . . ∨ . ∨ . ∨ .
(z|x) −◦ (z|y) = (z|x) | (z|y) = (z | x) | (z|y) = (z |z) | (x |y) = | (x |y) = >|(x −◦ y) [ 42 ]

Hence x ≤
= y implies z|x ≤
= z|y. Not only does z| preserve order, it also reflects it.
[ 39 ] Left as an easy exercise: a map that preserves midpointing and ⊥ also preserves order.
[ 40 ] For reasons to become clear in the next section.
[ 41 ] Again, see the next section.
[ 42 ] See Section 46, p??–?? for a subscoring.

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Contravariance of dotting is immediate:

. .
u −◦ v = v −◦ u.

For each semiquation we obtain the dual semiquation by replacing the terms with their
duals and reversing the semiquation.
A few more formulas worth noting are:
> −◦ x = x
.
x −◦ ⊥ = x
. ∨
x −◦ x = x
.
. ∧
x −◦ x = x
∨ ∨
Note that −◦ x = >|x hence x = > iff ≤ = x. The lemma we needed (and proved) for
∨ ∨ ∨
transitivity, that u = > = v implies u | v = >, is now an easy consequence of the covariance
of midpointing.
It is immediate from the definition and absorbing laws that ⊥ ≤
=x≤
= > all x.
We obtain a swap-and-dot lemma for semiquations:
4.2 Lemma:
. .
u|v ≤= w|x iff u| w ≤ = v |x
.∨ . . ∨ . ∨ .
Because (u|v) −◦ (w|x) = (u|v) | (w|x) = (u| v) | (w|x) = (u|w) | (v | x) =
. .∨ . . .
(u| w) | (v |x) = (u| w) −◦ (v |x). (See Section 46, p??–?? for a subscoring.)
An important fact: > is an extreme point in the convex-set sense, that is, it is not the
midpoint of other points (see Section 41, p??, for a discussion on this definition):
4.3 Lemma: x|y = > iff x = > = y.
Because x|y ≤= >|y (without any hypothesis), hence x|y = > implies > = x|y ≤
= >|y ≤
= >
forcing >|y = >, hence y = >c
|y = >
b = >.
4.4 Lemma: Zooming is covariant.
The covariance of >-zooming (and, hence, ⊥-zooming) requires work. In constructing this
theory an equational condition was needed that would yield the Horn condition that u ≤ =v
∧ ∧ ∧ ∧
implies u ≤ = v. (The equation, for example, (u −◦ v) −◦ (u −◦ v) = > would certainly
suffice. Alas, this equation is inconsistent with even the axioms of minor scales: if we replace
u with > and v with it becomes the assertion that ≤ = ⊥.)
The fact that > is an extreme point says that it would suffice to have:
∧ ∧ ∨ ∨
u −◦ v = (u −◦ v) || (u −◦ v).

Indeed, this equation implies that the two zooming operations collectively preserve and reflect
the order.
Finding a condition strong enough is, as noted, easy. To check that it is not too strong, that
is to check the equation on the standard model, it helps to translate back to more primitive
terms: . . ∨
.∨ ∧ ∨ ∧ | ∨ ∨ ∨ . ∨ ∧ | ∧. ∨ ∨
u | v = (u | v) | (u | v) = (u | v) | (u | v).

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Since dotting is involutory this is equivalent to the outer equation but without the dots:
∨ ∨ ∨ ∧ ∧ ∨ ∨
u | v = (u | v) || (u | v),
to wit, the dual of the scale identity. And it was this that was the first appearance of the scale
identity (and its first serious use—the three previous direct applications that have appeared
here served only to replace what had, in fact, once been axioms, to wit, the variable-free
∨ ∨ ∧ ∧
equation ⊥ = ⊥ and the two one-variable equations, x | x = x and d |x = ⊥| x, which three
laws are much more apparent than is the scale identity).
∧ ∨
Among the corollaries are the covariance of the binary operations | , | and the important
semiquations:
4.5 Lemma:
∧ ∨
x = x ≤
≤ = x
∧ ∧ ∧ ∨ ∨ ∨
Because x = x | x ≤
= >| x = x = x| ⊥ ≤
= x | x = x.
Further corollaries: x −◦ y is covariant in y and contravariant in x. (If a/ x is viewed as a
binary operation then it is covariant in x and contravariant in a.)
4.6 Lemma: the convexity of >-zooming:
∧ ∧
|v ≤
ud = u || v
∧ ∨ ∧ ∧ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧
= (>| u) | (>| v) =
Because u | v = (u | u) | (v | v) ≤ > | (u | v) = u|v .
The dual semiquation:
4.7 Lemma: ∨ ∨ ∨
u| v ≥
= u|v
∧
Define a binary operation, temporarily denoted x  y, as x | (x −◦ y). The law of balance
says, in particular, that the -operation is commutative and consequently covariant not just
∧ ∧
in y but in both variables. Note that if x ≤ = y then x  y = x | (x −◦ y) = x | > = x, which
together with commutativity says that whenever x and y are comparable, x  y is the smaller
of the two. As special cases we obtain the three equations: x  > = >  x = x  x = x.
These three equations together with the covariance are, in turn, enough to imply that x  y
is the greatest lower bound of x and y: from the covariance and x  > = x we may infer that
xy ≤ = x  > = x and, similarly, x  y ≤
= y; from covariance and z  z = z we may infer that
x  y is the greatest lower bound (because z ≤ = x plus z ≤
= y implies z = z  z ≤
= x  y).
All of which gives us the lattice operations (using duality for x ∨ y):
4.8 Lemma: ∧ ∧ .∨ [ 43 ]
x ∧ y = x | (x −◦ y) = x | (x | y)
.
x ∨ y = x ∨| x
d |y
We will extend the notion of duality to include the lattice structure. (But note that we do
not have a symbol for the dual of −◦ .)
Direct computation now yields what we will see must be known by an oxymoronic name;
∨ ∧
it is the “internalization” of the (external) disjunction: x = > or x = ⊥.
[ 43 ] It behooves us to figure out just what the term x|(x −
−◦ y) is before it is >-zoomed. We know that it is commutative and
∨ ∧
−◦ y)) | (x | (x −
covariant in both variables. The law of compensation says that it is equal to (x | (x − −◦ y)). We know now what
∧ ∨
the right-hand term, x | (x −
−◦ y), is. For the left-hand term, x | (x −
−◦ y), note that its covariance implies that it is always at
∨ ∨
−◦ ⊥) = ⊥ | > = >. Hence x|(x −
least ⊥ | (⊥ − −◦ y) = >|(x ∧ y).

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4.9 Lemma: the coalgebra equation:

.
∨ ∧
x∨ x = >
. .d. .
∨ ∧ ∨∨ ∨ ∧ ∨ ∧ .
∨ ∨ \ .
∨ ∨ b ∨ ∨
because: x ∨ x = x | x | x = x | (x | x) = x | x = (x | x)∨ = ( )∨ = >.
.
If we replace x with x|y we obtain the internalization of the disjunction:
(x ≤
= y) or (y ≤= x).
4.10 Lemma: the equation of linearity:
[ 44 ]
(x −◦ y) ∨ (y −◦ x) = >

Indeed it says that what logicians call the disjunction property (the principle that a
disjunction equals > only if one of the terms equals >) is equivalent with linearity:
4.11 Lemma: The following are equivalent for scales:
Linearity; The disjunction property; The coalgebra condition.
We will need:
4.12 Lemma: the adjointness lemma:
∧
= v −◦ w
u ≤ iff u| v ≤
= w
∧ ∧ ∧
Because if u ≤
= v −◦ w then v | u ≤
= v | (v −◦ w) = v ∧ w ≤
= w. And if u | v ≤
= w then
. .∨c
u ≤
= v ∨ u = v | v|u = v −◦ u|v
c ≤ = v −◦ w. [45]
We close this section with a few interval isomorphisms. For any b < t the interval [b, t| is
order-isomorphic with an interval whose top end-point is >, to wit, the interval [t −◦ b, >]. The
∧ ∧
isomorphism is t −◦ (−). Its inverse is t | (−). (The fact that the composition t | (t −◦ x) = x
∧
for all x ∈ [b, t] is just the fact that t | (t −◦ x) = t ∧ x = x. The fact that the composition
∧ .
t −◦ (t | x) = x for all x ∈ [t −◦ b, >] is just the fact that t = t −◦ ⊥ ≤ = t −◦ b ≤
= x and
∧ .∨ ∧ .
hence that t −◦ (t | x) = t | (t | x) = t ∨x = x. These are not just order-isomorphisms. With the
forthcoming Theorem 15.1 (p??) it will be easy to prove that they preserve midpointing and
when—in Section 17 (p??)—we note that all closed intervals have intrinsic scale structures
it will be easy to see that they are scale isomorphisms.[46]
[ 44 ] .
The coalgebra equation is obtainable, in turn, from the equation of linearity by replacing y with x.
[ 45 ] Using the linear representation theorem from Section 8 (p??–??) one can show the initially surprising fact that the binary
∧
operation | is associative. Any poset may be viewed as a category and this associativity together with the adjointness lemma
∧
allows us to view a scale as a “symmetric monoidal closed category” with | as the monoidal product and − −◦ as the “closed”
structure. The monoidal unit is >. A scale is, in fact, a “?-autonomous category”: the “dualizing object” is ⊥.
∧
A straightforward verification of the associativity of | on a linear scale entails a lot of case analysis. Perhaps it is best to
use the equational completion that will be proved in Section 15 (p??–??); it suffices to verify it on just one non-trivial example.
The easiest we have found is to take the unit interval—not the standard interval—and to verify the dual equation,
∨ ∨
the associativity of | . On the unit interval x | y is addition truncated at 1, easily seen to be associative.
With the associativity in hand there’s a nicer proof for the adjointness, even better, for the internal version of the adjointness:
. ∨ .∨ . ∨. ∨ ∧ .∨ ∧
u−−◦ (v − −◦ w) = u | (v | w) = (u | v) | w = (u | v) | w = (u | v) − −◦ w.
[ 46 ] The construction of the dilatation operator can be motivated by this material. For any a, the function (a|>) − −◦ (−)
sends the image of a|− to the interval [ , >], quite enough to suggest that (a|>) − −◦ (a|−) is the same as >|−, hence (using
the unital law) that (a|>\ −◦ (a|x) = x. The function (a|>\
)− −◦ (−) is a/ −.
)−
The construction for dilatation came rather late for me, but not as late as interval isomorphisms. It arose from the observation
that each of the “quarter-intervals” on [−1, 1], that is, the intervals [−1, − 12 ], [− 12 , 0 ], [ 0, 12 ], [ 12 , 1], are isomorphic to the
entire interval via a pair of zoomings, hence if we could move the image of a| to a quarter-interval we’d be done. We can do just
that with a contraction that sends the top of the image of a| to the top of a quarter-interval. The top of the image is (a|>).
.
The contraction at (a|>) sends it to the top of [− 21 , 0 ].

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5. Diversion: Lukasiewicz vs. Girard

On the unit interval the formula for −◦ has a prior history as the Lukasiewicz notion of
many-valued logical implication. A traditional interpretation of Φ ≤
= Ψ is “Ψ is at least as
likely as Φ.” Then Φ −◦ Ψ becomes the “likelihood of Ψ being at least as likely as Φ.” [47]
The unit interval viewed as a ?-autonomous category (resulting from its scale-algebra
structure) reveals Lukasiewicz inference as a special case of Girard’s linear logic when we
∧ ∨ &
write | as ⊗ and its “de Morgan dual” | as (“par”). (The midpoint operation is an
&
example of a “seq” operation—it lies between ⊗ and .)
If we interpret the truth-values as frequencies (or probabilities) we can not infer, of course,
the frequency of a conjunction from the individual frequencies. But we can infer the range
of possible frequencies. If Φ and Ψ are the individual frequencies then the maximum possible
frequency for their disjunction occurs when they are maximally exclusive: if their frequencies
add to 1 or less and if they never occur together then the maximal possible frequency of the
disjunction is their sum; if their frequencies add to more than 1 then the maximal possible
frequency of the disjunction is, of course, 1. That is, the maximum possible frequency for their
&
disjunction is Φ Ψ. The minimal possible frequency for their conjunction likewise occurs
when they are maximally exclusive and similar consideration yields Φ ⊗ Ψ. This works best
if we understand that separate observations are made, one for Φ and one for Ψ (hence Φ ⊗ Φ
&
is the minimal possible frequency that Φ occurs in both observations, Φ Φ the maximal
possible frequency that Φ occurs in at least one of the two observations).
The “additive connectives,” likewise, have such an interpretation. The minimal possible
frequency for their disjunction occurs when they are minimally exclusive, that is, when the less
probable event occurs only when the more probable event occurs, hence the minimal possible
frequency for the disjunction is Φ ∨ Ψ. Similar computation yields that the maximal possible
frequency of their conjunction is Φ ∧ Ψ. The range of frequencies possible for the conjunction
&
is the interval [Φ ⊗ Ψ, Φ ∧ Ψ] and the range for the disjunction is [Φ ∨ Ψ, Φ Ψ]. The midpoint
&
of Φ and Ψ is also the midpoint of Φ ⊗ Ψ and Φ Ψ (using the law of compensation) and
the midpoint of Φ ∧ Ψ and Φ ∨ Ψ (using the forthcoming linear representation theorem of
Section 8 (p??–??). Note that we have in descending order:
1
&
Φ Ψ
Φ∨Ψ
Φ|Ψ
Φ∧Ψ
Φ⊗Ψ
0
When Φ −◦ Ψ = 1 it is possible (just knowing the frequencies of Φ and Ψ) that whenever
Φ occurs Ψ will occur. More generally Φ −◦ Ψ gives the maximal possible probability that a
single pair of observations will fail to falsify the hypothesis “if Φ then Ψ.” The adjointness
lemma, Φ ≤ = Ψ −◦ Λ ⇔ Φ ⊗ Ψ ≤ = Λ, then says that Φ is possibly less frequent than Ψ
appearing to imply Λ iff the frequency of the conjunction of Φ and Ψ is possibly less than
the frequency of Λ. [48] We constructed the meet operation as Φ ⊗ (Φ −◦ Ψ). That is, the
maximal possible frequency for the conjunction of two events is equal to the minimal possible
[ 47 ]
. ∨ ∨
We may interpret ⊥-zooming using the equation Φ − −◦ Φ = Φ: given a sentence Φ it says that Φ is the likelihood that
.
. ∧
Φ is at least as likely as not. Using the companion equation Φ −
−◦ Φ = Φ we see that in the Lukasiewicz interpretation the
∨ ∧ T
coalgebra condition ( Φ = T or Φ = ) says that for any statement either it or its negation is at least as likely as not.
[ 48 ] This would not, of course, be heard as an acceptable sentence in ordinary language. But few translations from the

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frequency of the conjunction of another pair of events, the first of which remains the same
and the second is the maximal possible frequency of failing to refute the hypothesis that the
first implies the second. (Surely someone previously must have observed all this.)
When the coalgebra condition is interpreted we obtain the interval rule for linear logic:
&
Φ Ψ=1 or Φ⊗Ψ=0

(Either it is possible for one to succeed or it is possible that both fail.) Alternatively we may
.
replace Φ with Φ so that the coalgebra condition becomes

Φ≤
=Ψ or Ψ≤
=Φ

delivering a theory of linear linear (or planar?) logic.

Missing above are Girard’s modal unary operations, of-course and why-not, which he
denoted with a ! and a ?. [49] In Section 19 (p??–??) on “chromatic scales” we introduce
the (discontinuous) “support” operations on scales. Using chromatic-scale notation one may
argue that !Φ = Φ and ?Φ = Φ.

6. Diversion: The Final Interval Coalgebra as a Scale

The final interval coalgebra, I, comes equipped, of course, with the two constants and >
∧ ∨
⊥, and the two zooming operations x and x. As previously noted in Section 1 (p??–??) we
. . .
may define x via the unique coalgebra map I→ I where I is the coalgebra obtained by
swapping the two constants and the two zooming operations. The order on I is definable via
the observation that x < y iff there is a sequence of zooming operations (∧, ∨) that carries x
to ⊥ and y to >.
There is, indeed, a useful interval coalgebra structure on I × I so that its unique coalgebra
map to I is the midpoint operation,[50] but, alas, this coalgebra structure on I × I requires
the midpoint operation for the construction of its two zooming operations: hu, vi is sent by
∨∧∧ ∧∧∨ ∨∨∧ ∧∨∨
>-zooming to h u | v , u | v i and by ⊥-zooming to h u | v , u | v i. [51]

We must eventually come to grips with the notion of co-recursion but will settle now for a
quick and dirty proof that for u, v ∈ [ 0, 1] the binary expansion of u|v is forced by the scale
axioms. Recall the earlier quick and dirty proof. In this case it says that we should iterate
∨
(forever) a procedure equivalent to: If u | v = > then emit “1” and replace u|v with
∧ ∨
u | v else emit “0” and replace u|v with u | v. We need, obviously, to expand.
∨ . ∧ ∨
We will use that u | v = > iff u ≤= v and we will attack the computation of u | v and u | v
∧ ∨
by using the scale identity. We need a single procedure for the three cases u|v, u | v, u | v.

mathematical notation to ordinary language yield acceptable sentences—else who would need the math? Note that the next step
(provided by [??], p??) would be the internalization: Φ −
−◦ (Ψ − −◦ Λ) = (Φ ⊗ Ψ) − −◦ Λ. Anyone want to try a translation?
[ 49 ] Hollow men pronounce these as bang and whimper.
[ 50 ] The word “useful” is important here. Given any functor T with a final coalgebra F → T F then for any retrac-
x y
tion F → A → F = 1F there is a (not very useful) coalgebra structure on A that makes y a coalgebra map, to wit,
y Tx
A → F → T F → T A.
[ 51 ] Imagine stumbling across this use of the scale identity, the initial discovery of which was in answer to a very different

question.

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Hence we iterate (forever) a procedure that takes an ordered triple hu, s, vi n

as input where
∨ ∧
o
u and v are elements of [ 0, 1] and s is an element of the set of three symbols | , | , | .

If s = | then
. ∧
if u ≤ = v then emit “1”; replace hu, |, vi with hu, | , vi.
∨
else emit “0”; replace hu, |, vi with hu, | , vi.
∧ ∧ ∧ ∨ ∧ ∧
else if s = | then if u = ⊥ then emit “0”; replace hu, | , vi with hu, | , vi.
∧ ∧ ∧ ∧ ∨
else if v = ⊥ then emit “0”; replace hu, | , vi with hu, | , vi.
∧ ∧ ∧
else replace hu, | , vi with hu, |, vi.
∨ ∨ ∧ ∨ ∨
else if u = > then emit “1”; replace hu, | , vi with hu, | , vi.
∨ ∨ ∨ ∨ ∧
else if v = > then emit “1”; replace hu, | , vi with hu, | , vi.
∨ ∨ ∨
else replace hu, | , vi with hu, |, vi.
.
For a proof that this is forced by the axioms for midpointing note first that u ≤
= v implies
∨
≤
= u|v, hence u | v = > which means that the first digit is 1 and the remaining digits are
∧ ∧
determined by u | v. For u | v we use the scale identity:
∧ ∨ ∧ ∧ ∧ ∧ ∨
u | v = (u | v) | (u | v)
∧
When u = ⊥ this becomes:
∧ ∨ ∧ ∧ ∧ ∨ ∨ ∧ ∧
u | v = (u | v) | (⊥ | v) = (u | v)|⊥
∧
hence, by the absorbing law, (u | v)∧ = ⊥ which
 ∨ ∧ means
∨ that the first digit is 0 and the
∧ ∨ ∧ ∨ ∧ ∧
remaining digits are determined by (u | v) = (u | v)|⊥ which by the unital law is u | v.
∧ ∧ ∧ ∨ ∨
A similar argument holds for the case v = ⊥. If neither u nor v are ⊥ we have u = v = > and
the scale identity and unital law yield
∧ ∧ ∧ ∧ ∧ ∧ ∧
u | v = (> | v) | (u | >) = v | u
∨
which returns us to the case s = |. The dual argument holds for the case s = | .

As previously commented ([??], p??) there are contexts in which I ∼

= I ∨ I is a pushout:
⊥
• →I
>↓ ↓ >|x
I→I
⊥|x

We can use the scale structure on I to effect that comment. Working in the category whose
objects are scales but whose maps are arbitrary functions between them let f⊥ , f> : I → S
be such that f⊥ (>) = f> (⊥). (We’re
 in a full subcategory of the category of sets; there are
f>
 ∨ ∧

no other conditions.) Then : I → S is the map that sends x to c / f⊥ ( | f> (
x) x)
f⊥
where c = f⊥ (>) = f> (⊥).

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7. Congruences, or: >–Faces

One of our first aims is to prove that every scale can be embedded in a product of linear
scales. Put another way: we wish to find, on any scale, a lot of quotient structures that are
linearly ordered. And for that we must get an understanding of quotient structures.
As for any equational theory, the quotient structures of a particular algebra correspond
to the “congruences” on that structure, that is, the equivalence relations that are compatible
with the operators that define the structure. For some well-endowed theories the congruences
correspond, in turn, to certain subsets. Such is the case for scales.[52]
Given a congruence ≡ define its kernel, denoted ker(≡), to be the set of elements
congruent to >. Clearly, ≡ can be recovered from ker(≡) (because x ≡ y iff both x −◦ y
and y −◦ x are in ker(≡)). We need to characterize the subsets that appear as kernels.
Borrowing again from convex-set terminology, we say that a subset is a “face” if it is
not just closed under midpointing but has the property that it includes any two elements
whenever it includes their midpoint. Saying that an element is an extreme point, therefore,
is the same as saying that it forms a one-element face. (See Section 41, p??, for a discussion
of these definitions.) We will be interested particularly in those faces that include >. Thus
we define a subset, F, to be a T-face, “top-face,” if:
>∈F
x|y ∈ F iff x ∈ F and y ∈ F
Because inverse homomorphic images of faces are faces and because {>} is a face it is clear
that ker(≡) is a >-face for any congruence. We need to show that all >-faces so arise.
Given a >-face F define x  y (mod F ) to mean x −◦ y ∈ F and define
x ≡ y (mod F ) as the “symmetric part” of , that is, x ≡ y iff x  y and y  x.
It is routine that x ≡ > iff x ∈ F.
Clearly ≡ is reflexive (because x −◦ x = >) and it is symmetric by fiat. Transi-
tivity requires a little more. First note that a >-face is an updeal, that is,
x ∈ F plus x ≤ = y implies y ∈ F (immediate from the law of balance). Second, in the
∨ ∨ ∨ . .
dual of the convexity of >-zooming, u | v ≤ = u | v, replace u with w |x and v with x |y to
obtain:
= >|(w −◦ y) [ 53 ]
(w −◦ x) | (x −◦ y) ≤
.∨ .∨ . ∨ . . ∨ . ∨ .
because (w −◦ x)|(x −◦y) = (w | x)|(x | y) ≤
= (w |x) | (x |y) = (x |x) | (w |y) = | (w |y) =
. ∨
>|(w | y) = >|(w −◦ y). Hence if w  x and x  y then (w −◦ x)|(x −◦ y) ∈ F forcing
>|(w −◦ y) ∈ F and, finally, (w −◦ y) ∈ F.

Thus ≡ is an equivalence relation. It is a congruence with respect to dotting because

. . . .
w −◦ x = x −◦ w, hence w  x iff x  w. In the verification that y| is covariant we used the
equation (y|w) −◦ (y|x) = >|(w −◦ x) which quite suffices to show that w  x iff y|w  y|x
and consequently that ≡ is a congruence with respect to midpointing. Finally, to see that ≡
[ 52 ] Almost all well-endowed theories in nature contain the theory of groups. Two exceptions (besides scales): the theory of

division allegories and (its better-known special case) the theory of Heyting algebras.
[ 53 ] If both sides of this semiquation are >-zoomed we obtain

∧
(w −
−◦ x) | (x −
−◦ y) ≤ (w −
−◦ y).

When this semiquation is viewed as a map in a monoidal closed category:

(w −
−◦ x) ⊗ (x −
−◦ y) → (w −
−◦ y)

its name is the “composition map.”

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∧ ∧
is a congruence with respect to zooming it suffices to show that w  x implies w  x. We
∨ ∨
may as well show that it implies w  x at the same time. The scale identity, in the form
∧ ∧ ∨ ∨
w −◦ x = (w −◦ x) | (w −◦ x) does just that.
Given an element s in a scale we will need to see how to construct ((s)) the
principal T-face it generates, that is, the smallest >-face containing s.
= (>|)n x for all large n.
7.1 Lemma: The principal >-face ((s)) is the set of all x such that s ≤
((>|)n is the nth iterate of the contraction at >. ) Clearly this set includes > and is closed
= (>|)n (x|y)
under midpointing; for the other direction, suppose it includes x|y; then from s ≤
we may infer s ≤ = (>|) (x|y) ≤
n
= (>|) (>|y) = (>|) y for sufficiently large n and y is clearly
n n+1

in the set. For the other direction note that in any quotient where s becomes > the equality
s≤= (>|)n x clearly forces (>|)n x to become >, after which n applications of >-zooming will
force x itself to be >. That is, if s ≤= (>|)n x for any n, then x must be in any >-face that
contains s.

8. The Linear Representation Theorem

We wish to prove:
8.1 Theorem: Every scale can be embedded in a product of linear scales.
An algebra (for any equational theory) is said to be subdirectly irreducible, or SDI for
short, if whenever it is embedded into a product of algebras one of the coordinate maps is
itself an embedding. Every algebra (for any equational theory) is embedded in the product
of all of its sdi quotients (we will repeat the proof for this case). But first:
8.2 Lemma: If a scale is an sdi then it is linearly ordered.
A homomorphism of scales is an embedding iff its kernel is trivial. A scale is an sdi iff the
map into the product of all of its proper quotient scales fails to be an embedding. Hence it
is an sdi iff the intersection of all non-trivial >-faces is non-trivial. Let s < > be an element
in that minimal non-trivial >-face. Then for every element a < >, its principal >-face, ((a)),
must contain s. Thus an sdi scale has an element s < > such that for all a < > it is the
case that a < (>|)n s for almost all n. (This may be rephrased: a scale is an sdi iff there is a
sequence of the form {(>|)n s}n cofinal among elements below >. ) If x and y are both below
> then clearly x ∨ y < (>|)n s almost all n, in particular x ∨ y is below top. That is, sdi scales
satisfy the disjunction property which, as has already been observed in Lemma 4.11, implies
linearity via the equation of linearity, (x −◦ y) ∨ (y −◦ x) = >. [54]
The fact that all scales can be embedded in a product of linear scales is now easily obtain-
able: for each element s < > use the axiom of choice to obtain a >-face, Fs , maximal among
>-faces that exclude s; it is routine that in the corresponding quotient scale the element in the
image of s becomes equal to > in every proper quotient thereof; hence it is, as just argued,
linearly ordered. The intersection of all the >-faces of the form Fs is clearly trivial. (Note that
the structure of this proof of the linear representation theorem is forced: if the result is true
then necessarily every sdi is linear and the theorem is equivalent to sdi being linear.)
An immediate corollary:
[ 54 ] There’s a softer proof that uses an easy lemma about principal >-faces: ((x)) ∩ ((y)) = ((x ∨ y)); if x, y are elements in a

scale S such that x ∨ y = > then the map S → S/((x)) × S/((y)) is monic. If, further, S is sdi then either ((x)) or ((y))
must be ((>)). [Added 2008–12–31]

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8.3 Corollary: Every equation, indeed every universal Horn sentence, true for all linear
scales is true for all scales.
It should be noted that the axiom of choice is avoidable for purposes of this corollary.
Given a Horn sentence,
(s1 = t1 ) & · · · & (sn = tn ) ⇒ (u = v)
suppose there were a counterexample in some scale, A. The elements used for the
counterexample generate a countable subscale, A0. The term (u −◦ v) ∧ (v −◦ u) evaluates
to an element b < >. We can construct a >-face F in A0 maximal among those that exclude
b without using choice since A0 is countable. The image of the counterexample in the linear
scale A0 /F remains a counterexample.
When working with a linearly ordered set it is completely trivial that covariant functions
automatically distribute with the lattice operations.[55] Hence for all scales we have:
x|(y ∧ z) = (x|y) ∧ (x|z)
x|(y ∨ z) = (x|y) ∨ (x|z)
∧ ∧
x[∧z = x∧z
∧ ∧
x[∨z = x∨z
∨ ∨
(x ∧ z)∨ = x∧z
∨ ∨
(x ∨ z)∨ = x∨z
x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)
[??]
x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)
[56]
It is now easy to check that if  is any binary operation on a linear scale satisfying the
“dilatation equation,” a  (a|x) = x, and if, further, for any fixed a it is covariant in x, then
a  x = a/ x, in particular dilatations are self-dual.[57]
Using the linear representation theorem we obtain a proof for a lemma that we will need
later:
8.4 Lemma: The image of the central contraction, |, is the sub-interval [⊥| , |>].
We need to show that if |⊥ ≤ =x≤ = |> then we can solve for x = |y. We remove the
existential to obtain a Horn sentence by setting y = / x. Thus we need to show that in
any linear scale |⊥ ≤ =x≤ = |> implies x = |( / x). Linearity allows us to reduce to the
two cases x ≤= and ≤ = x. Symmetry allows us to concentrate on the case ≤ = x, hence
∨ ∧ ∧ ∧
we can assume x = x | x = >| x. From x ≤ = |> we infer that x ≤= |> =
d hence that
[ 55 ] One may, of course, de-trivialize by—instead—establishing the lemma that any function from a linear lattice is covariant

iff it is a lattice homomorphism.

[ 56 ] The last two equations—the definitions of distributive lattices—are, of course, equivalent for any lattice. Note that the

distributivity of a lattice is equivalent with its having an embedding into the product of its linearly ordered quotients: given
elements a, b with a not bounded by b, distributivity implies that the {⊥, >}-valued characteristic function of a filter maximal
among those that contain a but not b is a lattice homomorphism. Its target is not only linear, it has just two elements. (This
also all works for Boolean algebras: thus we teach truth tables.)
“ . ”∨∧ “ . ”∧∨
[ 57 ] As promised, we now have (a |⊥)|x = (a |>)|x . There are two other corollaries of interest (see Section 46, p??–??
“ “ . ””
for subscorings). First, any dilatation is definable using just central dilatation: a/ x = / / (a | )|x because the one
“ “ . ”” “ “ . ””
appearance of x is in a covariant position and / / (a | )|(a|x) = / / (a |a)|( |x) = / ( |x) = x.
“ “ . ””.
Second, central dilatation is definable using (twice) any one dilatation: / x = a / a/ ( |a)| x because the one
“ “ .””. “ “ . ””. “ “ . ””.
appearance of x is in a covariant position and a / a/ ( |a)|( |x) = a / a/ ( |a)|( | x) = a / a/ |(a| x) =
“ “ . . ””. “ “ . . ””. . . .
a/ a/ (a| a)|(a| x) = a/ a/ a|(a | x) = a/ (a | x) = a/ (a|x) = x.Hence any one dilatation can be used to construct all
other dilatations.

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∨ ∧ ∨ ∨ ∨ ∨
∧ ∧ ∧ ∧ ∧ ∧ ∧
x = x | x = x |⊥ which combines to give x = >|(⊥| x) = (>|⊥)|(>| x) = |(>| x). (So
∨
∧
y = >| x.) It is routine now that |( / x) = x. [58]
The fact that >-zooming distributes with meet has an important application: the lattice
of congruences is distributive. Recall, first, that in any lattice a set is called a “filter” if
it is hereditary upwards and closed under finite meets.[59] By a zoom-invariant filter we
mean a filter closed under the zooming operations. Since ⊥-zooming is inflationary a filter is
zoom-invariant iff it is closed under >-zooming. An important lemma:
8.5 Lemma: A subset of a scale is a >-face iff it is a zoom-invariant filter.
Because: suppose that F is a filter invariant with respect to zooming; from
x ∧ y = (x ∧ y) | (x ∧ y) ≤ = x|y we know that F is closed under midpointing; that
it is a face follows immediately from c ≤
x|y = x|
d > = x.

The other direction is an immediate consequence of the facts that zoom-invariant filters
are preserved under inverse images of homomorphisms and that any >-face is the inverse
image of a one-element zoom-invariant filter, to wit, {>}. (It is not hard to give a direct
proof: we have already noted (p??) that the law of balance says that a >-face is an updeal,
that is, if x is an element of a >-face F, and if x ≤
= y then y ∈ F; the law of compensation
∧
easily implies that x ∈ F; and if x and y are both in F we finish with x|y
c ≤ = x|
d > = x and
similarly x|y ≤
c = y hence x|y ≤
c = x ∧ y.)
8.6 Theorem: The congruence lattice of any scale is a “spatial locale.”
The pre-ordained name for the space in question is the spectrum of S, denoted Spec(S).
First, the lattice of filters in any distributive lattice is itself a distributive lattice and
the argument continues to work when we replace “filter” with “zoom-invariant filter” for
the simple reason that the set of zoom-invariant filters is closed under arbitrary meets and
joins. The main observation (for both proofs) is that the join of filters F and G is the
set { x ∧ y : x ∈ F, y ∈ G }. (It is clearly closed under meet and if x ∧ y ≤ = z then, using
distributivity of the lattice, z = (x ∨ z) ∧ (y ∨ z) where, of course, x ∨ z ∈ F and y ∨ z ∈ G.)
To see that (F ∨ G) ∩ H ⊆ (F ∩ H) ∨ (G ∩ H) (the reverse containment holds in any lattice)
we note that an arbitrary element in the left-hand side is of the form x ∧ y where x ∈ F,
[ 58 ] ∧
Let x denote the central dilatation / x. We could take x as primitive and define x as (⊥| )|x. There is something to
∧ ∨
be said for this choice. x (unlike x and x ) appears as an innate operation on almost any graphic calculator. At first glance it
looks like we could reduce by one the number of axioms. We would take the single |x = x and use the previous footnote to
obtain the two unital laws.
The important reason for not using this definition is that the origin of the notion of scales would be belied. But there is
another: even when the scale identity is translated into this language the equations are not complete. They do not fix the
primitive operation, x. For a separating example take any scale and define x as the standard central dilatation with one
exception: redefine > any way one chooses. Thus a further equation—besides the translation of the scale identity—is needed to
∧
fix the primitive x operation as defined from x. (Without such, note that there is no way of proving that the primitive operation
x is covariant, hence no way of showing that >-faces arise from congruences and no way of obtaining the linear representation
theorem.)
One could redo the notion of minor scale using x as the primitive. After the constant law add the three equations |x = x,
>|(>|x) = > , and ⊥|(⊥|x) = ⊥. The proofs that the elements named by the graded terms are closed under dotting and
midpointing remain unchanged. That they are closed under x one need only verify > = >|(>|>) = >, ⊥ = ⊥|(⊥|⊥) = ⊥,
>|(⊥|x) = (>|⊥)|(>|x) = >|x, and, similarly, ⊥|(>|x) = ⊥|x. All of which implies that the previous argument that the theory
has a unique consistent equational completion still holds.
[ 59 ] It’s worth noting—in the context of scales—an alternative definition: F is a filter if

>∈F
x∧y ∈F iff x ∈ F and y ∈ F

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y ∈ G and x ∧ y ∈ H. But the last condition implies that both x and y are in H. Hence,
x ∈ F ∩ H and y ∈ G ∩ H, thus x ∧ y ∈ (F ∩ H) ∨ (G ∩ H).
Distributivity, recall, is quite enough to establish that a lattice of congruences is a locale,
that is, finite meets distribute with arbitrary joins. It is always a spatial locale: the points are
the “prime” congruences, that is, those that are not the intersection of two larger congruences.
Translated to filters: F is a point if it has the property that whenever x ∨ y ∈ F it is the case
that either x ∈ F or y ∈ F. Put another way, of course, the points of Spec(S) are the linearly
ordered quotients of S. We’ll show (in Section 42, p??–??) that it is compact normal (but not
always Hausdorff). We can obtain (just as in the ancestral subject for spectra) a representation
of an arbitrary scale (instead of an arbitrary Noetherian ring without nilpotents) as the scale
of global sections of a sheaf of linear scales (instead of domains).[60]
We pause to obtain a “pushout lemma” for scales:
8.7 Lemma: Let A → B be monic and A → C a quotient map. Then in the pushout
A→B
↓ ↓
C→D

the map C → D is monic (and, as in any category of algebras, B → D is a quotient map).

Because, if we view A as a subscale of B and take F = ker(A → C) then we obtain a >-face
of B, to wit, F ⇑ = { b ∈ B : ∃a∈A a ≤
= b }. It is easy to check that F ⇑ is zoom-invariant and
⇑ ⇑
that A ∩ F = F. Define D = B/F. The map A → B → D has the same kernel as A → C
and we obtain an embedding C → D. It is easily checked to yield a pushout diagram.[61]

9. Lipschitz Extensions and I-Scales

Given an equational theory T, we may say that an extension T 0 is “co-congruent” if
congruences for the operations in T remain congruences for all the operations in T .0 If all new
operations are constant then the extension is automatically co-congruent (e.g., the theory of
monoids is a co-congruent extension of the theory of semigroups). A more interesting example
is the next step: a monoid congruence is automatically a congruence with respect to the entire
group structure (because x ≡ y implies x-1 = x-1 yy -1 ≡ x-1 xy -1 = y -1 ). [62]
[ 60 ] Recall that the space of points of a spatial locale is always “sober” (most easily defined as a space maximal among T
0
spaces with the given locale of open sets). There is often a minimal space, one with the fewest points (the pre-ordained name
for this condition is “spaced-out”). For any distributive congruence lattice this minimal space does exist: its elements are the
congruences of the sdi quotients. (Could this connection between universal algebra and Stoned theory be new?)
[ 61 ] This pushout lemma fails in most equational theories. In the category of groups consider the pushout square—as above—

where B is the alternating group of order 12, A is the Klein group (the unique subgroup of B of order 4), A → B its inclusion
map, and A → C an epimorpism where C is a group of order 2. Then the pushout, D, is of order 3. The map C → D is
clearly not epic. A more dramatic example is to enlarge B to the alternating group of order 60; D then collapses to a 1-element
group.
[ 62 ] There are several similar examples that involve a unary involutory operation that delivers something like an inverse. A

lattice-congruence on a Boolean algebra is automatically a congruence for negation: if x ≡ y then ¬x = ¬x ∧ (y ∨ ¬y) ≡

¬x ∧ (x ∨ ¬y) = ¬x ∧ ¬y = (¬x ∨ y) ∧ ¬y ≡ (¬x ∨ x) ∧ ¬y = ¬y. A ring-congruence on a von Neumann strongly regular ring
is automatically a congruence for the “pseudo-inverse” (to wit, the unary operation that satisfies x2 x∗ = x = x∗∗ ): if x ≡ y
then (using that xx∗ = x∗ x is a consequence of the axioms) x∗ = x∗2 x ≡ x∗2 y = x∗2 y 2 y ∗ = x∗2 y(y 2 y ∗ )y ∗ = x∗2 y 3 y ∗2 ≡
x∗2 x3 y ∗2 = x∗ (x∗ x2 )xy ∗2 = x∗ x2 y ∗2 = xy ∗2 ≡ yy ∗2 = y ∗. (See Section 46, p??–?? for subscorings.)
A congruence on a acale with respect “ . to midpointing“ . and .the
” two zoom
“ . .operations is automatically ”a congruence
“ . . for dotting:
. . .” . ” “ . . . ”
if u ≡ v then u = / ( | u) = / (v |v)| u ≡ / (v |u)| u = / (v | u) | (u| u) = / (v | u) | = / (v | u) | (v |v) =
“. . ” “. . ” . .
/ v |(u |v) ≡ / v |(u |u) = / (v | ) = v . (These three sequences of equalities can be cut in half with the observation
that it suffices in showing two terms are congruent that one of them is congruent to a term that’s invariant when the terms are
interchanged.)
But these three examples are misleading. In each case the new operation is unique—when it exists—given the old structure. As
we will see such is not the case for all the co-congruent extensions of the theory of scales. (Indeed the potentially most powerful

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An extension of the theory of scales is co-congruent iff the equivalence relation determined
by any >-face respects the new operations that appear in the extended theory. We will use
x ◦−−◦ y to denote (x −◦ y)∧(y −◦ x). A new unary operator f is co-congruent if every >-face
F that contains the element x ◦−−◦ y also contains the element f x ◦−
−◦ f y. If we take F to be the
principal >-face ((x ◦−
−◦ y)) then for co-congruence to hold we must have, for some integer n,

n
x ◦−
−◦ y ≤ = >| (f x ◦− −◦ f y)
When interpreted on the standard interval this becomes the assertion that f is Lipschitz
= 2n ):
continuous (with Lipschitz constant ≤

= 2n |x − y|
|f x − f y| ≤ [ 63 ]

If we move to the free algebra on two generators for the extended theory we see that
co-congruence requires the existence of an n that works in all models. The argument for
unary operations easily extends to arbitrary arities. Hence for extensions of the theory of
scales we will use the phrase Lipschitz extension instead of “co-congruent extension.”
The first application is that Lipschitz extensions of the theory of scales all enjoy the linear
representation theorem. (More important will be the consequence of the section to come:
every consistent Lipschitz extension has an interpretation on the standard interval I.)
If M is a monoid, we understand an M-scale to be a scale on which M acts. We treat the
elements of M as naming unary operations on the scale. Given m ∈ M and x in the scale we
use the usual convention of denoting the values of the corresponding unary operation as mx.
We will not require that the M -actions be endomorphisms of the entire scale structure or
anything else in particular. In all the cases to be discussed, however, the actions will preserve
midpointing and the center:

m(x|y) = mx|my m =

Every scale S is canonically an I-scale determined by the induction scheme (as described
. .
in Section 3, p??–??): >x = x, ⊥x =x, (>|q)x = x|(qx) and (⊥|q)x = x |(qx). [64]
We will be particularly interested in two cases: when M = I and when M is the submonoid
of rationals in the standard interval. The theory of I-scales is obtained by adding for all
r ∈ I and q ∈ I with q ≥= r the equation [65]

q ≥
= r>

and if q ≤
=r
q ≤
= r>
theorem provides existence proofs—on the standard interval—and the most valued such proofs are precisely those for which
there is no uniqueness to force the construction. See p??.) Nor is uniqueness sufficient. A meet semi-lattice has at most one
lattice structure but notice that on the four-element non-linear lattice the equivalence relation that smashes the three elements
below the top to a single point is a meet- but not a lattice-congruence. A lattice has at most one Heyting-algebra structure
but the only non-trivial variety of Heyting algebras in which lattice-congruences are automatically Heyting congruences is the
variety of Boolean algebras: any non-Boolean Heyting algebra contains a three-element subalgebra and the equivalence relation
on the three-element Heyting algebra that smashes the bottom two elements to a point is a lattice- but not a Heyting-algebra
congruence. This works also as a one-variable example: to wit, it is a congruence for the lattice structure but not a congruence
for the negated semi-lattice structure (as described in Section 34, p??–??).
[ 63 ] On the unit interval x ◦− •
−◦ y (the dotting operation applied to x ◦− −◦ y) is |x−y|. The dual semiquation
n • •
of x ◦− −◦ y ≤ (>|) (f x ◦−−◦ f y) is (⊥|)n (f x ◦−−◦ f y) ≤ x ◦− −◦ y.
[ 64 ] The set of operations that preserve midpointing and form a closed midpoint algebra and we can see this action of I on
arbitrary scales as a consequence of the fact that I is the initial closed midpoint algebra.
[ 65 ] Bear in mind that in the presence of a lattice operation any semiquation is equational: x ≤ y is equivalent with x = x ∧ y.

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The theory is not as it stands Lipschitz.[66] So we impose the further condition

x ◦− = rx ◦−
−◦ y ≤ −◦ ry

There is obviously a unique I-scale structure on the standard interval I.

9.1 Lemma: Any consistent theory of scales may be conservatively extended to include the
theory of I-scales.[67]
The “compactness argument” is just what is needed: given a non-trivial model S of a given
theory every finite set of equations in the extended theory may be modeled on S itself (for
each relevant r ∈ I we can find q ∈ I whose action on S will satisfy the finite number of
equations that involve r) and such suffices for consistency.
It is a consequence of the results in the next section that every non-trivial model of a
Lipschitz theory of scales has a quotient isomorphic to I with its unique I-action.

10. Simple Scales and the Existence of Standard Models

10.1 Theorem: A scale is simple iff the sequence ⊥, >|⊥, >|(>|⊥), . . . , (>|)n ⊥, . . . is cofinal
among all elements below >.
The cofinality clearly implies simplicity: if F is a non-trivial

n >-face then necessarily there
exists x ∈ F, x 6= > and the cofinality says that >| ⊥ ∈ F for some n, hence that ⊥ ∈ F,
which means—of course—that F is entire. Conversely, a simple algebra is necessarily an sdi,
hence necessarily linear; but we have much more. We may take the element s used above in
the characterization of sdi s to be the element ⊥. Then, since ⊥ is included in every non-trivial
>-face (because in a simple scale the entire set is the only non-trivial >-face) we know that

n
the sequence { >| ⊥} is cofinal among all elements below >.
A scale satisfying this condition is called, of course, Archimedean.[68]
10.2 Lemma: A scale is simple iff between any two elements there is a constant (that is, an
element from the initial subscale).
One direction is immediate:
if between > and any x < > there is some constant, then there
n
must be one of the form >| ⊥. The other direction requires a little work. Given u < v, let n

n + 1
be minimal such that v −◦ >| ⊥, The argument requires induction on n. If n = 0, that
.
is if v −◦ u < then the two semiquations u = > −◦ u ≤ = v −◦ u and v = v −◦ ⊥ ≤ = v −◦ u
[ 66 ] Order the polynomial ring R[ε] by stipulating P (ε) ≥ 0 iff P (1/n) ≥ 0 for all sufficiently large n. Its standard interval

is a scale and the map that sends P (ε) to P (2ε) is a scale-automorphism thereon, hence easily satisfies all equations so far
(with r = 1). It is not Lipschitz.
[ 67 ] A standard equational-theory consequence is that any scale is embedded in its I-scale reflection: given a scale S use the

equational theory obtained by adding to the theory of scales a constant for each element of S and adding as equations all the
variable-free equations in these constants that hold for S.
[ 68 ] There are those who say that the property should be known as “Eudoxian.” Euclid wrote about it in the Elements and

Proclus said that the idea was due to one Eudoxus but a case may be made for Archimedes. Euclid’s Definition 5 of Book V:
Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any
equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth,
the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively
taken in corresponding order.
But Archimedes in his Quadrature of the Parabola does a better job of isolating the salient point:
The excess by which the greater of (two) unequal areas exceeds the less, can by being added to itself be made to
exceed any given finite area.
Which is how in the absence of the word positive one states that any positive quantity when repeatedly doubled becomes
arbitrarily large. (Surely such is equivalent to the assertion that any quantity when repeatedly halved becomes arbitrarily small.)
Archimedes, note, did not actually claim originality in this; immediately after the line quoted above he writes:
The early geometers have also used this lemma.

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(consequences of the contravariance of −◦ in the first variable and covariance in the second)
yield u < < v and we are done. If n > 0 we consider the three cases: u < v ≤= , u< <v
∧ ∧
and ≤
= u < v. In the 1st case, we have u = ⊥ = v, hence (using the scale identity)
∨ ∨ ∧ ∧ ∨ ∨ ∨ ∨
n
v −◦ u = (v −◦ u) | (v −◦ u) = (v −◦ u)|> and we obtain v −◦ u < >| ⊥. Hence by
∨ ∨
the inductive assumption there is a constant r ∈ I such that u < r < v, thus for q = r|⊥ we
∨ ∧ ∨ ∨ ∨ ∧
have u = u | u = u |⊥ < q < v |⊥ = v | v = v. In the 2nd case we take, of course, q = . In the
. . .
3rd case we use the 1st case to obtain v < q <u and finish with u < q < v.
There are two remarkable facts. The first is that there are many simple scales, so many
that every non-trivial scale has a simple quotient: use Zorn’s lemma on the set of >-faces that
do not contain ⊥.
The second is that there are very few simple scales. Because there is a constant between
any two elements we know that elements are distinguished by which constants appear below
(or for that matter, above) them, hence there can not be more elements in a simple scale
than there are sets of constants: therefore a simple scale has at most 2ℵ0 elements.[69]
Since there is no flexibility on what homomorphisms do to constants:
10.3 Proposition: Given a pair of simple scales there is at most one map from the first to
the second.
Thus the full subcategory of simple scales is a pre-ordered set. It has a maximal element
and the name of that maximal element is the closed interval, I. Non-constant maps from
simple scales are embeddings, hence every simple scale is uniquely isomorphic to a unique
subscale of I. (And for an algebraic construction of I take any simple quotient of a coproduct
over the family of all simple scales.)
Combining the two remarkable facts we obtain
10.4 Theorem: Every non-trivial scale—indeed, any non-trivial model of any Lipschitz
extension of the theory of scales—has a homomorphism to I.
One quick corollary: add any set of constants to the theory of scales and any consistent set
of equations thereon. Necessarily there is an interpretation for all the constants in I satisfying
all the equations. (Recall that if all the new operations in an extension of the theory of scales
are constants then it is automatically a Lipschitz extension.) As an example, adjoin just one
constant, a, and a maximal consistent set of axioms of the form q −◦ a = > and a −◦ q = >
where the q s are restricted to constants, (that is, names of elements of I). Such a maximal
consistent extension is, of course, called a “Dedekind cut” and this quick corollary of the
standard models theorem (to wit, that any such set of conditions can be realized in I) is, of
course, called “Dedekind completeness.”
Because every consistent Lipschitz theory of scales can be enlarged to a consistent Lipschitz
theory of I-scales (Section 9, p??–??) we obtain, as an immediate consequence of the existence
of simple quotients of non-trivial I-scales:
10.5 Theorem: on the existence of standard models
Every consistent Lipschitz extension of the theory of scales has an interpretation on I.
An immediate corollary (logicians would call it a “compactness theorem”): A Lipschitz
extension has a model on I iff each finite set of its equations has a model on I.
[ 69 ] Compare with the theory of groups: no quotient of the rational numbers—viewed as a group under addition—is simple.

On the other hand there are simple groups of every infinite cardinality; indeed, every group can be embedded in a simple group.
(For infinite G, compose its Cayley representation with the quotient map that kills all permutations that move fewer elements
than the cardinality of G.)

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Note that every non-trivial linear scale
has a unique map to I (the kernel of the map is
n
the >-face of all elements larger than all >| ⊥). We may rephrase this: consider the category
∧ ∨
of non-trivial scales that satisfy the coalgebra condition, to wit, x = ⊥ or x = > for all x.
This category has a final object. (This fact is, of course, much much weaker than the usual
statement of the coalgebraic characterization of I, but it is comforting to see it arise in such
a purely algebraic manner.)
Since each simple scale has a unique map to I:
10.6 Lemma: The maps from a scale to I are known by their kernels.
Later we will use the fact that the maximal--T-face spectrum of A, denoted Max(S)
is canonically equivalent with the set of maps (A, I). One immediate application:
10.7 Theorem: The standard interval, I, is an injective object in the category of scales.
Because, given a subscale S 0 of S and an I-valued map f 0 from S 0 we seek an extension to
all of S. Let F ⊆ S be the >-face generated by ker(f 0 ), (that is, the result of adding all
elements in S larger than an element in ker(f 0 )) and note that it remains a proper >-face,
hence S/F is non-trivial and we may choose a map S/F → I. Define f to be S → S/F → I.
The kernel of S 0 → S → I must, of course, contain ker(f 0 ). But ker(f 0 ) is maximal, hence
f 0 has the same kernel as S 0 → S → I and as we just noted, maps to I are known by their
kernels: thus f is an extension of f.0 [70]
Bear in mind that all these special properties of I are maintained for any Lipschitz
extension of the theory of scales.
Following the language of ring theory we define the Jacobson radical of a scale to be
the intersection of all its maximal >-faces and we say that a scale is semi-simple if its
Jacobson radical is trivial. (The name used in the theory of convex sets for maximal proper
faces is “facet,” hence we could say that the Jacobson radical is the intersection of all the
“top-facets.” Doing so, of course, means that one must not be bothered by etymology.) A scale
is semi-simple iff its representations into simple quotients are collectively faithful. (Hence, a
better term for both rings and scales would have been “residually simple.”)

n
10.8 Theorem: A scale is semi-simple iff supn >| ⊥ = >.

n
In any simple scale the cofinality of >| ⊥ implies the weaker condition that its supremum is >
and such remains the case in any cartesian product of simple scales. To see that the condition
implies semi-simplicity we need to show that it implies for every x < > that there is a simple
quotient in which x remains less than >. The
condition tells us that there is n such that the
equality (>|)n ⊥ < x fails, that is. (>|)n ⊥ −◦ x < >. By the linear representation theorem


we may find a linear quotient in which that failure is maintained, that is, (>|)n ⊥ −◦ x
remains below >, hence in which x ≤ = (>|)n ⊥. Now take its (unique) simple quotient.
10.9
Lemma: The Jacobson radical of a scale is the set, R, of all x such that
n
>| ⊥ ≤
= x for all n.

R is in the kernel of every simple quotient. For the converse
It is clear
we need to show that
(⊥|)m > −◦ x ≡ > (mod R)
for if all m, then x ∈ R. But (⊥|)m > −◦ x ≡ > (mod R)


says, of course, that (⊥|)m > −◦ x ∈ R hence (>|)n ⊥ ≤ = (>|)m ⊥ −◦ x all n. In particular


(>|)n ⊥ ≤= (>|)n ⊥ −◦ x all n. Use the adjointness lemma to obtain (\ >|)n ⊥ ≤
= x all n, hence
n−1
that (>|) ⊥ ≤ = x all n, which, of course, says x ∈ R.
[ 70 ] The injectivity of I is quite enough to yield the existence of a map to I from every non-trivial scale since we know that

any non-trivial scale contains a copy of at least one scale with a map to I. to wit, I.

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Our definition of the Jacobson radical as the intersection of all the maximal >-faces relied
on the axiom of choice. But note that this construction of the Jacobson radical is choice-free.
(So it would have been better—with both rings and scales—to use the choice-free construction
as the definition.)

11. A Few Applications

For the most algebraic construction of I, take the co-product of all one-generator simple
scales and reduce by the Jacobson radical. Put another way: start with a freely generated
scale and for each generator reduce by a maximal consistent set of relations that involve only
that generator (and, of course, the primitive constants of the theory of scales). Do this in such
a way that every possible such set of relations appears for at least one generator. Now take
any simple quotient. It is necessarily a copy of I. But we do not need the axiom of choice:
there is only one simple quotient and it is the result of reducing by the Jacobson radical.
(We are in a case where semi-simplicity implies simplicity.)
Footnote [??] (p??) suggested a way of handling limits of sequences
Q in I. Let’s redo it,
this time without using the axiom of choice. Again let IN = N I denote the scale of all
sequences in I. The first step is to identify sequences that agree almost everywhere to obtain
the quotient scale IN/E (where E is the >-face of all sequences that are eventually constantly
equal to >). The next step (a step we could not take before) is to reduce by the Jacobson
radical. (IN/E)/R. As already observed, there is never more than one map from I to a semi-
∆
simple scale hence the map I → IN → (IN/E) → (IN/E)/R is the unique map from I to
(IN/E)/R. Let
C → IN
↓ ↓
I → (IN/E)/R
be a pullback where we view both horizontal maps as inclusions. C is the subscale of all
convergent sequences. The (vertical) map C → I is the unique map from C to I that
respects almost-everywhere equivalence. Its standard name is “ lim ”.
n→∞
N
We said in Section 0 (p??–??) that C ⊆ I could be defined as the joint equalizer of
all the closed midpoint maps from IN/E to I. Since pullbacks of equalizers are equalizers it
suffices, obviously, to show that the (unique) map I → (IN/E)R is such a joint equalizer.
Define the simple part of any semi-simple scale to be the joint equalizer of all maps to
I; since, by definition, those maps are jointly faithful any map from the simple part to I
is necessarily an embedding and the simple part is, indeed, simple; conversely any simple
subscale has a unique map to I, hence is in the simple part. All of which says that any map
from I to a semi-simple scale, e.g., the diagonal map from I to IN followed by the quotient
map to (IN/E)R, is automatically its simple part.[71]
[ 71 ] In Section 0 (p??–??) we made the joint-equalizer assertion not for scale maps but for closed midpoint maps. For a proof,

let s1 , s2 , . . . , be a sequence in I. Note first that if q ∈ I is an upper bound for almost all sn , then for any closed midpoint
map, f : IN/E → I we have f (s) ≤ q, hence f (s) ≤ lim sup s and similarly f (s) ≥ lim inf s. In particular, for any convergent
s we have f (s) = lim s. Next, if a is an accumulation point of s then we may take infinite N0 ⊆ N such that s restricted
0
to N0 converges to a. The inclusion map N0 → N induces a scale map IN/E → IN /E that carries s to a convergent sequence.
The axiom of choice gives us an I-valued scale map, hence any accumulation point of s appears as a value of an I-valued
scale map—a fortiori, a closed midpoint map—from IN/E to I. Since I-valued closed midpoint maps are closed under convex
combinations we may conclude that the closed interval [lim inf s, lim sup s] is the set of all such values. (But the only values of
the scale maps are the accumulation points: if b is not an accumulation point we may chose ` < b < u such that sn 6∈ (`, u)
for almost all n; then (sn − −◦ `) ∨ (u −
−◦ sn ) = > for almost all n; hence (f (s) −
−◦ `) ∨ (u −
−◦ f (s)) = > and we may conclude
f (s) 6∈ (`, u). So every closed subset of I so appears: surely it’s an old exercise that any separable closed subset in any space is
the set of accumulation points of a sequence in that space.)

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Even before we took limits of sequences we defined C(G) as the set of continuous maps and
promised that “ In Section 11 (p??–??) we will obtain a totally algebraic definition.” The
scale of uniformly continuous I-valued maps on a uniform space, X, is easier to construct.
Consider, first, I,X×X the scale of all functions from X ×X to I. Let E be the >-face of those
functions that are equal to > on some element of the given uniformity and let R be the
Jacobson radical of IX×X/E. The pair of projection maps from X ×X to X induces a pair of
maps from IX to IX×X which, in turn, yields—by composition—a pair of maps from IX to
(IX×X/E)/R. The scale of uniformly continuous I-valued maps on X is the equalizer of this
final pair of maps.
For ordinary continuity let X be a topological space and IX = X I denote the set of
Q
all functions (continuous or not) from X to I. For x ∈ X identify functions that agree on a
neighborhood of x to obtain IN/Ex (where Ex is the >-face of all sequences that are equal to
N
> on some neighborhood of x ). Then reduce by the Jacobson radical (I /Ex )/R. Let

Cx→ IX
↓ ↓
I → (IX/Ex )/R
be a pullback where Cx → IX is an inclusion. Cx is the subscale of all functions from X to
I that are continuous at x, or, put another way:
11.1 Proposition: The functions in IX that are continuous at x is the pullback of the
simple part of (IX/Ex )/R.
Finally: \
C(X) = Cx ⊆ IX [ 72 ]

x∈X

12. Non–Semi-Simple Scales and the Richter Scale

The Richter scale, R, is defined as the interval from h−1, 0i through h1, 0i in the lexicograph-
ically ordered D-module D ⊕ D. [73] The set-valued “Jacobson-radical functor” is represented
by the Richter scale, that is, the elements of the Jacobson radical of a scale A are in natural
one-to-one correspondence with the scale maps from R to A. Mac Lane’s “universal element”
(the most important concept in Algebra !) may be taken to be h1, −1i: for every element, x,
in the Jacobson radical of A there is a unique map that carries h1, −1i to x. [74]
The Richter scale is not, of course, simple. But it just misses. It has just one quotient
neither entire nor trivial, to wit, its semi-simple reflection, I. Hence the Richter scale
appears as a subscale of every non–semi-simple scale: the necessary and sufficient condition for
semi-simplicity is that a scale not contain a copy of the Richter scale.
[ 72 ] A quite different way of getting at C(X) appears in [??] (p??).
[ 73 ] If we view D ⊕ D as the ring D[~]/(~2 ), ordered so that ~ is “infinitesimal,” then the Richter scale is just the standard
interval in the “ring of dyadic dual numbers.”Its Jacobson radical is the set of all of its pairs of the form h1, qi (note that q is
necessarily non-positive).
[ 74 ] For a proof, start with the following:
∧
f h1, −m2-n i = (>|)n ((x | )m >)
-
f h0, −m2 i = ⊥|f h1, −m21−n i
n
.
f h0, +m2-n i = (f h0, −m2-n i)
-n
f hq, ±m2 i = / (q|f h0, ±m2-n i)
These formulas can be used, at least, for the uniqueness of f , but the fact they describe a scale map requires a bit of work. When
we have in hand the representation theorem of Section 20 (p??–??) for the free scale on one generator an easier proof will be
available. (And note, in passing, that Aut(R) ∼ = Z.)

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A non-simple scale, S, is sdi iff there is an embedding R → S such that R? is cofinal in S?

(where the “lower star” means “remove the top”).[75]
It is worth having at our disposal examples of arbitrarily large sdis. Let L be any linearly
ordered D-module.[76] Order D⊕L⊕D lexicographically and define the Scoville scale, Sv[[L]]
as its closed interval with h1, 0, 0i as > and its negation as ⊥. We may regard R as the subscale
of Sv[[L]] of elements of the form hx, 0, yi. Every element in Sv[[L]] not in this copy of R is
less than an element in R’s Jacobson radical to wit, the elements for the form h1, 0, yi (where
necessarily y ≤ = 0) and any element x < > in that radical is such that {(>|)n x} is cofinal
among the elements below > in all of Sv[[L]]. The existence of such an element, recall, is
equivalent with Sv[[L]] being an sdi (Section 8, p??–??).

13. A Construction of the Reals

Among the many ways of constructing the reals perhaps the nicest is as the set of “germs of
midpoint- and -preserving self-maps” on I, that is, a real is named by such a map defined
on some open subinterval containing ∈ I; two such partial maps name the same real if
their intersection is also such a partial map. For each real there is a canonical name, to wit,
the partial map with the largest domain.
The entire ordered-field structure is inherent: 0 is named by the constant map; 1 by

the identity map; negation by dotting; r + s is characterized by (r + s)x = / (rx)|(sx) ;
multiplication is, of course, defined as composition (with reciprocation obtained by inverting
maps); and r ≤= s iff rx ≤ = sx for all positive x near . The standard interval in this ring
has a (unique) scale-isomorphism to I, to wit, the map that sends r to r> (as defined by
the canonical name for r).
To fill in the details we need:
13.1 Lemma: Any midpoint-preserving partial self-map on I with an interval as domain is
monotonic.
It suffices to show that if f preserves midpoints then it preserves betweenness. Suppose that
a < b < c and that f (b) is not between f (a) and f (c). We will regularize the example by
.
replacing f with f, if necessary, to ensure that f (a) < f (c). Either f (b) < f (a) < f (c)
..
or f (a) < f (c) < f (b). We may—further—replace f x with f x, if necessary, to ensure
the latter. It suffices to show that there is another point b0 between a and c which is not
only a counterexample, as is b, but doubles, at least, the extent to which it is a counterexam-
ple, that is, we will obtain the semiquation f (b0 )−f (c) =
≥ 2(f (b)−f (c)). This suffices because
if we iterate the construction this distance will eventually be greater than the distance from
c to >. The construction of b0 is by cases:

0 c/ b if b > a|c
b =
a/ b if b < a|c
In the first case we have that f (b) = f (c)|f (b0 ), hence f (b0 ) − f (c) = 2(f (b) − f (c)). In the
second case we have f (b) = f (a)|f (b0 ), hence f (b0 ) − f (c) = (f (b0 ) − f (a)) + (f (a) − f (c)) =
= 2f (b) − 2f (c). [77]
2(f (b) − f (a)) + (f (a) − f (c)) = 2f (b) − (f (a) + f (c)) ≥
[ 75 ] We will see later that all such R → A are “essential extensions” as defined in Section 24 (p??–??).
[ 76 ] One example: let J be a linearly ordered set and L the lexicographically ordered set of functions J → D, each with only
finitely many non-zero values.
[ 77 ] The monotonicity of midpoint-preserving maps requires linear ordering. Consider the non-constant midpoint-preserving
.
map, I × I → I that sends hu, vi to u| v. It sends both h>, >i and h⊥, ⊥i to and any monotonic map that collapses top
and bottom must, of course, be constant. But simplicity—not just linear ordering—is also required: the non-constant map on
the Richter scale that sends hx, vi to h0, vi preserves midpoints and, again, sends both ends to .

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13.2 Corollary: Any midpoint-preserving partial self-map on I with an interval as domain

is determined by its values at any two of its points..
It is easy enough to construct the midpoint operation on reals: given partial midpoint- and
-preserving maps r and s define (r|s)x = (rx)|(sx). (The medial law is just what is needed
to show that r|s preserves midpoints.) The idempotent, commutative and medial laws for
real midpointing are automatic. Using that the image of the central contraction, |, is the
“middle half,” to wit, the sub-interval [⊥| , |>] (Lemma 8.4, p??) we name the real 2 with
the map / defined on the middle half.
(To check that it preserves midpoints it suffices to
check that
|(2(x|y)) = | (2x)|(2y) .) We construct r+s as 2(r|s) and check that it satisfies
| (r+s)x = (rx)|(sx) for all x near . We easily verify—using self-distributivity and the
cancellation law—the medial law for addition: ( |((r+s)+(t+u))) = (|((r+s)|(t+u)))) =
( |(r + s))|( |(t + u)) = (r|s)|(t|u) and similarly ( |((r + t) + (s + u))) = (r|t)|(s|u). By fiat
0 is a unit for addition. Associativity is then a consequence of the medial law: (r + s) + u =
(r + s) + (0 + u) = (r + 0) + (s + u) = r + (s + u). For commutatity of multiplication it
suffices to verify it for reals named by contraction at elements of I (because by repeated
central contractions we can reduce any two arbitrary reals to such). For the commutativity
of the multiplicative structure on I it suffices to verify it on an order-dense subset, to wit, I.
Finally, for distributivity: r(s + t) = r(2(s|t)) = (r2)(s|t) = (2r)(s|t) = 2(r(s|t)) =


2 (rs)|(rt) = (rs) + (rt).

[78]
14. The Enveloping D-module
Given a scale, A, we construct its enveloping D-module, M , as the direct limit of:

| | |
A → A → A → ···
More explicitly, its elements are named by pairs, hx, mi, where x is an element of A and m is
a natural number. The pair hy, ni names the same
element iff ( |)n x = ( |)m y. Addition is
defined by hx, mi+hy, ni = h ( |)n x | ( |)m y , m+n+1i. It is routine to check that the


definition is independent of choice of name. Commutativity is immediate. The zero-element
is named by h , 0i and it is routine to see that it is a unit for addition. The medial law can
be verified by straightforward application of the definitions. Associativity is—once again—a
consequence: (a + b) + c = (a + b) + (0 + c) = (a + 0) + (b + c) = a + (b + c). The negation of
.
hx, mi is named by hx, mi. Scalar multiplication by 1/2 sends hx, mi to h |x, mi.
[79]
Embed A into M by sending x to hx, 0i. We order M by hx, mi ≤
=
[ 78 ]As noted, we usually pronounce “D-module” as “dy-module.”
[ 79 ]We have used so far only that A is a closed symmetric midpoint algebra, which, recall, was defined to be a model of the
three equations for midpointing, the one equation for dotting and the (non-equational) Horn condition of cancellation.
With a little more work one may drop dotting and obtain a representation for plain closed midpoint algebras as follows.
Given an object A with a binary operation satisfying the idempotent, commutative and medial laws and the Horn condition of
cancellation, define a congruence on the cartesian square A×A by hu, vi ≡ hw, xi iff u|x = v|w. Reflexivity and symmetry
are immediate. For transitivity suppose, further, that hw, xi ≡ hy, zi. Then from u|x = v|w and w|z = x|y we may infer
(u|x)|(w|z) = (v|w)|(x|y) hence (w|x)|(u|z) = (w|x)|(v|y) which by cancellation wields hu, vi ≡ hy, zi. That it is a congruence
is immediate: hu, vi ≡ hw, xi easily implies hy|u, z|vi ≡ hy|w, z|xi. Let A × A → S be the quotient structure. The three
equations automatically hold in S but the cancellation condition requires verification (if hy|u, z|vi ≡ hy|w, z|xi then from
(y|u)|(z|x) = (z|v)|(y|w) we may infer (y|z)|(u|x) = (y|z)|(v|w) and use cancellation in A to yield hu, vi ≡ hw, xi.) Define
.
dotting on S by hu, vi = hv, ui and verify the constant law. Map A into S by choosing an element c and sending x ∈ A to
hx, ci ∈ S. The cancellation law must be used one more time to prove that this is a faithful representation.
In [??] (p??) when verifying that I is the initial scale it was pointed out that there are cyclic closed symmetric midpoint
algebras. We may eliminate them by imposing a torsion-freeness condition, to wit, by adding the further Horn conditions
[(x|)p y = y] ⇒ [x = y], one such condition for each odd prime p. Then one may prove that the enveloping D-module will be
torsion-free. If one starts with a finitely generated torsion-free symmetric or plain midpoint algebra one ends in a finite-rank

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hy, ni iff ( |)n x ≤= ( |)m y. We obtain a midpoint-isomorphism from A to the set of all
elements in M from h⊥, 0i through h>, 0i. That is, h⊥, 0i ≤ = hx, ni ≤
= h>, 0i implies there is
y ∈ A such that hx, ni = hy, 0i and that is because the two M -semiquations translate to the
two A-semiquations: ( |)n ⊥ ≤ =x≤ = ( |)n >. When we showed that the image of the central
contraction is the sub-interval from |⊥ through |>, hence that the image of the nth iterate
of the central contraction is the sub-interval [( |)n ⊥, ( |)n >], we showed—precisely—that it
is possible to solve for hx, ni = hy, 0i.
We may thus infer:
14.1 Lemma: Every scale has a faithful representation as a closed interval in a lattice-
ordered D-module. [80]

15. The Semi-Simplicity of Free Scales

15.1 Theorem: An equation in the theory of scales (I-scales) holds for all scales iff it holds
for the initial scale, I (I).
15.2 Corollary: The theory of scales (I-scales) is a complete equational theory.
We need to show that if an equation in the operators for scales fails in any scale it fails
in I. It suffices, note, to find a failure in I since the operators are continuous—if a pair
of continuous functions disagree anywhere on In they must disagree somewhere on In. For
reasons to become clear later, we will settle here for a failure in between: we will find a failure
on the standard rational interval, I ∩ Q. Given an equation in the operators for scales we
already know that if there is a counterexample then there is a counterexample in a linear scale
and consequently in a closed interval in an ordered D-module. There are only finitely many
elements in the counterexample, hence the ordered D-module may be taken to be finitely
generated. The ring D is a principal ideal domain, hence every finitely generated D-module is
a product of cyclic modules, to wit, copies of D or finite cyclic groups of odd order. But the
existence of a total ordering rules out the finite cyclic groups. We are thus in an interval in a
totally ordered finite-rank free D-module, hence, a fortiori, a totally ordered finite-dimensional
Q-vector space. We need to move the counterexample into a totally ordered one-dimensional
Q-vector space which, of course, will be taken to be Q with its standard ordering.
We first translate the given counterexample into a set of Q-linear equalities and semiqua-
tions. Besides the variables x1 , x2 , . . . , xn that appear in the counterexample we treat > and ⊥
as variables. The equation that fails is replaced with a strict semiquation, namely, the strict
semiquation that results when the given counterexample is instantiated. For each i we add
the two semiquations ⊥ ≤ = xi ≤ = >. And, of course, we add the strict semiquation ⊥ < >. We
eliminate the scale operations by iterating the following substitutions (where A and B are
terms free of scale operations, that is, are linear combinations of the variables): replace A|B
. ∧
with (A + B)/2; replace A with ⊥ + > − A; replace A with either ⊥ or 2A− >, whichever is
∧
correct for the given counterexample and if A = ⊥ add to the set of conditions to be satisfied
the condition 2A ≤ = ⊥ + >.
Next, replace each weak semiquation either with an equality or strict semiquation,
free D-module (using that D is a principal ideal ring). And one may then embed that module into R, all of which yields a
completeness result, to wit, every universally quantified first-order assertion about the (symmetric) plain midpoint algebra I is
true for all torsion-free (symmetric) midpoint algebras. Continuity considerations suffice for the same result with I replaced by I.
[ 80 ] One immediate consequence is the generalization to all contractions of the lemma just used about central contractions.

That is, the image of the contraction a| on a scale is the subscale from a|⊥ through a|>. (Because the semiquations allow one
to prove that a/ x = 2x − a.) An equational proof that a|⊥ ≤ = x ≤ = a|> implies x = a|(a/ x) must therefore exist but which
proof would appear to be quite incomprehensible.

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depending, once again, on which obtains for the given counterexample. We now eliminate
the equations by using the standard substitution technique to eliminate for each equation,
one variable (and one equation). We thus are given a finite set of strict linear semiquations
and we know that there is a simultaneous solution in a totally ordered finite-dimensional
Q-vector space. We wish to find a solution in Q.
We know two proofs, one geometric and the other syntactical.[81]
The geometric argument takes place in Euclidean space. We first standardize the strict
linear semiquations to a set A of “positivities,” that is a set of linear combinations of the
variables to be modeled as positive elements. We are given a totally ordered finite-dimensional
R-vector space with an (n+2)-tuple of points one for each variable ⊥, >, x1 , x2 , . . . , xn . We
wish to find an orthogonal projection onto a 1-dimensional subspace L so that all of the linear
combinations in A are sent to the same side in L of the origin. To that end, let P be the
polytope whose vertices are precisely those linear combinations in A. Our one use of the total
ordering on the vector space is the knowledge that P does not contain the origin. Given that
fact simply take L to be the 1-dimensional subspace through the point in P nearest to the
origin. The image of P on L lies on only one side of the origin which, of course, we declare its
positive side. The image of the variables on that line give us an R-instantiation as needed.[82]
The syntactical proof uses an induction on the number of variables ⊥, >, x1 , x2 , . . . , xn . Let
C be the set of strict semiquations that do not involve the variable xn . Recast each remaining
equality in the form either

a⊥ ⊥ + a> > + a1 x1 + a2 x2 + · · · + an−1 xn−1 < xn

or
xn < a⊥ ⊥ + a> > + a1 x1 + a2 x2 + · · · + an−1 xn−1
where the ai s are rational. Let L be the set of linear combinations that are to be modeled as
strictly less than xn , and R the set to be modeled as strictly larger that xn . If L is empty use
the inductive hypothesis to find an instantiation that models all the semiquations in C and
then choose an instantiation of xn less than all the modeled values of the combinations in
the set R. Dually if R is empty. If neither is empty, model all the semiquations in C and all
semiquations of the form L < R where L ∈ L and R ∈ R. Then model xn strictly between
the largest of the modeled values of the forms in L and the smallest in R.
A consequence is that the maps from a free scale into I are collectively faithful because
if two terms can not be proved equal then necessarily there is a counterexample in the free
scale, hence in I, a fortiori in I. That is, there are elements of I which—when they are
used to instantiate the variables—produce different values for the two terms. Such, of course,
describes a scale homomorphism from the free scale that separates the two terms. Hence:
15.3 Theorem: Every free scale appears as a subscale of a cartesian power of I.
An immediate corollary is the semi-simplicity of free scales:
15.4 Theorem: Every free scale is embedded into the product of its simple quotients.
Note that the constructions used for counterexamples of equations in I can as easily be
used for counterexamples of universal Horn sentences: the constructions not only preserve
strict semiquations but (in the process of elimination of variables) any number of equalities.
[ 81 ]I learned the latter from Dana Scott, who claimed he was only following the lead of Alfred Tarski.
[ 82 ]For later purposes, we will need to know that there is a Q-instantiation. Since the problem has been reduced to modeling
strict semiquations we may use the continuity of the operations to insure rationality, even dyadic rationality.

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Thus given a sentence of the form

(s1 = t1 ) & · · · & (sn = tn ) ⇒ (u = v)
with a counterexample anywhere the constructions produce counterexamples in I, indeed, in
the rational points in I. (There are Horn sentences true for some non-trivial scales that do
not hold for I. A universal Horn sentence is consistent iff it holds for the initial scale, I. An
example of such that does not hold for I is [x|(x|⊥) = >|⊥] ⇒ [y = z].) We may add one
∨ ∧
non-equational condition, x = > or x = ⊥ (which for good historical reasons will be called
the coalgebra condition) that yields a completeness theorem for the entire universally
quantified first-order theory: such a sentence is a consequence of the defining equations for
scales plus the coalgebra condition iff it true for I.

16. Diversion: Harmonic Scales and Differentiation

The theory of harmonic scales is given by a binary operation whose values are denoted
with catenation, xy, satisfying the equations:
>x = x = x>
x = = x
x (y|z) = (xy)|(xz)
(x|y) z = (xz)|(yz)
(u ◦−
−◦ v)|(x ◦−
−◦ y) ≤
= >|(ux ◦−
−◦ vy)
We’ll refer to the top row as the “unit condition,” the next three rows as as the “bilinear
condition.” The bottom row is, of course, the Lipschitz condition.[83] Standard multiplication
is the unique interpretation of these equations on I, hence there is at most one interpretation
on any semi-simple scale.
A few lemmas we’ll need: 16.1 Lemma:
. .
u v = (uv)
.
⊥v = v
|(uv) = ( |u)v
. . .
For the 1st equation use cancellation on (uv)|(u v) = (u| u)|v = |v = = (uv)|(uv). For the
. . .
2nd : ⊥ v => v = (>v) = v. For the 3rd : ( |u)v = ( v)|(uv) = |(uv).
The harmonic structure greatly extends our descriptive power. Among many other things
it allows us to identify not just any algebraic number in I but many transcendentals. As just
one example, we will see (p??) that if we add a unary operator f satisfying the condition:
. . 
2 .
(f u) |(f (v) | (f v)(u| v) = (u| v)4
  
≤
then for any q ∈ I it is the case that q(f ) < > |( |(f >)) is provable (indeed,
provable using only the rules for equational logic [84] ) iff q <
> e/4 (as defined in the
standard interval).
The standard interval in any ordered unital ring with 12 satisfies the first six equations. Consider again the ring we
[ 83 ]

used before to show the necessity of a Lipschitz condition: the polynomial ring R[ε] ordered by P stipulating PP (ε) ≥ 0
iffPP (1/n) ≥ 0 for all sufficiently large n. If we use, instead, the nonstandard multiplication, ( an εn ) ◦ ( bn εn ) =
1
2
(a0 bn + an b0 − 2n an bn )ε,n then on I it is Lipschitz but on the standard interval of R[ε] it is not.
[ 84 ] A proof that one term is equal to another can always be obtained by a sequence of transformations each of which replaces

a subterm of the form f ht1 , t2 , . . . , tn i with one of the form ght1 , t2 , . . . , tn i where the t s are arbitrary terms and the equality
of f hx1 , x2 , . . . , xn i and ghx1 , x2 , . . . , xn i is an axiom.

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The harmonic structure allows us to identify values of derivatives. Suppose f is a unary

operator on I and a, b ∈ I are constants:
16.2 Proposition: For any natural number, n:
. . . .  .
= (u| a)2 implies f 0 a = b.
= ( |)n ((f u) |f a) | (b(u| a)) ≤
[(u| a)2 ] ≤


Switching to usual notation, this is saying that if

f u − f a − b(u−a)
(u−a)2
is bounded then f 0 a = b. We may rewrite: if there is a constant K such that
fu − fa
−K|u−a| ≤
= −b ≤
= K|u−a|
u−a
then
fu − fa
lim −K|u−a| ≤
= lim −b ≤
= lim K|u−a|
u→a u→a u−a u→a

hence
= f 0a − b ≤
0 ≤ = 0

Therefore, if f and g are unary operators on I satisfying the conditions

. . . .  .
= ( |)n ((f u) |f v) | ((gv)(u| v)) ≤
[(u| v)2 ] ≤ = (u| v)2


we have f 0 = g and—since g is bounded—that f is Lipschitz. We will need that last fact, not
just on simple scales, but for all scales. Even better, g is also Lipschitz:
16.3 Lemma: If f and g are unary operations on a scale and n ∈ N such that
. . . .  .
= ( |)n ((f u) |f v) | ((gv)(u| v)) ≤
[(u| v)2 ] ≤ = (u| v)2


then f and g are Lipschitz.

Because | is a homomorphism with respect to dotting and midpointing we may replace f
with ( |)n f and g with ( |)n g and reduce to the case
. . . .
.
= ((f u) |f v) | (gv)(u| v) ≤
[(u| v)2 ] ≤ = (u| v)2
We may then apply ( |) twice more and replace f with ( |)2 f and g with ( |)2 g thus
reducing to the case

. . . .
.
= ((f u) |f v) | (gv)(u| v) ≤
( |)2 [(u| v)2 ] ≤ = ( |)2 (u| v)2

As often when working with harmonic scales we resort to the standard notation appropriate
to the enveloping D-module (and, in this case, also negate the terms):
− 14 (u−v)2 ≤
= f u − f v − (gv)(u−v) ≤
= 1
4
(u−v)2
Because g was replaced with ( |)2 g we have the additional information − 41 ≤ = 14 , hence
=g≤

− 14 |u−v| − 14 (u−v)2 ≤
= fu − fv ≤
= 1
4
(u−v)2 + 14 |u−v|
1
If we now use the equality 4
(u−v)2 = 21 |u−v| we obtain
≤
− 43 |u−v| ≤
= fu − fv ≤
= 3
4
|u−v|

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which, of course, says that f is Lipschitz.

The proof for g is easier. After replacing f and g with ( |)n f and ( |)n g and moving to
standard notation we have the two semiquations:
−(u−v)2 ≤ = (u−v)2
= f u − f v − (gv)(u−v) ≤
By switching u and v we also have (obviously):
−(v−u)2 ≤ = (v−u)2
= f v − f u − (gu)(v−u) ≤
When we add the two rows we obtain:
−2(u−v)2 ≤ = 2(u−v)2
= (gu − gv)(u−v) ≤
hence
|gu − gv| ≤
= 2|u−v|

Going back to the condition that allows us to identify e/4, it says that in any model on I
we have f = f 0 hence that f u = Aeu for some constant A and thus ef = f > in any model
on I. Since f is necessarily Lipschitz we know that in any simple quotient and any q ∈ I
we have q(f ) < > |( |(f >)) iff q < > e/4, hence such is the case in any linear quotient. The
linear representation theorem says, therefore, that in any model, linear or not, and any q ∈ I
we have q(f ) < > |( |(f >)) iff q < > e/4. We can infer more: let F be the initial algebra
for the theory of harmonic scales with a unary operator satisfying the condition we used to
identify e/4 plus an equation to fix the value of f ; then its semi-simple reflection, F/R, is
simple (to wit, the smallest harmonic subscale of I closed under the action of f ). It suffices
to show that F/R is its own simple part: but for any element, a, we have either q ≤ = a or
a≤ = q for all q ∈ I, forcing a to be in the equalizer of all maps to I, that is, forcing a to be
in the simple part.[85]
A function may have√a derivative at a without [−f u + f a + (f 0 a)(u−a)]/(u−a)2 being
3
bounded (e.g., f (u) = u 4 with a = 0) but not when the derivative is Lipschitz; we have
the converse of the last lemma:
16.4 Lemma: If the derivative of f : I → I is f 0 : I → I and, if, further, f 0
is Lipschitz then there is n ∈ N such than
. . . .  .
= ( |)n ((f u) |f v) | ((f 0 v)(u| v)) ≤
[(u| v)2 ] ≤ = (u| v)2


For a proof let

1 f 0u − f 0v 1 f 0u − f 0v
L = inf U = sup
2 u6=v u−v 2 u6=v u−v
We wish to prove the two semiquations:
= f u − f v − (f 0 v)(u−v) ≤
L(u−v)2 ≤ = U (u−v)2
We’ll prove the first semiquation the second can then be obtained by replacing f and f 0 with
their negations. For each a ∈ I let
ha (u) = f u − f a − (f 0 a)(u−a) − L(u−a)2
[ 85 ] In standard notation we thus have that there are non-trivial functions f such that ((1 − x + y)f x − f y)2 ≤
(x − y)4 for all x, y ∈ [−1, 1]. For any such f it is the case that f 1/f 0 = e. (Could this really be new?) By
2
using e-x instead of e x we needn’t restrict to the standard interval: there are non-trivial real functions f such that
((2x − 2xy + 1)f x − f y)2 ≤ (x − y)4 for all x, y ∈ R. For any such f it is the case that f 0/f 1 = e.
2

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The first semiquation is equivalent to 0 ≤ = ha for arbitrary a. Clearly ha a = h0a a = 0. It thus

suffices to show that a is an absolute minimal point for ha 0and0 for that it suffices to show
that h0a u ≤
= 0 for u < a and h0a u ≥
= 0 for u > a. Since 2L =≥ f u−f
u−a
a
we have for u > a that

f 0u − f 0a
h0a u = f 0 u − f 0 a − 2L(u−a) =
≥ f 0u − f 0a − (u−a) = 0
(u−a)

Note that this (very last) semiquation requires (u − a) ≥= 0. The argument for u < a can
be obtained by using the present result with f (x) and f (x) replaced with f (−x) and f 0 (−x).
0

The last two lemmas yield:

16.5 Theorem: f : I → I is differentiable with a Lipschitz derivative f 0 : I → I iff there
is n ∈ N such that:
. . . .  .
= ( |)n ((f u) |f v) | ((f 0 v)(u| v)) ≤
[(u| v)2 ] ≤ = (u| v)2


Consequently the harmonic structure allows an exploration of differential equations.

In case one wishes to identify π besides e, add to the theory of harmonic scales a unary
operator g subject to the condition:
. .
2 .
2(1|v 2 )(gu|(gv) ) | (v| u) = (u| v)4
≤
.
Then for any q ∈ I it is the case that q < > (g >)|(g ⊥) is provable iff q < > π/4,
0 2 -1
as defined in the standard interval. (The condition implies g (v) = (1 + v ) in any model in
the standard interval.)[86] Again, if we add an equation to fix g we may proceed to show
that the semi-simple reflection of the free model for these operators is simple.[87]
The increase in expressive power comes, as usual, with a cost: whereas the first-order theory
of harmonic I-scales is decidable (as a consequence of Tarski’s proof of the completeness of the
first-order theory of real closed fields), the addition of further Lipschitz equational structure
can make it possible to capture all of first-order number theory.[88]
[ 86 ] Using standard notation (and twice the g), if g 0 v = 2/(1+v 2 ) then using that the derivative of 2/(1+v 2 ) can be seen
˛ 2 -1
˛ 2
to lie between −2 and 2 (most easily so on any graphing calculator)
˛ we have ˛gu − gv − ˛ 2(1+v )2 (u−v)2 ≤ (u−v) 2 for all
˛
2 2
u, v ∈ R. If we multiply by both sides by (1+v ) we obtain (1+v )(gu−gv) − 2(u−v) ≤ (1+v )(u−v) ≤ 2(u−v) , which
˛ ˛
semiquation is quite enough to imply g 0 (v) = 2/(1 + v 2 ) hence that g(v) = 2 arctan(v) + g(0) and, finally, g(1) − g(−1) = π.
[ 87 ] It may be the case that the free model is semi-simple (hence simple). To show otherwise we need a term provably greater

than (>|)n ⊥ all n ∈ N but not provably equal to >. The same goes for the free model used for identifying e.
[ 88 ] One way is as follows: the set of positive elements of I under ordinary multiplication and a non-standard

“addition” characterized by (x+̃y)(x|y) = |(xy) is isomorphic—via reciprocation—to the real half-line [1, +∞) (with
its usual multiplication and addition). We can identify the reciprocals of positive integers by using the differential equation
x4 h00 x − 10x3 h0 x + (30x2 + 1)hx = 0 : the further equations h(±π -1 ) = 0 and h0 (±π -1 ) = 2-6 π -4 identify hx as (x/2)6 sin x-1
[unfortunately the last exponent did not appear in the TAC version]; hence x ∈ I is the reciprocal of a positive integer iff
x > 0 and h(π -1 x) = 0. Given π -1 it thus suffices to add three unary operations h, h0, h00 and four conditions:

h(π -1 ) = 0

h0 (π -1 ) = 2-6 π -4
“ ”. “ ” “ ”
32
x h x | 16
1 4 00 5 3 0
x h x = 16 x | 32
15 2 1

“ . . ”2 “ . . ”2 “ . ”4
((hu) |(hv)) | ((h0 v)(u| v)) ∨ ((h0 u) |(h0 v)) | ((h00 v)(u| v)) ≤ u| v

where (following Section 3, p??–??):

1
32
= >|(⊥|(⊥|(⊥|(⊥| ))))
5
16
= >|(⊥(|>|(⊥| )))
15
16
= >|(>|(>|(>| )))

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[Added 2018–11–13]
We close this section with a few implementations for computing numbers that can be iden-
tified with recursively presented equational theories of Lipschitz extensions. To get started, we
consider the binary expansions of e/4 and π/4. In Section 1 (p??–??) we gave the (quick and
dirty) algorithm for computing the binary expansion of an element in an interval coalgebra
as iterating (forever) the algorithm:
∨ ∧
If x = > then emit “1” and replace x with x
∨
else emit “0” and replace x with x.
When we start with a positive element x in the standard interval we move to the unit
∧
interval by replacing it with x. Then iterate (forever) the following (less quick and less dirty)
algorithm in which the notation ` · · · = · · · means that · · · = · · · is provable (i.e. true
in the initial model):
 ∨
  ∧

If ` x = > If ` x = ⊥
emit “1”; emit “0”;
   
||
   
replace x with replace x with
   
   
∧ ∨
x x

We are using the fact that e and π are not dyadic rationals. We need some explanations.
We are assuming that we are given a recursively presentable Lipschitz extension of the
equational theory of scales with the special property that there is only one simple quotient
of its initial model. If an element x is not the center then since in each linear model x is on
the same side of the center (it is on the same side in each linear model as it is in the unique
simple quotient) and since we noted in Lemma 4.11 that linear models are interval coalgebras
∨ ∧
it must be the case that either x= > in all linear models or x= ⊥ in all linear models. From
∨ ∧
that we may infer that either x= > or x= ⊥ in the initial model and from that we may infer
∨ ∧
that either ` x = > or ` x = ⊥.
The condition that x is not the center is critical. If the initial model is not simple then
there could be models with x on different sides. Fortunately there is a solution and its name
is “signed binary expansions.” We have a lot say about this in Section 65 (p??–??) where
we’ll have a theorem that uses something called the “The Triumvirate of Open Halves.” Here,
though, we use
16.6 Lemma: the triumvirate of closed halves
Given a recursively presentable Lipschitz extension of the equational theory of scales with the
property that there is only one model on the standard interval, [−1, +1], then the initial model
is the union of the three closed halves: ∨ ∧
∨ ∨ ∧ ∧
{x : ` x=>} {x : ` x=> ∧ x=⊥} {x : ` x= ⊥}

For the signed-binary expansion of any term x iterate (forever) the non-deterministic
procedure:
   ∨ ∧   
∨ ∨ ∧
 If ` x = >   If ` x = > ∧ x = ⊥   If ` x ≤ = 0 

 emit “+1”;  || 
  emit “0”;  || 
  emit “−1”; 

 replace x with   replace x with   replace x with 
∧ ∨
x /x x

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17. Subintervals
Given scale elements b < t the interval of all elements x such that b ≤ =x≤ = t is, as usual,
denoted [b, t]. It is, of course, closed under midpointing, but not, in general, under dotting
or zooming. It does have an induced scale-structure. Define a new dilatation operation that
sends x, y ∈ [b, t] to b ∨ (x / y) ∧ t (in a distributive lattice b ∨ z ∧ t is unambiguous
when b ≤ = t). For elements in [b, t] this enjoys the characterizing properties of dilatations, to
wit, it is covariant in y and undoes contraction at x (the “dilatation equation” in Section 8,
p??–??). We then define the zoom operations on [b, t] as the dilatations at b and t. Dotting
.
is obtained by dilitating “into” its center: x = x/ (b|t).
The verification of the scale axioms is most easily dispatched by using the semi-simplicity
of free scales. If there were a counterexample anywhere there would be one in I. It is easy
to see that the induced structure on any non-trivial subinterval of I makes it isomorphic to
I. [89]
Consider the example used in [??] (p??). We considered the set, F, of functions from the
standard interval to itself, such that |f (x)| ≤ = |x| for all x. If we view F as a subset of all
functions from the standard interval to itself it is an example of a twisted interval. It may
be described as the set of functions whose values “lie between the identity function and its
negation.” Given any scale and elements b, t therein we can formalize the notion by defining
the twisted interval [[b, t]] as the set of all elements, x, such that in every linear representation
it is the case that x is between b and t. This results, easily enough, in the ordinary interval
[b∧t, b∨t]. But the scale structure we want on [[b, t]] is different: the bottom is to be b not b∧t
and the top is to be t not b ∨ t. We simply repeat the construction as for ordinary intervals
but with that one change—the new dilatation operator is still obtained by contracting the
output of the ambient operation to the ordinary interval [b ∧ t, t ∨ b] it being understood that
the top and bottom are not the standard endpoints but rather b and t. We know such yields
a scale because we know that on every linear quotient it does so (albeit that on some of those
linear quotients the order is not the induced order but its opposite).

18. Extreme Points

There is a curious similarity between idempotents in rings and extreme points in scales. First:
18.1 Proposition: The following are equivalent:
x is an extreme point
∧
x = x
∨
x = x
.
x∨ x = >
.
x ∧ x = ⊥
 . . 
∃v (x ∨ v) = > = (x ∨ v)
 . . 
∃v (x ∧ v) = ⊥ = (x ∧ v)

Because the law of compensation easily shows, first, that fixed-points for either
[ 89 ] But the subintervals of I are not all isomorphic to each other. We constructed isomorphisms between subintervals of the

same length at the end of Section 4, p??–?? (actually, we constructed an isomorphism between any interval and the interval of
the same length of the form [b, >]). It is easy to see that subintervals are isomorphic if the ratio of their lengths is a power of 2.
The odd part of the numerator of the dyadic rational that measures the length is a complete isomorphism invariant: clearly if a
pair of subintervals have the same odd part of their length we may construct an isomorphism; for the converse note that universal
∨
Horn sentences of the form [(x | )n (⊥) = ] ⇒ [y = z] hold for the induced scale-structure of a subinterval of I iff n does
not divide the odd part of its length.

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>- or ⊥-zooming are automatically fixed-points for the other and, second, that extreme points
are fixed-points. Conversely, to show that fixed-points are extreme suppose that x|y is a fixed
∨ ∨ ∧ ∧
point; the only cleverness needed is x = x | ⊥ ≤= x | y = x|y = x | y ≤
= x | > = x.
Before dispatching the remaining conditions note that the interval coalgebra condition,
∨ ∧
x = > or x = ⊥, known to be equivalent in scales with linearity, implies that there are just
. .
the two fixed-points, > and ⊥. On a linear scale either equation x ∨ x = > or x ∧ x = ⊥ is
thus clearly equivalent with x being extreme and therefore a fixed-point, hence such is the
case in any scale. And, similarly, on a linear scale the existence of a complement is also clearly
equivalent with being a fixed-point and that dispatches the last of the conditions.
We will use B(S) to denote the set of fixed-points/extreme-points of a scale, S. The set
of extreme points of any scale is closed under dotting and the lattice operations and we will
regard B as a (covariant) functor from scales to Boolean algebras.
Recalling that C(X) denotes the scale of I-valued continuous functions on a Hausdorff
space, X, we see that B(C(X) is isomorphic to the Boolean algebra of clopens in X. Following
ring-theoretic language, we will say that a scale, S, is connected if B(S) has just two
elements, > and ⊥.
If A and B are scales then in A×B the elements h>, ⊥i and h⊥, >i are a complementary
pair of extreme points. Every complimentary pair of extreme points arises in this way: first,
note that if e is an extreme point in a scale S then the principal >-face ((e)) generated by e is
the interval [e, >] (indeed, any subset that is both an interval and a face must have extreme
points as endpoints). The quotient structure S/((e)) is isomorphic to the induced scale on
.
the interval [⊥, e] via the map that sends x to e ∧ x. The map S → [⊥, e] × [⊥, e] that
.
sends x to he ∧ x, e ∧ xi is an isomorphism; its inverse sends hx, yi to x ∨ y. This is,
of course, just the analog of Peirce decomposition for central idempotents.[90] (The fact that
e ∧ x describes a homomorphism is most easily dispatched using the linear representation
∧
theorem. For example, its preservation of midpointing is the Horn sentence [ e = e ] ⇒
e ∧ (x|y) = (e ∧ x) | (e ∧ y) , a triviality in any linear scale since its only extreme points are
 
⊥ and >. )

The atoms of B(S) thus correspond to the connected components of S. One consequence
is the uniqueness of product decompositions. If S is finite product of connected scales B(S)
is finite; its atoms yield the only decomposition into indecomposable products it has. All of
this is just as it is for central idempotents in the theory of rings.[91]
If X is totally disconnected then every >-face in C(X) is generated by the extreme points
it contains; the lattice of >-faces is canonically isomorphic to the lattice of filters in B(C(X)).
Q
In a product of connected scales J Sj the extreme points are the characteristic functions
of subsets of J. An ultrafilter of B = 2J generates a maximal >-face of the product. The
quotient structure is usually called an ultraproduct. It has the wonderful feature that any
first-order sentence is modeled by the ultraproduct iff it is modeled by enough Sj s, that is,
iff the set of j such that the sentence is modeled by Sj is one of the sets in the ultrafilter.[92]
[ 90 ] I am a terrible speller myself, but the great first American mathematician deserves to have his name spelled correctly. And

pronounced correctly—he and his family rhyme it with terse.

[ 91 ] But there are a few isomorphisms that are not reminiscent of Peirce decomposition. [⊥, e] is isomorphic to [e, .
>] via
. . .
the map that sends x ∈ [⊥, e] to e ∨ x ∈ [e, >]. The inverse isomorphism sends y ∈ [e, >] to e ∧ y ∈ [⊥, e]. The product
decomposition arising from an extreme point e can be re-described as the isomorphism to [⊥, e] × [e, >] that sends x to
.
he ∧ x, e ∨ xi. Its inverse sends hx, yi ∈ [⊥, e]×[e, >] to x ∨ e ∧ y (necessarily x ≤ y).
[ 92 ] One may show that any linear scale may be embedded in a subinterval (with its induced scale-structure) of an ultrapower
of I.

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In a twisted interval [[b, t]] let b ∧ t and b ∨ t denote the elements as defined in the
ambient scale (since b is bottom according to the intrinsic ordering on the twisted
interval the use of the intrinsic—instead of the induced—lattice operations would be
unproductive). b ∧ t and b ∨ t are a complementary pair of extreme points in the twisted
interval and the pair yields an isomorphism [[b, t]] → [b ∧ t, b]◦ ×[b, b ∨ t] where ◦ denotes the
opposite scale: the one obtained by swapping >- with ⊥-zooming and top with bottom. But
any scale is isomorphic to its opposite via the dotting operation hence [[b, t]] is isomorphic
to [b ∧ t, b]×[b ∧ t, t] via the map that sends x ∈ [[b, t]] to h((x/ (t|b)) ∧ b), x ∧ ti (using
(x ∧ b)/ ((b ∧ t)|b) = (x/ (t|b)) ∧ b). And that yields the isomorphism from [[b, t]] to [b ∧ t, b ∨ t]
that sends x to ((x/ (t|b)) ∧ b) ∨ (x ∧ t). All of which totally obscures the geometry of the
opening section’s construction of derivatives.

19. Diversion: Chromatic Scales

A chromatic scale [93] is a scale with a (non-Lipschitz, indeed, discontinuous) unary support
operation, whose values are denoted x, satisfying the equations:

⊥ = ⊥
∧
x = x
x∧x = x
x∧y = x∧y

Note that the first three equations have a unique interpretation on any connected scale
and the 4th equation holds iff the connected scale is linear.[94]
These equations say, in concert:
19.1 Lemma: x is the smallest extreme point above x.
(The 3rd equation says, of course, that x ≥ = x; next, if e is an extreme point then e =
. . . . .
= e ∧ (e ∨ e) = e ∧ > = e; [95]
e ∨ ⊥ = e ∨ e ∧ e = e ∨ (e ∧ e) = (e ∨ e) ∧ (e ∨ e) = e ∧ (e ∨ e) ≥
third, if e is an extreme point above x then since the 4th equation implies that the support
operation is covariant we have e = e =≥ x).
Note that it follows that the support operator distributes not just with meet but with join
(its covariance yields x ∨ y ≤
= x ∨ y and the characterization of x ∨ y as the smallest extreme
point above x ∨ y yields x ∨ y ≤= x ∨ y). The co-support, x, of x is the largest extreme point
.
.
below x. It is easily constructable as x = x.
19.2 Theorem: A scale is a simple chromatic scale iff it is linear. Every chromatic scale
is semi-simple, that is, any chromatic scale is embedded (as such) in a product of linear
scales. The defining equations for the support operator are therefore complete: any equation—
indeed any universal Horn sentence—true for the support operator on all linear scales is a
consequence of the four defining equations.
We have already noticed that connected scales have a support operator iff they are linear,
so among chromatic scales connectivity and linearity are equivalent. But connected easily
[ 93 ] To some extent, chromatic scales are to measurable functions as scales are to continuous. See, in particular,

Section 25 (p??).
.
[ 94 ] The 2nd equation becomes redundant if the 3rd equation is replaced with x∧ x = ⊥. See Section 28 (p??–??).
[ 95 ] See Section 46, p??–?? for a subscoring.

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implies simple: if a congruence is non-trivial then its kernel has an element, x, below >. But
then x 6= > hence ⊥ = x ≡ > = >.
If F ⊆ B is a filter in the Boolean algebra of extreme points then the >-face it generates
in the scale, F ⇑ , consists of all elements x such that x ∈ F (because clearly the set of such
x is a zoom-invariant filter).
The support operator is discontinuous on the standard interval but among discontinuous
operations it appears to play something of a universal role. The Heyting arrow operation,
u → v can be constructed as u −◦ v ∨ v (the defining equations for the operation hold for this
construction on linear scales, hence the representation theorem implies that they hold on all
chromatic scales).[96] We observed in Section 5 (p??–??) that Girard’s !Φ is Φ and his ?Φ is Φ.

If F is an ultrafilter, then for extreme points e and e0 if e ∨ e0 ∈ F either e ∈ F or e0 ∈ F

consequently if x ∨ x0 ≡ > (mod F ⇑ ) then either x ≡ > or x0 ≡ > forcing the quotient scale
to be linear. The scale map to the quotient structure clearly preserves the co-support—and
hence the support—operation.
Given any x 6= > we can find an ultrafilter of extreme points excluding x hence x remains
below > in the corresponding quotient structure.[97]
Given any scale S and Boolean algebra B let S[B] be the scale generated by S and the
elements of B subject to relations that, first, make those elements fixed-points and, second,
obey all the lattice relations that obtain in B; then the maps from S[B] to any scale T
are in natural correspondence with the pairs of maps, one a scale-homomorphism S → T,
the other a Boolean-homomorphism B → B(T ). For the special case S = I we have that the
functor I[−] from Boolean algebras to scales is the left adjoint of B(−) from scales to Boolean
algebras.
It is the case that the adjunction map from B to B(S[B]) is an embedding and in the case
that S is connected, B → B(S[B]) is an isomorphism. This and a number of related issues
are discussed in Section 38 (p??–??).

[ 96 ] The support operator is definable, in turn, from the Heyting operation, indeed, just from the Heyting negation:
.
x = (x → ⊥). (Hence the less colorful alternate name, “Heyting scales.”) See Section 34 (p??–??).
[ 97 ] The analogous material for rings and idempotents is the following equational theory (which I have assumed for at least 40

years must already be known):

Define a support operation on a ring to be a unary operation satisfying:

0 = 0
( x2 = x )
xx = x
xy = xy

(The 2nd equation is, in fact, redundant. It is present only to emphasize the analogy with chromatic scales. See [??], p??.)
For one source of examples take any strongly regular von Neumann ring and take x = xx∗. For a better source note that the
first three equations say, in concert, that a connected ring has a unique support operator and it satisfies the 4th equation iff the
ring is a domain (that is, a ring, commutative or not, without zero divisors). These equations are complete for such examples: any
equation—indeed any universal Horn sentence—true for all domains is a consequence of these equations because any algebra is
embedded (as such) into a product of domains. To prove it, first note that if x2 = 0 then x = xx = xx x = xx2 = x0 = x0 = 0.
Any idempotent is central ( (1−e)xe is 0 since its square is 0 hence xe = exe and similarly ex = exe). For any idempotent e
we have e = e (because e = e + (1−e)0 = e + (1−e)(1−e)e = e + (1−e)(1−e) e = e + (1−e)e = e + e − ee = e + e−e = e).
(See Section 46, p??–?? for a subscoring.) The equations characterize x as the smallest element in the Boolean algebra of
idempotents that acts like the identity element when multiplied by x. If A is an ideal in that Boolean algebra then the ideal it
generates in the ring consists of all elements x such that x ∈ A (if e ∈ A then ex = ex ∈ A and if, further, e0 ∈ A then
(ex + e0 x0 ) = (e ∨ e0 )(ex + e0 x0 ) where e ∨ e0 = e + e0 − ee0 ∈ A which gives e ∨ e0 ≥ ex + e0 x ∈ A). When A is a maximal
Boolean ideal then the ring ideal it generates is prime (since xy is in it iff x y is and a maximal Boolean ideal is a prime ideal
in the Boolean algebra) hence the corresponding quotient is a domain. The ring-homomorphism down to the quotient is easily
checked to be a homomorphism with respect to the support operation. Finally, given any x 6= 0 we can find a maximal Boolean
ideal containing 1−x hence x remains non-zero in the corresponding quotient.

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20. The Representation Theorem for Free Scales

In this section I and I will always refer to standard intervals [−1, +1].
20.1 Theorem: The free scale (I-scale ) on n generators is isomorphic to the scale of all
continuous piecewise D-affine (affine) functions from the standard n-cube, In, to I.
We need some definitions.
Given a scale, S, let S[x1 , . . . , xn ] denote the scale that results from freely adjoining n
new elements, traditionally called “variables,” x1 , . . . , xn , to S. (The elements of S[x1 , . . . , xn ]
are named by scale terms built from the elements of S and the symbols x1 , . . . , xn with two
terms naming the same element iff the equational laws of scales say they must.)
The free scale (I-scale ) on n generators is thus I[x1 , x2 , . . . , xn ]. Each of its simple
quotients is the image of a unique map to I, hence is determined by where it sends the gen-
erators x1 , x2 , . . . , xn . And, of course, each xi may be sent to an arbitrary point in I. Thus
Max(I[x1 , x2 , . . . , xn ]) may be identified with I.n Because we now have the semi-simplicity of
I[x1 , x2 , . . . , xn ] we have a faithful representation I[x1 , x2 , . . . , xn ] → C(In ). In this section we
will henceforth treat I[x1 , x2 . . . , xn ] as a subscale of C(In )
We will show that any function in I[x1 ] is what is traditionally called “piecewise linear,”
that is, a function f : I → I for which there is a sequence ⊥ = c0 < c1 < · · · < ck = > such
that f is an affine function whenever it is restricted to the closed subinterval from ci through
ci+1 . We need to generalize the notion to higher dimensions.
We are confronted with a terminological problem. Tradition has it that need both notions.
(In Section 26, p??–??, free lattice-ordered abelian groups piecewise affine functions be called
“piecewise linear” but we will will be represented as the continuous functions on Rn that—
informally stated—allow a dissection of Rn into a finite number of polytopal collections of
rays on each of which the function is linear.) Free scales lead not to piecewise linear but
piecewise affine functions. Informally: a piecewise affine function is one whose domain may
be covered with a finite family of closed polytopes on each of which the function is affine.
So let us start at the beginning. An affine function from a convex subset of Rn to R can
be defined as a continuous function that preserves midpoints.[98] (When the domain is all of
Rn such is equivalent to the preservation of affine combinations, that is, combinations of
the form ax+by where a+b = 1. Continuity is then automatic.) It is routine, of course, that a
function f : Rn → R is affine iff there are constants a0 , a1 , . . . , an such that f hx1 , x2 , . . . , xn i =
a0 + a1 x1 + · · · + an xn . If f preserves the origin, equivalently if a0 = 0, it is said to be linear.
We will say that an affine function is D-affine, pronounced “dy-affine,” if it carries dyadic
rationals to dyadic rationals. It is routine that such is equivalent to the ai s all being in D.
For our purposes the simplest—and technically most useful—definition is that
f : In → I is continuous piecewise affine if it is continuous and if there exists an
affine certification, to wit, a finite family, Af = {A1 , A2 , . . . , An }, of global affine functions,
such that f (x) ∈ {A1 (x), A2 (x), . . . , An (x)} all x ∈ In. Given f and an affine certification,
Af , we construct its canonical polytopal dissection, P, as follows: starting with the
interior of In remove all hyperplanes that arise as equalizers of two members of Af ; the dense
open set that remains then falls apart as the disjoint union of a finite family of open convex
polytopes; we take Pf to be the family of closures of these open polytopes.
[ 98 ] One need not require continuity if the target is bounded: we pointed out in Section 13 (p??–??) that if a midpoint-

preserving function lies in a closed interval then it is monotonic, in particular, continuous. Hence if f is a midpoint-preserving
map from a convex subset, C, of Rn to I, then f restricted to any slice through C is continuous and perforce preserves affine
combinations. But the preservation of affine combinations is, by definition, something that takes place on slices.

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◦
Given any P ∈ P we know that the functions in Af nowhere agree on P, the interior of P.
We will use the following observation several times: If a function g is continuous piecewise
affine on a connected set and the functions in its certification nowhere agree on that set then
g is not just piecewise affine but affine This observation does not use anything about affine
functions other than their continuity: the equalizers of g and the functions in its certification
form a finite family of closed subsets that partition the domain, hence each is a component.
◦
Thus f agrees with one element of Af on P and—by continuity—on all of P. We denote that
affine function as AP .
Note that there is unique minimal affine certification for f (given any certification retain
only those elements of the form AP ). The resulting canonical polytopal dissection will, in
general, be simpler for smaller certifications.[99]
By a cpda function we mean a continuous piecewise affine function whose certification
consists only of D-affine functions.
The fact that cpda functions are closed under the scale operations is easily established:
suppose that g is another cpda function and that Ag is its affine certification; then the finite
.
family A | A0 : A ∈ Af , A0 ∈ Ag certifies f |g; easier is that −A : A ∈ Af certifies f
 
 ∧
and {−1} ∪ 2A−1 : A ∈ Af certifies f . Hence I[x1 , x2 , . . . , xn ] viewed as a subset of
C(In ), consists only of cpda functions. (Clearly the generators and constants name D-affine
functions.) What we must work for is the converse: that every cpda function from In to
I so appears. It is fairly routine (but a bit tedious) to verify that −1 ∨ A ∧ +1 [100] is in
I[x1 , x2 , . . . , xn ] for every D-affine A. [101]
Let f : In → I be an arbitrary cpda function and P the canonical polytopal dissec-
tion for its minimal certification Af . We will construct for each pair P, Q ∈ P a function
fP, Q ∈ I[x1 , x2 , . . . , xn ] such that:
fP, Q (x) ≥
= f (x) for x ∈ P ;

fP, Q (y) ≤
= f (y) for y ∈ Q.
(Note that it follows that fP, P (x) = f (x) for x ∈ P.)
Then necessarily:
!
_ ^
f = fP, Q
P Q

[ 99 ] Note, though, that canonical polytopal dissections are not necessarily minimal. With n = 2 consider the piecewise D-affine

function (x1 ∧ x2 ) ∨ 0. Its minimal certification is, obviously, {x1 , x2 , 0} which yields a canonical polygonal dissection with 6
polygons. But there are many (indeed, infinitely many) dissections with only 4 convex polygons.
[ 100 ] In any distributive lattice ` ∨ (a ∧ u) = (` ∨ a) ∧ u whenever ` ≤ u.
[ 101 ] For a proof, say that A is “small” if its values on In lie in I, that is, if −1∨A∧+1 = A. The set of small affine functions

is clearly closed under dotting and midpointing. We first show that every small D-affine function is in I[x1 , x2 , . . . , xn ] and we
will do that by induction. Given a small D-affine f hx1 , x2 , . . . , xn i = a0 + a1 x1 + a2 x2 + · · · + an xn we will say that it is of
type m if all the ai s can be expressed as dyadic rationals with denominator at most 2m . There are only 2n + 3 small D-affine
functions of type 0, to wit, 0, ±1, ±x1 , ±x2 , . . . , ±xn . Suppose that we have obtained all small D-affine functions of type m.
Given f of type m+1 let {σi }i and {bi }i be such that ai = σi bi 2-(m+1) where σi = ±1 and bi is a natural number. The
smallness condition is equivalent to the semiquation b0 + b1 + · · · + bn ≤ 2m+1 . Let {ci }i and {di }i be sequences of natural
numbers such that bi = ci + di each i and c0 + c1 + · · · + cn ≤ 2m , d0 + d1 + · · · + dn ≤ 2m . Define the small D-affine functions
ghx1 , x2 , . . . , xn i = σ0 c0 2-m + σ1 c1 2-m x1 + · · · + σn cn 2-m xn and hhx1 , x2 , . . . , xn i = σ0 d0 2-m + σ1 d1 2-m x1 + · · · + σn dn 2-m xn .
Then f = g|h . (There are many ways of finding the ci s and di s, none of which seems to be canonical. Perhaps the easiest to
specify is obtained by first defining ei = 2m ∧ ij=0 bj , then ci = ei − ei−1 and di = bi − ci .)
P

Finally, given arbitrary D-affine A let m be such that 2-m A is small. Then −1 ∨ A ∧ +1 = ( /)m (2-m A).

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We take fP, P (of course) to be −1 ∨ AP ∧ +1.

Otherwise let DP denote the set of the functions of the form AR −AS that are non-negative
on P and let D1 , D2 , . . . , Dk be the functions in DP that are non-positive on Q. Define

fP, Q = −1 ∨ AP, Q ∧ +1

where
AP, Q = AP + m(D1 + D2 + · · · + Dk )
for suitably large integer m.
≥ 0 such that:
The two semiquations for the fP,Q s are thus reduced to finding m =

AP, Q (x) ≥
= AP (x) for x ∈ P

= AQ (y) for y ∈ Q
AP, Q (y) ≤

The 1st semiquation holds regardless of m ≥ = 0. The 2nd requires a little work. The subsets
P and Q are intersections of closed half spaces. Each non-contant function in D yields a
closed half-space. to wit, the set of points with non-negative values. The intersection of these
half-spaces is the set P and among them the half-spaces defined by D1 , D2 , . . . , Dk are those
that do not contain Q.
The interior of Q is disjoint from P hence has a negative value for at least one Di . Any
Q-vertex y for which (D1 + D2 + · · · + Dk )(y) = 0 is a vertex shared with P . We can
choose m large enough so that for any vertex y of Q not shared with P it is the case that
AP,Q (y) ≤
= AQ (y). Since each y ∈ Q is a convex combination of its vertices we obtain that
AP.Q (y) ≤
= AQ (y)
(One immediate application of all this is a construction for the Richter scale that empha-
sizes its role as the representor for the Jacobson-radical functor. Start with the free scale
on one generator, x, and reduce by the >-face F generated by all elements of the form
((>|)n ⊥) −◦ x (it produces the minimal congruence that forces x into the Jacobson radical).
It is easy to verify that F is the set of all cpdas that are constantly equal to > on some non-
trivial interval ending at >; the congruence induced by F thus identifies two cpdas precisely
when they represent the same germ at >. The congruence class of a cpda f is determined by
the value f (>) and the left-hand derivative of f at >. Note the curious reversal of sign that is
needed to establish an isomorphism with our previous construction of the Richter scale: the
element h1, −1i corresponds to a cpda with a positive left-hand derivative at >. )

21. Finitely Presented Scales, or: How Brouwer Made Topology Algebraic
A finitely generated scale, f.p.scale is, of course, a scale that appears as a quotient of a
finitely presented free scale where the kernel is a finitely generated >-face.
By a closed piecewise D-affine subset of Euclidean space we mean a subset of the form
h-1 (1) where h is a continuous piecewise D-affine function. If X is closed piecewise D-affine
we will denote the scale of cpda I-valued functions on X as CP DA(X). (In the last section
we showed that CP DA(In ) is isomorphic to I[x1 , x2 , . . . , xn ].)
21.1 Theorem: Given a closed piecewise D-affine subset X of In we obtain a scale homo-
morphism CP DA(In ) → CP DA(X) that is onto. Moreover CP DA(X) is an f.p.scale, and
all f.p.scales so arise.
As should be expected, we may remove all the “D-”s:

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By a closed piecewise affine subset of Euclidean space we mean a subset of the form
h-1 (1) where h is a continuous piecewise affine function. If X is closed piecewise affine we
will denote the scale of cpa I-valued functions on X as CPA(X). (In the last section we
showed that CPA(In ) is isomorphic to I[x1 , x2 , . . . , xn ].)
Because the evaluation maps for points in X are thus collectively faithful we will obtain
the immediate corollary:
21.2 Theorem: All finitely presented scales (I-scales) are semi-simple.[102]
First note that a finitely presented scale needs just one relation, equivalently any finitely
generated >-face is principal: it suffices to note that the >-face generated by elements
a1 , a2 , . . . , ak is generated by the single element a1 ∧ a2 ∧ . . . ∧ ak (one may use | instead
of ∧). When we view the free scale on n generators as the scale of cpda functions on the
standard n-cube it is easily seen that a D-affine function, f , is in the >-face generated by h
iff h-1 (1) ⊆ f -1 (1), hence functions f and g are congruent mod ((h)) iff they behave the
same on the closed D-affine set S = h-1 (1). If we had the lemma that cpda functions on
closed D-affine subsets of the cube extend to cpda functions on the entire cube we would
be done. (We will, in passing, prove such to be the case since we will show that any such
function is given by an element in the f.p.scale, hence is describable by a term in the free scale
and any scale and any such term describes a cpda function on the standard cube, indeed,
on the entire Euclidean space.)
So we must redo the previous proof, this time not for the n-cube but for an arbitrary
closed D-affine subset thereof. The only serious complication is that the polytopes of interest
are no longer all of dimension n. This complication turns out to be mostly in the eye of the
beholder.
Let h be a cpda function with certification Ah = {A1 , A2 , . . . , Ak }, let X = h-1 (1) and
let f be an arbitrary cpda function on X with certification Af = {A01 , A02 , . . . , A0l }. We seek
a term in I[x1 , x2 , . . . , xn ] that describes f on X. Given s ∈ X let Ds be the set of functions
of the form Ai − Aj and A0i − A0j that are non-negative on s. Note that the negation of some
functions in Ds can also be in Ds (to wit, all those that are zero on s).
Define \ 
Ps = ≥0
x : D(x) =
D∈Ds
◦
and let Ps now be—not the interior but—the inside of Ps , that is, the points not contained
in any of its proper sub-faces.
◦
Note that if two functions in D agree anywhere on Ps , they agree everywhere on Ps (and
if such happens, we know that Ps is of lower dimension than the ambient Euclidean space).
For convenience let As ⊆ Ah be a minimal certification of h on Ps . Using the lemma from
the previous section, we know that h is affine on Ps and since h(s) = 1 we know that h is
constant on Ps . That is, Ps ⊆ X. Similarly choose a minimal certification A0s ⊆ A0 of f on
Ps . The same lemma says that f agrees with an element of A0s everywhere on Ps .
If two polytopes of the form Ps overlap, their intersection appears as a face of each and is
itself of the form Ps . Let P be the polytopes of form Ps not contained in any other. For each
P ∈ P chose a function AP in Af that agrees with f on P . We may now proceed with the
construction just as before, starting with the definition of the functions denoted fP,Q (where
◦
we understand DP to be Ds for s ∈Ps ).
[ 102 ] The injectivity of I implies that this corollary can be strengthened to all “locally f.p.scales,” indeed, it suffices for

semi-simplicity that each element be contained in an f.p.subscale.

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We may put this material together to obtain:

21.3 Theorem: The full subcategory of finitely presented scales is dual to the category of
closed piecewise D-affine sets and cpda maps.
A quite different notation presents itself, one that emphasizes the pivotal role played by
I (to use a popular phrase—avoided by those bothered by etymology—it is the duality’s
“schizophrenic object”). Tradition insists that we change notation. Instead of CP DA(X)
we’ll use X.? The functor that sends X to X ? may be viewed as an contravariant algebra-
valued representable functor, that is, we could also denote X ? as (X, I) where I is viewed as
an object in the category of closed piecewise D-affine sets equipped therein with its structure
as a scale (just as can any equational theory in any category with finite products) thereby
endowing (X, I) with a scale structure for any X and endowing (f, I) with the status of scale
map for any f : X → X.0
When S is a scale, let S ? denote the space of I-valued scale-homomorphisms on S. Because
I-valued scale maps are known by their kernels (and because all simple scales are uniquely
embeddable in I) S ? is naturally equivalent to Max(S)
The fact that this pair of functors is an equivalence of categories is equivalent to the
“adjunction maps” X → X ?? and S → S ?? being isomorphisms. To establish the first
isomorphism suppose that X is a subset of In of the form h-1 (>) where h is continuous
piecewise D-affine. Then X ? may be taken as Fn /((h)) where Fn is the free scale on n gener-
ators, Ihx1 , x2 , . . . , xn i. The adjunction map X → X ?? sends x ∈ X to the evaluation map
that sends f ∈ X ? to f (x). The semi-simplicity of Fn /((h)) says that X → X ?? is monic.
To prove that it’s onto, let g ∈ X,?? that is, g : X ? → I. We know that there’s x ∈ In such
that Fn → X ? → I is the evaluation map at x. We need only show that x is in X. Suppose
not. It suffices to find k ∈ X,? that is, k : X → I such that g(k) 6= k(x). It’s easy: take k = h.
The argument for S → S ?? is essentially the same.
It is worth noting that the standard notion of homotopy translates rather nicely into this
setting. The “co-cylinder” over S ? is S[U]? where S[U] is the “polynomial scale” over S, that
is, terms in a fresh variable U built from elements in S. A pair of maps f⊥ , f> : T → S gives
rise to a pair of “co-homotopic” maps from S ? to T ? iff there is a map H : T → S[U] such
H ve fe
that for e = ⊥, > we have T → S[U] → S = T → S where ve is the map that evaluates
a polynomial at e. Brouwer’s simplicial approximation theorem [103] is just what is needed for:

21.4 Theorem: The homotopy category of continuous maps between finitely triangulable
spaces is dual to an algebraically defined quotient category of the category of f.p.scales.

A closing comment: the transitivity of homotopy uses—directly—the coalgebra structure.

If f⊥ is co-homotopic to f via H⊥ and f to f> via H> then we need to put H⊥ and H>
together to create H such that for e = ⊥, >
H ve fe
T → S[U] → S = T → S.
Using the comment on p??, H may be constructed as:
   
[ 104 ]
f / v ∨ (H⊥ ) v ∧ (H> ) .
U U
[ 103 ] Everyone seems to agree that Brouwer intended, with this theorem, to transform topology into something we could get
our hands on. Or, as we would say it now, something worthy of the name “algebraic topology.”
[ 104 ] Yes—to my amazement—this is a scale homomorphism. If t ∈ T then H (t) is, for each U, either f (t) or f (t).
⊥ >

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22. Complete Scales

Consider the smallest full subcategory of scales that includes I and is closed under the
formation of limits. The construction of left-limit–closures of subcategories can be compli-
cated, but the injectivity of I in the category of scales makes the job easier: it is the full
subcategory of scales that appear as equalizers of pairs of Qmaps between powers of I. That
is, first take the full subcategory of objects of the form I and then add all equalizers of
pairs of maps between them. It is clear that such are closed under the formation of products.
To see that the fullQ subcategory
Q of such is closed under equalizers let X be theQequalizer Qof a
pair of maps from I I to J I and Y the equalizer of a pair of maps from K I to L I.
Let f, g be a pair
Q of mapsQ from X to Y . The injectivity of I allows us to extend f and g
to maps from I I to K IQ . The equalizer
Q Qof f, g is constructable as the equalizer of the
resulting pair of maps from I I to J I × K I.
Any locally small category constructed as the left-limit–closure of a single object is
automatically a reflective subcategory. We will call its objects complete scales. They have
a number of alternative characterizations.
Say that a map of scales, A → B is a weak equivalence (a phrase borrowed from
homotopy theory) if it is carried to an isomorphism by the set-valued functor (−, I), or—put
another way—if every I-valued map from A factors uniquely through A → B. A scale S
is complete iff (−, S) carries all weak equivalences to isomorphisms. The “lluf subcategory”
(that is, one that contains all objects) of equivalences falls apart into connected components,
one for each isomorphism type of complete scales. They are precisely the objects that appear
as weak terminators in their components.
The most algebraic description of the category of complete scales is as a category of
fractions, to wit, the result of formally inverting all the weak equivalences. (All full reflective
subcategories are so describable: they are always equivalent to the result of formally inverting
all the maps carried to isomorphisms by the reflector functor.)
A quite different description of complete scales—one that appears not to be algebraic—is
in terms of a metric structure. In this setting it is useful to take I to be the unit interval
[0, 1] and I the dyadic rationals therein. The intrinsic pseudometric (of diameter one) on a
scale S is most easily defined—in the presence of the axiom of choice—by taking the distance
from x to y as supf :S→I |f (x) − f (y)|. Such is a metric (not just a pseudometric) iff S is semi-
simple. It is, further, a complete metric iff S is a complete scale (indeed, the reflection of an
arbitrary scale into the subcategory of complete scales may be described—metrically—as the
usual metric completion of the scale viewed as a pseudometric space).
The intrinsic metric may be defined directly without recourse to the axiom of choice. Define
the intrinsic norm of x ∈ S as ||x|| = inf q ∈ I : x ≤ = q and the distance between x and
• •
y as ||x ◦−
−◦ y|| (recall that ◦−−◦ is the dotting operation applied to ◦− −◦ ). (It is easy to verify
•
that on the unit interval x ◦− − ◦ y = |x−y|.) To see that this definition agrees with the previous
in the presence of the axiom of choice we need to show that ||x|| = supf :S→I f (x). Clearly
= ||x|| for all f : S → I. If ||x|| = 0 we are done. For the reverse semiquation when
f (x) ≤
||x|| > 0 it suffices to find a single f : S → I such that f (x) = ||x|| and for that it suffices—in
the presence of the axiom of choice—to find a proper >-face that contains {qn −◦ x}n for a
strictly ascending sequence {qn }n of dyadic rationals approaching ||x||. The >-face generated
by {qn −◦ x}n is the ascending union of the principal >-faces { ((qn −◦ x)) }n hence it suffices
to show that each ((qn −◦ x)) is proper. It more than suffices to find a linear quotient in which
qn ≤
= x. The linear representation theorem says that if there were no such linear quotient then
Hence for each U it is equal to a scale homomorphism.

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x < qn , directly disallowed by the choice of the qn s.

23. Scales vs. Spaces

The main goal of this section is to show that the category of compact-Hausdorff spaces is dual
to the category of complete scales. Using the notation already introduced, the equivalence
functor from compact Hausdorff spaces to complete scales sends X to C(X), the scale of
continuous I-valued maps on X. The equivalence functor from complete scales to compact
Hausdorff spaces sends S to Max(S), the set of maximal >-faces on S, topologized by the
standard “hull-kernel” topology.
As with the duality between f.p.scales and closed piecewise D-affine sets, tradition insists
that we change notation. Instead of C(X) we use X.? Again, the functor that sends X to X ?
may be viewed as an contravariant algebra-valued representable functor, that is, we could
also denote X ? as (X, I) where I is viewed as a scale algebra in the category of topological
spaces. It is clear from the metric characterization that X ? is a complete scale.
If S is a scale let S ? denote the set of I-valued scale-homomorphisms on S, topologized
by taking as a basis all sets of the form, one for each s ∈ S:
f ∈ S ? : f (s) < >

Us =
The fact that Max(S) and S ? describe the same space rests on the fact that I-valued scale-
homomorphisms are known by their kernels. (The fact that the hull-kernel topology describes
the same space is easily verified: Us corresponds to the complement of the hull of the principal
>-face ((s)).)
We will find useful the formulas:
Us ∩ Ut = Us ∨ t
Us ∪ Ut = Us ∧ t
Us = Uŝ
U> = ∅
U⊥ = S?

23.1 Lemma: Spaces of the form S ? are compact-Hausdorff.

For the Hausdorff property let f, g be distinct elements of S ? and chose a ∈ S such that
f (a) 6= g(a). We may assume without lose of generality that f (a) < g(a). Let q ∈ I be such
that f (a) < q < g(a). Then f ∈ Uq −◦ a and g ∈ Ua −◦ q . [105] The equation of linearity
0
yields Uq −◦ a ∩ Ua −◦ q = U(q −◦ a)∨(a −◦ q) = U> = ∅. For compactness  let S be a0 subset
of S. The necessary and sufficient condition that the family of sets Us : s ∈ S be a
? 0 ?
cover of S is that the >-face generated by S is entire (because the elements f ∈ S not
in any Us are precisely those such that S 0 ⊆ ker(f ) ). A >-face is entire iff it contains ⊥.
Hence there must be s1 , s2 , . . . , sn ∈ S 0 such that ⊥ is the result of applying >-zooming a
finite number of times to the element s1 ∧ s2 ∧ . . . ∧ sn . And that is enough to tell us that
Us1 ∪ Us2 ∪ · · · ∪ Usn = Us1 ∧s2 ∧···∧sn = U⊥ = S ?
For each s ∈ S we obtain the “evaluation map” from S ? to I that sends f ∈ S ? to f (s) ∈ I.
Yet another description of the topology on S ? is as the weakest topology that makes all these
evaluation maps continuous: given q < r ∈ I the inverse image of the open interval (q, r) ⊆ I
is U(s −◦ q)∨(r −◦ s) (and, of course Us is the inverse image of I \{>}).
[ 105 ] There is actually a canonical choice: take q to be of minimal denominator (it’s unique).

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For an arbitrary space X and element x ∈ X the evaluation map in X ?? that sends f ∈ X ?
to f (x) ∈ I is clearly a homomorphism. The natural map X → X ?? that sends each point
in X to its corresponding evaluation map is continuous (the inverse image of Uf ⊆ X ?? is
the f -inverse-image in X of the open subset I \{>}.)
23.2 Theorem: If X is compact-Hausdorff then X → X ?? is a homeomorphism
It is monic because of the Urysohn lemma. To see that it is onto, let H : X ? → I be an arbitrary
scale-homomorphism. Because ker(H) is maximal it suffices to find x such that ker(H) is
contained in the kernel of the evaluation map corresponding to x, that is, it T suffices to find
x such that H(f ) = > implies f (x) = >, or put another way, to find x in f ∈ker(H) f -1 (>).
First note that if f -1 (>) = ∅ then there is q ∈ I such that f < q < >, hence H(f ) < q
forcing f 6∈ ker(H). The compactness of X therefore says that  it suffices to show that the
finite-intersection property holds for the family of closed sets f -1 (>) : f ∈ ker(H) . But
this family is closed under finite intersection: f -1 (>) ∧ g -1 (>) = (f ∧ g)-1 (>) and ker(H) is
clearly closed under finite intersection.
The natural map S → S ?? that sends each element in S to its corresponding evaluation
map is a homomorphism.
23.3 Theorem: If S is a complete scale then S → S ?? is an isomorphism.
The proof is an immediate consequence of
23.4 Theorem: If X is compact-Hausdorff then the necessary and sufficient condition for
a subscale, S, of X ? to be dense (under the intrinsic metric) is that S separates the points of
X, that is, for every two points x, y ∈ X there exists f ∈ S such that f (x) 6= f (y).
Note that the necessity uses the Urysohn lemma. The proof is much easier than its model,
the Stone-Weierstrass theorem (or is it—in this case—Stone-without-Weierstrass?). We first
establish that S has the “two-point approximation property,” that is, for every pair a, b ∈ I
and every pair of distinct points x, y ∈ X there is f ∈ S with f (x) = a and f (y) = b. If
a = b we can, of course, take f to be that constant. Otherwise we can assume without loss of
generality that a < b. Start with any f such that f (x) 6= f (y), as insured by the hypothesis.
.
If f (x) > f (y) replace f with f . Let c ∈ I be such that f (x) < c < f (y). There exists n
such that (c/)n f (x) = ⊥ and (c/)n f (y) = >. Replace f with (c/)n f to achieve f (x) = ⊥ and
f (y) = >. Finally, replace that f with a ∨ (f ∧ b) to achieve f (x) = a and f (y) = b.
We can now repeat the Stone argument. Let X be a compact space and S a sublattice in
X ? with the two-point approximation property. Given any h ∈ X ? and ε > 0 we wish to find
f ∈ S such that the values of f and h are everywhere within ε of each other. For each pair
of points x, y ∈ X let fx,y ∈ S be such that fx,y (x) is within ε of h(x) and fx,y (y) is within ε
of h(y). Define the open set

Ux,y = z ∈ X : fx,y (z) < h(z) + ε .
It is best to regard I here as a fixed closed interval in R. Since y ∈ Ux,y we know that for fixed x
the family {Ux,y }y is an open cover. Let y1 , y2 , . . . , ym be such that
Ux,y1 ∪ Ux,y2 ∪ · · · ∪ Ux,ym = X and define fx = fx,y1 ∧ fx,y2 ∧ · · · ∧ fx,ym . Then fx (x) > h(x) − ε
and for all z we have fx (z) < h(x) + ε Now for each x define the open set

Ux = z ∈ X : fx (z) > h(z) − ε .
Since x ∈ Ux we know that the family {Ux }x is an open cover. Let x1 , x2 , . . . xn be such that
Ux1 ∪ Ux2 ∪ · · · ∪ Uxn = X. Finally define f = fx1 ∨ fx2 ∨ · · · ∨ fxn . [106]
[ 106 ] Yes, the argument of this paragraph fails if X has only one element.

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23.5 Theorem: The full subcategory of complete scales is dual to the category of
compact-Hausdorff spaces.
For a remarkably algebraic definition of the category of compact Hausdorff spaces: start with
the category of scales; chose an object that has a unique endomorphism and is a weak termi-
nator for all objects other than the terminator; call that object I; formally invert all maps
that the contravariant functor (−, I) carries to isomorphisms; take the opposite category.
One can replace I with any non-trivial injective object (as will be seen in the next section
all injective scales are retracts of cartesian powers of I). Another choice is first to restrict to
the full subcategory of semi-simple scales and then formally invert the mono-epis (obtaining
what some might call its “balanced reflection”).[107]

24. Injective Scales, or: Order-Complete Scales

As complete scales are to the study of continuous maps we expect that injective scales will be
to measurable functions. As we will see, all injective scales come equipped with a (necessarily
unique) chromatic structure but—fortunately for that expectation—maps between them need
not preserve that structure. Let us lay out the groundwork.
If an object (in any category) is injective, then it is clearly an absolute retract, that
is, whenever it appears as a subobject it appears as a retract. The converse need not hold,
even for models of an equational theory.[108] As we will see though, absolute retracts in the
category of scales are, indeed, injective.
24.1 Lemma: A scale that is an absolute retract is order-complete (that is, it is not just a
lattice but a complete lattice).
Let E be an absolute retract and L ⊆ E an arbitrary subset, U ⊆ E the set of upper
bounds of L. Freely adjoin an element b to E to  obtain the “polynomial scale” E[b] and
let F be the >-face generated by the elements ` −◦ b : ` ∈ L and b −◦ u : u ∈ U .
Once we know that E → E[b]/F is an embedding we are done because then a retraction
f : E[b]/F → E necessarily sends b to a least upper bound of L. The fact that E → E[b]/F
is an embedding is equivalent to the disjointness of F? and E? (recall that the “lower star”
removes the top). If x ∈ E? were in F then a finite number of the generators of F would
[ 107 ] In the full subcategory of semi-simples a map is an epi iff its image is an order-dense subset of the target (and—as always

for models of equational theories—monos are one-to-one). But for arbitrary scales the characterization of epimorphisms is more
complicated. If we work with Q-scales, that is, scales for which multiplication by each rational in the standard interval is defined,
then the only epis are onto (hence all mono-epis are isos). For a proof, it suffices (as usual) to consider “dense subobjects,” that
is, those whose inclusion maps are epi. Suppose S 0 ⊆ S is dense. Let V be the enveloping Q-vector space of S (defined,
of course, analogously to the enveloping D-module) with its inherited partial ordering. For any s0 ∈ S \ S 0 use the axiom of
choice to find a map f : V → Q with S 0 in its kernel such that f (s0 ) 6= 0 . Partially order V ⊕ Q lexicographically and define
a new scale T as the interval from h⊥, 0i to h>, 0i with its induced scale structure. The two scale homomorphisms that send
s ∈ S to hs, 0i, in the first case, and hs, f (s)i, in the second, agree on S 0 and disagree on s0 . For the general scale case, the
condition for S 0 to be dense in S is that it be a “pure” subscale: ns ∈ S 0 implies s ∈ S 0 for all integers n > 0.
[ 108 ] The Sierpiński monoid ({0, 1} under multiplication) is an absolute retract in the category of commutative monoids:

whenever it is a submonoid the “least map” (least with respect to the order induced from {0, 1}) from the ambient monoid
to the Sierpiński monoid (to wit, the characteristic map of the subgroup of units) is a retraction. But the least map from the
one-generator monoid (the natural numbers) does not extend to a map from the one-generator group (the integers). Curiously
the Sierpiński monoid is injective in the full subcategory of finite commutative monoids, indeed, in the full subcategory of locally
finite commutative monoids (which in the commutative case is the same as saying each one-generator submonoid is finite).
The full subcategory of idempotent commutative monoids—semi-lattices as they are usually called—is both a reflective and
co-reflective subcategory of the category of all commutative monoids. If one restricts to locally finite commutative monoids then
not only is it both reflective and co-reflective but the same functor delivers both the reflection and co-reflection: the reflection
map sends each element of a locally finite commutative monoid to the unique idempotent element that appears in the sequence
of its positive powers. Since the functor also delivers co-reflections it preserves monomorphisms and hence the injectivity of
the Sierpiński monoid follows from its injectivity among semi-lattices. If one deems them meet-semi-lattices then maps into the
two-element meet-semi-lattice are the characteristic maps of filters and it is easy to see that any such characteristic map extends
to any larger meet-semi-lattice. (Note that—quite unusual for injective objects—the axiom of choice is not used in extending
maps to the Sierpiński monoid.)

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account for it. But given `1 −◦ b, `2 −◦ b, . . . , `m −◦ b and b −◦ u1 , b −◦ u2 , . . . , b −◦ un we may

easily obtain a retraction of E[b] back to E by sending b to u1 ∧ u2 ∧ · · · ∧ un . The kernel of
any retraction of E is, of course, disjoint from E? . But this kernel contains the listed finite
number of elements (indeed it contains ` −◦ b for all ` ∈ L).
24.2 Corollary: Absolute retracts are chromatic scales.
Easily enough: x = m (⊥/)m x and, hence, x = m (>/)m x.
W V

24.3 Lemma: Absolute retracts are semi-simple scales.

Semi-simplicity is equivalent to > being the least upper bound of I? . But a least upper bound
of I? is necessarily invariant under >-zooming. There is only one such element larger than
(check in any linear scale) and that element is >.
24.4 Corollary: Absolute retracts are injective scales.
An absolute retract, being semi-simple, can be embedded in a cartesian power of the injective
object I. [109] A cartesian power of an injective is injective, that is, every absolute retract can
be embedded in an injective which, of course, is the necessary and sufficient condition for an
absolute retract to be injective (any retract of an injective is injective).
As is the case for any equational theory, a model is an absolute retract iff it has no proper
essential extensions. We recall the definitions: a monic A → B is essential if whenever
A → B → C is monic it is the case that B → C is monic. For models of an equational theory
this translates to the condition that every non-trivial congruence on B remains non-trivial
when restricted to A. Note that for any monic A → B we may use Zorn’s lemma to obtain a
congruence maximal among those that restrict to the trivial congruence on A thus obtaining
a map B → C such that A → B → C is an essential extension. It follows that if A has no
essential extensions other than isomorphisms then A is an absolute retract. (The converse is
immediate.)
24.5 Lemma: A scale B is an essential extension of a subscale A ⊆ B iff A? is co-final
in B?
Because essentiality is clearly equivalent to every non-trivial >-face in B meeting A
non-trivially; and clearly a principal >-face, ((b)) ⊆ B meets A non-trivially iff there is
a ∈ A? such that b < a. The converse for Lemma 24.1 (p??):
24.6 Lemma: If a scale is order-complete then it is an absolute retract.
First a lemma: if A? is co-final in B? then for any a ∈ A and b ∈ B such that b < a there
exists a0 ∈ A with b < a0 < a because we may use the order-isomorphism a −◦ (−) from [b, a]
to [a −◦ b, >] [110] to obtain a00 ∈ A such that a −◦ b < a00 < >. The inverse isomorphism thus
∧
delivers an element in A (to wit, a0 = a | a00 ) strictly between b and a. The dual lemma: if
. .
a < b we may find a0 ∈ A such that a < a0 < b (simply apply the previous case to b < a).
Consider an order-complete scale A and essential extension A ⊆ B. Given b ∈ B let
a ∈ A be the greatest lower bound of the A-elements above b. Using the dual lemma we
reach a contradiction from the strict semiquation a < a ∨ b (because if a0 ∈ A were such
that a < a0 < a ∨ b then a would not be the greatest lower bound). Hence a = a ∨ b, that
is, b ≤
= a. Using the lemma (as opposed to its dual) we reach a contradiction from the strict
semiquation b < a (because if a0 ∈ A were such that b < a0 < a then a would not be a lower
bound of the A-elements above b). Thus a = b. That is, every element in B is in A.
[ 109 ] For the injectivity of I see Theorem 10.7 (p??).
[ 110 ] This was discussed at the end of Section 4 (p??–??).

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In great generality—in particular for the models of any equational theory—a maximal
essential extension of an object is injective, indeed, it is minimal among injective objects in
which the object can be embedded. Such maximal-essential/minimal-injective extensions are
called injective envelopes.[111] (If an object can be embedded in an injective then each of
its essential extensions must appear therein and we may find one that is maximal.)
For scales we have noted that A ⊆ B is essential iff A? is cofinal in B? . Semi-simple
scales, we know, can be embedded in injectives (to wit, cartesian powers of I) hence have
injective envelopes (which are unique up to—perhaps many—isomorphisms).[112]
24.7 Theorem: Every injective scale is the injective envelope of its subscale generated by
its extreme points.
Given an injective scale E and an element a < > we need n ∈ N and an extreme point
e such that a ≤= (>|)n e < >. Using the semi-simplicity of E we can specialize to the case
that E is a subscale of a cartesian power K I. [113] Take n to be such that (>|)n⊥ is not a
Q
lower
Qbound of a (if there were no such n then a would be >). We first describe an element
e ∈ K I by stipulating its value for each co-ordinate i ∈ K:

ei = > if (>|)n⊥ ≤
= ai else ⊥
e satisfies the three equations:
.
a −◦ (>|)n e e −◦ (>|)n⊥ −◦ a



> = e∨e = =

= (>|)n e.
The 1st equation says that e is an extreme point. The 2nd equation says that a ≤
The 3rd equation ensures that e < >, hence (>|)n e < >. Now use injectivity to obtain an
element in E satisfying the same three equations.
24.8 Theorem: Order-complete/injective scales are precisely those scales of the form C(X)
where X is an extremely disconnected compact Hausdorff space (that is, one in which the
closure of every open set is open or—as sometimes called—“clopen”).
Note first that any retract of a (metrically) complete scale is complete, hence injective
scales, being retracts of cartesian powers of I, are necessarily of the form C(X). We need to
show that the order-completeness implies that X is extremely disconnected. Let U ⊆ X
be open. Define F ⊆ C(X) to be the set of all continuous functions from X to I that are
constantly equal to ⊥ on the complement of U and let U ⊆ C(X) be the set of upper bounds
of F and let L be the set of lower bounds of U. The order-completeness of C(X) says that
there is a (necessarily unique) continuous g ∈ L ∩ U. The Urysohn lemma says that for every
x ∈ U there is an element in F that sends x to > hence every element in U sends all of U
to >. The Urysohn lemma also says that for every point x in the complement of U there
is a function in U that sends x to ⊥ hence we know that g is constantly equal to > on U,
therefore, on its closure U. And, of course, any function equal to > everywhere on U is is an
upper bound of L.
Dually, we know that for every x 6∈ U, the closure of U, there is an upper bound of L that
sends x to ⊥ hence the least upper bound of L must be constantly equal to > on U and ⊥ on
its complement. It is the (continuous!) characteristic function of U which means, of course,
that U is clopen.
[ 111 ] Sometimes “injective hulls,”
[ 112 ] If it is not semi-simple it will have essential extensions of unbounded cardinality, indeed, in Section 12 (p??–??) we saw
this phenomenon for the Richter scale when we saw, first, that is appears co-finally in every non–semi-simple sdi and, secondly,
that there are non–semi-simple sdis of unbounded cardinality.
[ 113 ] Yes, of course we could take K = Max(E) but we needn’t since its topology has no role in this proof.

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◦
The proof of the converse makes use of an unexpected subject. Let I denote the initial
scale with the endpoints removed and P a complete poset. Define Q(P ) to be the set of
◦
sup-preserving functions from I to P . We will use the fact that Q(P ) is also a complete
lattice. And that use will take advantage of the Urysohn method for obtaining continuous
I-valued functions.[114]
Given a extremely disconnected space X take P to be the boolean algebra of clopens.
Given U ∈ Q(P ) define S f : X → I by f (x) = inf { t : x -∈1 Ut } and
T obtain that for all
-1
s ∈ I that f [⊥, s) = r<s Ur is necessarily open and that f [⊥, s] = t>s Ut is necessarily
closed which, note, establishes the continuity of f . Moreover every f ∈ C(X) is obtainable
from a unique element in Q(P ), to wit, the element defined by taking Us to be the closure
of f -1 [⊥, r). All of which shows that C(X) is order-complete. (The correspondence between
C(X) and Q(P ) is contravariant.)[115]
[In the TAC version it was erroneously stated that “Order-complete scales are precisely
those (metrically) complete scales that are chromatic.” For a complete chromatic scale that’s
not order-complete see [??], p?? (where it appears as a footnote for a proof that the converse
implication—that order-complete/injective scales are chromatic—is correct).]
Since we have identified the injective objects in the full category of complete scales it
follows that we have identified the projectives in its dual category, the category of compact
Hausdorff spaces.[116] Given such a space X let Y be the set of ultrafilters on the (discrete)
set X. Y is, of course, the “compactification” of that discrete set, that is, the reflection of
the discrete space in the full subcategory of compact Hausdorff spaces. The canonical
Q map
from Y back to X is a retraction of X and, hence, C(X) is a retract of C(Y ) = X I which,
being a cartesian product of injective objects, is, of course, injective.
In the category of compact Hausdorff spaces Gleason constructed the minimal projective
cover of a space X as the Stone space of the Boolean algebra of regular closed sets of X (those
closed sets that are the closures of their interiors). That Stone space may be constructed as
the set of ultrafilters of regular closed sets. The covering map is clear: send such an ultrafilter
to the unique point in its intersection. It may be described also as the scale of “adjoint
pairs” of semicontinuous I-valued functions on X, that is pairs consisting of a lower- and an
upper semicontinuous function where the lower semicontinuous function is the largest lower
semicontinuous function less than the upper semicontinuous function, and dually. See Section
44 (p??–??).

25. Diversion: Finitely Presented Chromatic Scales

If we move to chromatic scales we remove the word “continuous” to obtain “piecewise
D-affine map” and the word “closed” to obtain “piecewise D-affine set.” The definitions
are no longer as simple (we can not, for example, get by just with the existence of a
D-affine certification). The proofs of the parallel theorems, however, are easier. The cate-
gory of finitely presented chromatic scales is dual to the category of piecewise D-affine maps
between piecewise D-affine sets. Any piecewise D-affine set is a disjoint union of “boundary-
[ 114 ] Urysohn showed that if P is the lattice of open sets of a space X and U ∈ Q(P ) has the “Urysohn property,” to

wit, the property that the closure of Us is contained in Ut for all s < t then we obtain continuous f : X → I such that
f (x) = inf { t : x ∈ Ut }. Moreover given any continuous f we obtain a Urysohn element by Us = f -1 [⊥, s).
[ 115 ] It’s worth noting just where this argument fails when P is taken as the lattice of open subsets: we lose the Urysohn

property when we take infinite joins.

[ 116 ] First done by Andrew Gleason Projective topological spaces. Illinois J. Math. 2 (1958), p482–489. But this result is an easy

consequence of the fact that the injective objects in the category of boolean algebras are the complete boolean algebras: Sikorski,
Roman A theorem on extension of homomorphisms. Ann. Soc. Polon. Math. 21 (1948), p332–335.

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less simplices,” to wit, those that result when all boundary points are removed from ordinary
closed simplices (note that the 0-dimensional boundaryless simplex is not empty but a sin-
gle point); an n dimensional piecewise D-affine set may be described with an (n + 1)-tuple
of natural numbers, hs0 , s2 , . . . , sn i specifying the number of boundaryless simplices of each
dimension (necessarily sn > 0). Define the Euler characteristic, χ, as s0 − s1 + · · · + (−1)n sn .
Non-empty spaces turn out to be determined up to isomorphism by just two invariants:
dimension and Euler characteristic.[117]
The chromatic scale corresponding to an n-dimensional space can not be generated with
fewer than n generators. If χ = 1 the corresponding chromatic scale may be taken to be the
free chromatic scale on n generators, I[x1 , x2 , . . . , xn ].
If χ > 1 then the corresponding scale may be constructed as I[x1 , x2 , . . . , xn ]/((t))
where t = (x1 −◦ q1 ) ∨ (x1 ◦− −◦ q2 ) ∨ · · · ∨ (x1 ◦−
−◦ qχ ), one −◦ , the rest ◦− −◦ s, and
⊥ < q1 < q2 < · · · < qχ = > are I-elements. As always for finitely presented chromatic
scales this is isomorphic to an interval in the free algebra, to wit, [⊥, t ]. The free algebra
splits as the product [⊥, t ] × [ t, >] and we may dispatch the case of negative characteristic
with the observation that the second factor is of the same dimension and the characteristics
of the two factors add to one. For χ = 0 we can use I[x1 , x2 , . . . , xn ]/((x1 )). Finally, when
n = 0 the only corresponding chromatic scales are the cartesian powers of I (bear in mind
that χ is necessarily non-negative and that the empty product is the one-element terminal
scale).
The only time that we need more generators than dimension is when n = 0 and χ > 1.
(The corresponding chromatic scale is constructable with one generator: I[x]/((v)) where v =
(x ◦−
−◦ q1 ) ∨ (x ◦−
−◦ q2 ) ∨ · · · ∨ (x ◦−
−◦ qχ ) and q1 < q2 < · · · < qχ .)

26. Appendix: Lattice-Ordered Abelian Groups By a lattice-ordered abelian

group, or LOAG for short, is meant, of course, an object with both an abelian-group and a
lattice structure in which the lattice ordering is preserved by addition (that is, x + (y  z) =
(x + y)  (x + z) for either lattice-operation ).[118]
[ 117 ] Any such set is piecewise D-affine isomorphic to one described by an (n + 1)-tuple where s = s = · · · = s
0 1 n−2 = 0
and either (sn−1 = 0) & (sn > 0) or (sn−1 > 0) & (sn = 1) (the first possibility occurs precisely when χ 6= 0 with signature
(−1)n ). First, any k-dimensional boundaryless simplex with k > 0 is the disjoint union of one (k −1)-dimensional and two
k-dimensional boundaryless simplices, which means that we may, without changing isomorphism type, increment by 1 any two
adjacent si s provided the right-hand one is already positive. A sequence of such increments—working from the top down—can
guarantee that all the si s are positive, indeed as big as we want them. We can then minimize the number of positive si s by
successively performing the reverse of such increments—working from the bottom up—until s0 = s2 = · · · = sn−2 = 0. We
then perform as many such “reverse increments” as we can on the pair sn−1 , sn . The result is as advertised.
(Exercise: show that any such set with positive χ is isomorphic to one boundryless simplex plus a bunch of isolated points.)
[ 118 ] We can simplify the definition by noting that we need only truncation at zero, 0 ∨ x, as a primitive. We will denote this

truncation here as bxc. The two axioms: trunc-1: x = bxc − b−xc trunc-2: bx − bycc = bbxc − bycc
Trunc-1 is justified by x + (0 ∨ −x) = (x + 0) ∨ (x + (−x)) = x ∨ 0. For Trunc-2 it suffices to justify byc + bx − bycc =
byc + bbxc − bycc. But byc + (0 ∨ (x − byc)) = (byc + 0) ∨ (byc + (x − byc)) = byc ∨ x = (y ∨ 0) ∨ x and byc + (0 ∨ (bxc − byc)) =
(byc + 0) ∨ (byc + (bxc − byc) = byc ∨ bxc = (y ∨ 0) ∨ (x ∨ 0). See Section 46, p??–?? for subscorings.)
Use the truncation operator to define: x ∨ y = x + by − xc The idempotence of ∨ is easily seen to be equivalent to
what we’ll call Trunc-0, b0c = 0, which can be proven by b0c = bb0cc − b0 − b0cc = bb0cc − bb0c − b0cc = bb0cc − b0c =
bbb0cc − b0cc − bb0c − bb0ccc = bb0c − b0cc − bbb0cc − bb0ccc = b0c − b0c = 0.
The fact that addition distributes with ∨ is immediate. For commutativity note that x ∨ y = y ∨ x translates to x + by − xc =
y + bx − yc and that rearranges to x − y = bx − yc − by − xc, an instance of Trunc-1.
For associativity note that by adding byc to both sides of Trunc-2 we obtain byc ∨ x = byc ∨ bxc, making it clear that
byc ∨ x = byc ∨ bxc = y ∨ bxc and hence (y ∨ 0) ∨ x = byc ∨ x = y ∨ bxc = y ∨ (0 ∨ x) which—together with distributivity with
addition—easily yields full associativity.
(Trunc-2 is stronger than associativity—it has Trunc-0 built into it. It was chosen as an axiom not for its strength but for its
simplicity: the truncation equation equivalent to associativity is bx + by − xcc = bxc + by − bxcc. For a separating example take
the positive rationals under multiplication and define the associative “join” operation to be ordinary addition. The truncation
operator is then just shifting by 1. Trunc-1, when rewritten, becomes x = (1 + x)(1 + x-1 )-1 which is satisfied and Trunc-2
becomes 1 + x(1 + y -1 ) = 1 + (1 + x)(1 + y -1 ) which is not.)
The induced ordering, that is, the one obtained by defining x ≤ y iff x ∨ y = y, is, of course, preserved under addition.

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There are many similarities and differences between the theories of loags and of scales.
Among the similarities (each of which, I trust, is to be found somewhere in the literature):
The theory of loags is a complete equational theory, that is, every equation on its operators
is either inconsistent or a consequence of its axioms (as exemplified in the last footnote). Every
consistent equation holds for the loag of integers, Z, because every consistent equation has a
non-trivial model, every non-trivial model has a positive element (e.g., (x ∨ 0) + ((−x) ∨ 0) for
any x 6= 0) and any positive element generates a sub-loag isomorphic to Z, all of which says
that the maximal consistent equational extension of the theory of loags is—precisely—the
theory of Z. To verify that the equations in hand already provide that maximal consistent
extension it thus suffices to show that every equation not a consequence of those axioms
has a counterexample in Z. It clearly suffices to find a counterexample in the rationals, Q,
because multiplying by a suitable positive integer would then yield a counterexample in Z
and because the operations are continuous, it clearly suffices for that to find a counterexample
in the reals, R. The previous proof for the theory of scales can be easily replicated for this
case. Or, if one wishes, we can reduce this case to that previous case. Given an equation with
a counterexample in some loag we can first tensor with the dyadic rationals, D, to obtain
a D-module and then chose an element we’ll call > large enough so that the computation of
the terms in the counterexample all lie in the interval [−>, >]. Replacing 0 with , −x with
.
x and x + y with / (x|y) we obtain a scale with a counterexample for the given equation
and from that we know that there is a counterexample in R.
The free loag on n generators is the loag of continuous piecewise integral-linear R-valued
functions on Rn. [119] The functions in question are necessarily “radial:” f (rx) = rf (x) for
any r > 0, hence are determined by their values on the faces of the “standard cube” and that
allows us to reduce to the result for free scales.
Every loag can be embedded in a product of linearly ordered abelian groups (TOAGs). The
proof that all loags can be embedded in a product of toags is—as is to be expected—
essentially the same as it was for scales: it is necessary and sufficient to show that every
subdirectly irreducible loag is a toag. Just as for scales (see Theorem 8.6 at p??), a
consequence is that loags are not just lattices but distributive lattices and that allows an
easy proof that the lattices of congruences Spec(L), for a given loag L, is distributive and—
as always for distributive lattices of congruences—therefore a spatial locale. When Spec(L)
is viewed not as lattice but as a space, its points are the congruences for non-trivial linear
quotients [120] and if we specialize to Max(L), the congruences of simple quotients, the points
are the congruences for non-trivial Archimedean linear quotients.
Among the differences:
• Not every non-trivial loag has a simple quotient. Consider the toag of integral poly-
nomials in one variable “ordered at infinity,” that is, f ≤
= g iff f (n) ≤
= g(n) for almost all
natural numbers n. For each quotient algebra there exists d ∈ N such that f and g name
the same element in the quotient iff degree(f−g) < d. None is maximal. There is no simple
quotient. Another huge difference is that Max(−) is not a functor into the category of spaces;
And from that we may infer that it is reversed by negation: x ≤ y iff x − (x + y) ≤ y − (x + y) iff −y ≤ −x. Hence
negation must convert least upper bounds into greater lower bounds, yielding what can only be called “De Morgan’s law”:
−(x ∧ y) = (−x) ∨ (−y). For a direct formula we have x ∧ y = x − bx − yc (because x ∧ y = −((−x) ∨ (−y)) =
−((−x) + b(−y) − (−x)c) = x − bx−yc).
The kernel of a loag homomorphism—besides being closed under truncation at zero—is distinguished by its “betweenness
property,” that is, it includes any element between any two of its elements. Note the semiquations: −b−ac ≤ bx+ac−bxc ≤ bac.
[ 119 ] That is, continuous functions f : Rn → R such that f agrees at each point with one of a finite set of (homogeneous)

linear functions with integer coefficients.

[ 120 ] Hence, as for scales, the points are the congruences of linear quotients and thus—again as for scales—every loag has a

representation as the group of global sections of a sheaf of toags.

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loag maps K → L, when not epimorphic, do not induce functions Max(K) → Max(L), only
partial functions.[121]
• Max(−) is not a representable functor. (Certainly every simple quotient may be embedded
in the reals, but not uniquely: a non-trivial map f : L → R certainly names a point in Max(L)
but rf names the same point for every r > 0.) No “schizophrenic object.”
• For a loag L neither Spec(L) nor Max(L) need be compact. It is the case that Max(L)
is Hausdorff (as in Section 42, p??–??). The easiest non-compact examples to describe
are the free loags on infinitely many generators. (But Max(L) is compact for all finitely
generated L.)
• No non-trivial injectives. Any loag may be embedded in another one in which it acquires
an upper bound.[122] Hence no absolute retracts.
• Max(Fn ), where Fn is the free loag on n-generators, is not—as is the case for free scales—
the n-cube. Far more interesting: it is the (n 1)-sphere.
1)-sphere It is not just the topology. To measure
the distance between two maximal ideals use the probability that the two orderings they
induce disagree whether an element of Zn ⊂ Fn is positive. To be precise, for each k ∈ N
define a k-walk to be a sequence of k+1 elements in Zn such that each element agrees with
the next on all but one coordinate, and that difference is 1. Let Dk be the proportion of all
k-walks starting at the origin that end on an element on which the two orderings disagree. The
probability of disagreement is limk→∞ Dk . Use de Moivre [123] and the rotational invariance
of Gaussian distributions to establish that the result is the standard sphere geometry. (If we
use Bernoulli [124] then with probability one we can compute the distance using the limiting
frequency of disagreement on an endless random walk. Add Pólya to the mix and it doesn’t
even matter where we start.[125] ) Those who insist on radians may multiply by π.

27. Appendix: Computational Complexity Issues

Finding which equations can be counterexampled in the theory of either scales or loags is
NP-complete. There is no substantive difference between the equational and universal Horn
theory problems. Indeed, if we change scales to linear scales and loags to toags we obtain
the result for the full first-order universal theory.
First, we observe that the “satisfaction” problem for Boolean algebras can be easily
converted to a satisfaction problem of roughly the same size in scales. The proof would be
∨.
straightforward if it were not the case that one of the variables in the formula x ∨ y = x | xd|y
appears twice. An uncaring use of this formula would lead to an exponential growth in the
length of the translation. Let B denote the Boolean algebra of extreme points in a scale.
We avoid the problem by using:
∧ ∨
27.1 Lemma: If x, y ∈ B then x ∧ y = x | y and x ∨ y = x | y.
(Check in any linear scale.)
[ 121 ] Since the domains will always be open we could obtain a functor by replacing it with Scone(Max(−)) as described in

Section 35 (p??–??).
[ 122 ] e.g., given a loag L take the initial model of the equational theory in which each element of L appears as a constant

plus one more constant b. Add to the theory of loags the (variable-free) equations that describe structure of L and the fact
that the b is their upper bound. If the map from L to the loag so constructed were not an embedding, that is, if there is a
constant a such that there’s a proof of a = 0 the join of the finite number of L-constants appearing in the proof could be
substituted for b in each of the equations appearing in the proof. Hence a is already 0.
[ 123 ] Central limit theorem [??] Law of large numbers

[[ 124
125 ]] We could make this look more complicated by measuring the frequency that the two orderings disagree, that is, how a

pair of random walks compare.

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Using these translations (plus the translation of negation into dotting) any Boolean term
may be converted to a scale term of the same size. We are not done until we restrict the
variables to B. Given a scale term A on variables v1 , v2 , . . . , vn any solution of the equation:
∧ ∧ ∧
(v1 −◦ v 1 )|(v2 −◦ v 2 )| · · · |(vn −◦ v n )|A = >

(associate at will) is a solution of A = > that necessarily lies entirely in B.

The complementary problem is also convertible. Every equation in the theory of Boolean
algebras is equivalent to an equation—of manageable size—in the theory of scales. (As in the
satisfaction problem the proof would be straightforward if it were not the case that one of
the variables in the formula for x ∨ y appears twice.)
∨.
For the conversion we will understand that x ∨ y is the term x | xd |y, the one in which y
∧ . ∨
appears once and, dually, x ∧ y is the term x | (x | y), again the one in which y appears once.
We will need a term M hv1 , v2 , . . . , vn i defined recursively by:
(n = 0) ⇒ (M = >)
.
M hv1 , v2 , . . . , vn i = (vn ∨ v n ) ∧ M hv1 , v2 , . . . , vn−1 i
Given terms A and B in the signature of Boolean algebras we produce terms hhAii and hhBii
such that A = B is true for Boolean algebras iff hhAii = hhBii is true for scales. The length of
hhAii will be bound by a constant multiple of the length of A times the number of variables
in A and B.
The conversion is defined recursively by the following rules in which v1 , v2 , . . . , vn are the
variables and M denotes M hv1 , v2 , . . . , vn i.
hh>ii = M
.
hh⊥ii = M
.
hhvi ii = M ∨(v. i ∧ M)
hh¬Aii = hhAii
.
hhA ∨ Bii = M ∧ [M /( hhAii | hhBii )]
.
hhA ∧ Bii = M ∨[M / ( hhAii | hhBii )]

Note that in a linear scale if any one of the variables is instantiated as then M = ,
.
otherwise M and M are distinct. In either case, an inductive argument shows that hhAii is
.
either M or M for every term A. Moreover hhA  Bii = hhAii  hhBii where  is either lattice
operation.
We may redo this construction for loags instead of scales.
Define M by
(n = 0) ⇒ (M = >)
M hv1 , v2 , . . . , vn i = (bvn c ∨ b−vn c) ∨ M hv1 , v2 , . . . , vn−1 i
and:
hh>ii = M
hh⊥ii = −M
hhvi ii = (−M ) ∨ (vi ∧ M )
hh¬Aii = −hhAii
hhA ∨ Bii = M ∧ ( hhAii + hhBii + M )
hhA ∧ Bii = (−M ) ∨ ( hhAii + hhBii − M )

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28. Appendix: Independence

The independence of all but the first two scale axioms is easy:
For the independence of the medial axiom consider the set {−1, 0, +1} with x|y defined
as “truncated addition,” that is
x|y = −1 ∨ (x + y) ∧ 1
. ∧
We take x = −x, x = x and > = 0. All defining laws of scales hold except for the medial law
(+1|0) | (+1| −1) 6= (+1| +1) | (0| −1).
For the independence of the unital and constant laws consider the set {0, 1} with ordinary
∧ .
multiplication for x|y and the identity function for x. If we take x = 1 − x and > = 1 then
∨
every equation is satisfied except for ⊥ | x = x. If, instead, we take > = 0 then every equation
∧ .
is satisfied except for > | x = x. If we take x = x and > = 1 then every equation is satisfied
.
except for the constancy of x |x.
For the independence of the scale identity consider I × I with the standard cartesian-
[
product structure except for >-zooming. The unital laws determine hx, yi only when x and
y are both non-negative. We maintain all the laws except for the scale identity, therefore, if
>-zooming is standard on just that top quadrant.

As promised in [??] (p??) we can do better. The absorbing laws determine hx, [ yi only
when x and y are both non-positive. We can maintain the minor-scale equations, therefore,
[
by keeping the standard definition of hx, yi just on the top and bottom quadrants, (that is,
the pairs hx, yi such that xy ≥
= 0).
∨ ∨ ∧ ∧
The first three uses of the scale identity were for > = >, x = x | x and x| d =x |⊥ (the
absorbing law, ⊥|x = ⊥, is a consequence of these). We may maintain the law of compensation
d
[
by stipulating hx, yi = hx, yi for xy < 0. Central distributivity requires a recursive definition.
Given hx, yi such that xy < 0 let n be the largest integer such that there exist u, v with
hx, yi = ( |)n hu, vi. If n = 0 then define hx, [ yi = hx, yi. For n > 0 recursively define
[ n−1 [
hx, yi = (( |) hu, vi)|h⊥, ⊥i. The scale identity itself fails (most easily seen by noting that
>-zooming no longer preserves order).
Also easy is the independence of the axioms for chromatic scales: if the support operation
is constantly > then only the 1st equation, ⊥ = ⊥, fails; if it is the identity function then only
∧
the 2nd equation, x = x, fails; if it is constantly ⊥ then only the 3rd equation, x ∧ x = x, fails;
if x = ⊥ when x = ⊥ else x = > then the 4th equation, x ∧ y = x ∧ y, fails when the scale is
non-linear but only it fails.[126]
As promised, we can eliminate the 2nd equation by strengthening the 3rd equation to
. . .
x ∧ x = ⊥. Show first that x is the Heyting negation, that is, y ≤ = x iff y ∧ x = ⊥ :
. . . .
if y ≤ = x then y ∧ x ≤ = x ∧ x = ⊥ ; if y ∧ x = ⊥ then y ∨ x = (y ∨ x) ∧ > =
. . . . . . . . . . .
(y ∨ x) ∧ ⊥ = (y ∨ x) ∧ y ∧ x = (y ∨ x) ∧ (y ∨ x) = (y ∧ y) ∨ x = ⊥ ∨ x = x . This im-
∨ ∨
. . ∨ . . ∨
plies, in particular, that x is an extreme point because x∧ x ≤ = x ∧ x = (x ∧ x)∨ = ⊥ = ⊥
[ 126 ] Similarly, the 1st , 3rd and 4th for support operations (see [??], p??) on rings are independent: if the support operation

is constantly 1 then only 0 = 0 fails; if it is constantly 0 then only xx = x fails; if x = 0 then x = 0 else x = 1
then xy = x y fails when the ring is not a domain but only it fails. (For the redundancy of the 2nd equation note first that it was
not used to show that x2 = 0 implies x = 0, hence (1 − x)x is 0 since its square is 0; finish with (1 − x)x = (1 − x)(1 − x)x =
(1 − x)(1 − x)x = (1 − x)0 = 0.)

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∨
. . . ∨. .
hence x ≤ = x and, consequently, x = x . Since x is an extreme point, so is x. We obtain
. .
the original 3rd equation by x ∧ x = (x ∧ x) ∨ (x ∧x) = (x ∧ x) ∨ x = ⊥ ∨ x = x.
We have not yet established the independence of the idempotence and commuta-
tive laws. We already proved in [??] (p??) that the commutative law may be
replaced with the single instance ⊥|> = >|⊥ and we promised in [??] (p??) that we could
.∧
remove the commutative law entirely by replacing the first unital law with ⊥ | x = x. To
do so, first establish the left cancellation law using a different construction for dilatation,
. . . . . . . . .
(((a | ⊥)|x)∧ )∨. Then (((a | ⊥)|(a|x))∧ )∨ = (((a | a)|(⊥ |x))∧ )∨ = (((⊥ |⊥)|(⊥ |x))∧ )∨ = ((⊥
|(⊥|x))∧ )∨ = (⊥|x)∧ = x. Hence a|x = a|y implies x = y and just as in the derivation of full
commutativity from the commutativity of > and ⊥ we obtain dot-distributivity. Then obtain
∨ . .
the involutory law from the second unital law written in full: x = ⊥ | x = (((⊥|x) )∧ ) =
. . ∧. . .
((⊥ | x) ) = (x). (Note in passing that. we now have the original first unital law.) Fin-
. . . . .
ish as before by first showing: that (x) = x implies x| x = (x) | x = hence the cen-
. .
trality of the center, |x = (x| |(x|x) = (x|x)|(
x) x|x) = x| . Finally use cancellation on
|(x|y) = ( |x) | ( |y) = ( |x) | (y| ) = ( |y) | (x| ) = |(y|x).

29. Appendix: Continuously vs Discretely Ordered Wedges

In Section 1 (p??–??) there appeared a quick and dirty procedure for computing the binary ex-
pansion of f (x) where f is the unique interval–coalgebra-map from a given interval-coalgebra
to the unit interval by iterating (forever):
∨ ∧
If x = > then emit “1” and replace x with x
∨
else emit “0” and replace x with x.

A numerical analyst will object to the very beginning: how does one determine when an
equality holds? There may be procedures that are guaranteed to detect when things are not
equal (assuming, of course, that they are, indeed, not equal) but in analysis there tend not
to be procedures that establish equality.[127]
Before considering computationally more realistic settings let us prove (in the classical
setting) that the unit interval is the final interval coalgebra. Using binary expansions the
interval coalgebra on [ 0, 1] is described with an automaton with three states L, U, and initial
state, M. It takes {0, 1}-streams as input and produces {0, 1}-streams as output:

Next State ⊥-Zoom Output >-Zoom Output

L M U L M U L M U
0 L L U 0 0 1 0 0 0
1 L U U 1 1 1 1 0 1

The blanks in the output tables will be called stammers. The output streams will always
be one digit behind the number of input digits.
[ 127 ] For just one example, suppose the given interval-coalgebra is, itself, the unit interval but that we know an element x only

by listening to its binary expansion. If that expansion happens to be .0 followed by all 1s we will never have enough information
∨ ∧
to conclude that x = >. But this is not the most telling example (we know in this case that x = ⊥). Suppose that we know x
only as the midpoint of a pair of binary expansions. If one sequence is the expansion of an arbitrary element y in the open unit
∨ ∧
interval and the other is the expansion of 1−y then we will never have enough information to know either x = > or x = ⊥.

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We need a (dual) pair of definitions: define ⊥  x if a finite iteration of ⊥-zooming carries

x to >, and x > if a finite iteration of >-zooming carries it to ⊥. In I these are unneeded
properties:  coincides with < . [128] For any I-valued coalgebra map, f, the two -relations
tell us how f (x) is situated with respect to the center, :
∧
f (x) > iff ⊥  x
∧ ∨
f (x) = iff neither ⊥  x nor x  >
∨
f (x) < iff x  >
∧ ∨ ∧
Note that ⊥  x implies x = > (since x 6= ⊥) hence in the 1st case the procedure produces
∧
a stream of binary digits starting with 1 followed by the stream for x which is precisely what
∧ ∨
is demanded by f (x) = (f x)∨ |(f x)∧ = >|f (x). The 3rd case is dual. In the 2nd case if x = >
∨
the procedure will produce a 1 followed by all 0s and if x 6= > a 0 followed by all 1s. That it
produces one of these two streams (and it doesn’t matter which) is just what is demanded
by f (x) = . [129]
?? For a computationally more realistic setting we are handed a guide to the needed modi-
fications. The Lawvere test of a definition for the reals in a topos is that working in Sh(X),
the category of sheaves on a space X, the definition yields the sheaf of continuous R-valued
functions on X, that is, the sheaf whose stalks are germs of continuous functions (as defined
in the topos of sets) from X to R. That experience leads us to view the sheaf of continuous
I-valued functions as the best candidate for the closed interval. We note immediately that
the disjunctive coalgebra condition fails. Given continuous g : X → I the “truth value” of the
∨
equation g = > (necessarily an open subset of X) is the interior of g -1 [ , >] and the value of
∧
g = ⊥ is the interior of g -1 [⊥, ]. Their union is not, in general, all of X. But it is a dense
subset. Hence we replace the discrete coalgebra condition:
∧ ∨
⊥ = x or x = >
with the weaker continuous coalgebra condition:
[ 130 ]
¬[ ⊥  x and x  > ]
(We will understand that both the discrete and continuous coalgebra conditions entail that
> and ⊥ are zooming fixed-points.)
We must, however, capture the detectability of semiquations. Hence we replace the
apartness condition:
⊥ 6= >
with this stronger separation condition:
[ 131 ]
⊥ 6= x or x 6= >
[ 128 ] < is the same as  in a scale iff the scale is semi-simple.
[ 129 ] This argument works even in the intuitionistic setting if we hold on to the computationally unrealistic coalgebra condition
∨ ∧
x = > or x = ⊥ (as in the Cantor—rather than Dedekind—closed interval).
[ 130 ] In footnote [??] (p??) about the modal operators 3 and 2 it was the continuous coalgebra condition

we invoked when we wrote “No one allows simultaneously both 3Φ 6= T and 2Φ 6= ” (less than completely
T

possible/tenable/conceivable/allowed/foreseeable but somewhat necessary/certain/known/required/expected). The discrete

condition would have been the stronger “All insist upon 3Φ = T or 2Φ = ” (everything is either totally
T

possible/tenable/conceivable/allowed/foreseeable or entirely unnecessary/uncertain/unknown/unrequired/unexpected.) The

continuous coalgebra conditions sound realistic, the discrete do not. (The last pair—the “Bayesian modality pair”—is in lieu of
a pair in which 2 means likely—there seems to be no English word that works for the corresponding 3, “not likely false.” The
pair should be viewed only as an approximation: among other ways of pronouncing this 2 are anticipated and foreseen.)
[ 131 ] In the presence of De Morgan’s law the two conditions are, of course, equivalent. In a topos the top and bottom of the

subobject classifier Ω (or as Grothendieck called it, “the Lawvere object) are always apart; they are separate only when De
Morgan holds throughout the topos. (Yes, they are apart—indeed separated—in the internal logic of the trivial topos.)

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Keeping in mind that the truth values in Sh(X) are open subsets of X, the extent to
which any characteristic map, χ A , is different from ⊥ is the interior of A, the extent to
which it is different from > is the “exterior” of A (the interior of its complement), hence
[⊥ 6= χ A ] ∨ [χ A 6= >] holds iff A is both open and closed.[132] The continuously ordered
wedge, as opposed to the discretely ordered wedge [133] of X and Y is
n o
hx, yi : ¬[x  > ∧ ⊥  y]

Its top is h>, >i, its bottom h⊥, ⊥i. If either X or Y satisfies the separation condition, so
does their ordered wedge. A continuous coalgebra structure on X is a map from X to the
continuously ordered wedge of X with itself.[134]
In preparation for establishing the nature of the final continuous coalgebra we’ll say that
an element c lies in the ∧
top open half when ⊥  c
∨
bottom open half when c  >

The top and bottom open halves do not form a cover (e.g., ∈ I is in neither of them)
but the continuous coalgebra condition is precisely the condition that they are disjoint.[135]
∧ ∨
We do have a notion of “middle open half,” to wit, when c is in the bottom open half and c
is in the top open half, that is, c lies in the
∨ ∧
∧ ∨
middle open half when c  > and ⊥  c

The critical lemma for the continuous case:

29.1 Critical Lemma: The Triumvirate of Open Halves
A continuously ordered wedge is the union of the three open halves: top, middle and bottom.
Because: we start with two instances of the separation condition
[ 132 ] Consider the sheaf of germs of all functions f : X → I, continuous or not, in the topos Sh(X). We wish to find the largest

separated subsheaf invariant under the zoom operators. So start by throwing away all germs that fail the separation condition.
The trouble now is that the resulting sheaf is not closed under the zoom operators; so throw away all germs for which there
is a zooming sequence α such that f α fails the separation condition. The resulting sheaf is the sheaf of germs of continuous
I-valued functions. (Suppose f is not continuous at x. To find a discerning α start with the fact that there are values of f
on arbitrarily small neighborhoods of x that are bounded away from f x. We may assume without loss of generality that those
values are below f x. Let ` ∈ I be such that ` < f x and for all neighborhoods of x there are values of f below `. Let u ∈ I
be such that ` < u < f x. Let α be such that `α = ⊥ and uα = >. Then for any open U ⊆ X such that f α 6= > on U it
must be the case that x 6∈ U and for any open V ⊆ X such α
˘ that f α 6=¯⊥ it must ¯ again, that x 6∈ V. That is
˘ be αthe case,
[⊥ 6= f α ] ∨ [f α 6= >] fails (because x is not in the union of y : ⊥ 6= f y and y : f y 6= > ).)
[ 133 ]
Sometimes “thick ordered wedge” as opposed to “thin ordered wedge.”
[ 134 ]
In footnote [??] (p??) we can replace the scale-algebras with interval coalgebras by adopting these intuitionistic modifi-
cations of the coalgebra definition to the classical setting. Given a set X with constants > and ⊥ and unary operations whose
∧ ∨
values are denoted x and x impose the conditions (where  is as defined on p??):
∧ ∨
h i
¬ ⊥  x and x  >
⊥x or x>
Q
The set, A , of sequences, IN = N I, reduced by almost-everywhere equality does not satisfy either condition (consider, for
example a sequence that is equal infinitely often to > and to ⊥ ). But there is a largest subset that does, to wit, the set of
sequences s such that for all zooming sequences α it is the case that
h i
¬ ⊥  sα∧ and sα∨  >
⊥  sα or sα  >
The resulting set is precisely the set of convergent sequences. If we now collapse to a point the set of its members such that
¬(s  >) and dually for ¬(⊥  s) we obtain an interval coalgebra. Its unique coalgebra map to I—as one must now have
surely learned to expect—is Lim. And the same modifications work for defining limits of functions at a point in a space and
for defining derivatives.
[ 135 ] It’s worth looking at the case when we’re working in the topos of sheaves on a space X. The extent to which something

is in a particular open half is given by an open subset of X.

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∨ ∨ ∧ ∧
∧ ∧ ∨ ∨
[⊥  c ] ∨ [ c  > ] and [ ⊥  c ] ∨ [ c  > ].
∨ ∧
Clearly ⊥  x is equivalent with ⊥  x for any x and dually for x  > and x  >. We
thus replace the two disjunctions with:
∨ ∧
∧ ∧ ∨ ∨
[⊥  c] ∨ [c  >] and [ ⊥  c ] ∨ [ c  > ].
The conjunction of these two disjunctions redistributes as a disjunction of four conjunctions:
∧ ∨ ∧ ∨
∧ ∨ ∧ ∨ ∧ ∨ ∧ ∨
[ ⊥  c ]∧[ ⊥  c ] or [ ⊥  c ]∧[ c  >] or [ c  > ]∧[ ⊥  c ] or [ c  > ]∧[ c  > ].
The second term is precisely what is prohibited by the definition of a continuously ordered
wedge and we can weaken the first and last conjunctions [136] to obtain:
∨ ∧
∧ ∧ ∨ ∨
[ ⊥  c ] or [ c  > ] ∧ [ ⊥  c ] or [ c  > ].
Exactly what we set out to prove.

30. Appendix: Signed-Binary Expansions: the Contrapuntal Procedure

and Dedekind Sutures
On July 31, 2000, I posted a note on the category net on how to obtain signed binary
expansions for elements from such coalgebras. Five days later Peter Johnstone posted a
note pointing out that—unlike the Dedekind-cut approach—my approach implicitly used
the axiom of dependent choice.[137] In fact, I was using a very weak version of dependent
choice [138] as will be explicated below. The approach can be modified without too much
trouble (in a way easily seen to be equivalent to using Dedekind cuts) but that modifi-
cation had not occurred to me at the time of the original publication. Because so many
“infinite-precision” programmers are quite happy (wittingly or not) with dependent choice I
did describe the approach in the printed version. We have much more inclusive motivation
now that we have the necessary modification.
Every element of the standard interval has a representation of the form
∞
X
an 2-n
n=1

where an ∈ {−1, 0, +1}. The sequence of an s is, of course, not unique: everything other than
the two endpoints has infinitely many expansions—indeed, everything not a dyadic rational
has continuously (or is it continuumly?) many.
The finite words on any alphabet may be viewed, of course, as a rooted tree and in the
case at hand every vertex has three branches each with a label from {−1, 0, +1}.
It needn’t be a tree. We get a diagram much easier to picture if we identify vertices when
the paths that reach them are of the same length and name the same dyadic rational. Choose
2 3
∧ ∧
[ 136 ] ∧ ∧ ∨ ∨
In fact they’re not weakenings but equivalences: [ ⊥  c ] ⇒ [ ⊥  c ] ∧ 4 [ ⊥  c ] ∨ [ c  > ]5 ⇒
2 3 2 3 2 3 2 3
∧ ∧ ∧ ∧ ∧
4[ ⊥  ∧ ∨ ∧ ∨ ∧ ∨ ∧ ∨ ∨
c ] ∧ [ ⊥  c ]5 ∨ 4[ ⊥  c ] ∧ [ c  > ]5 ⇒ 4[ ⊥  c ] ∧ [ ⊥  c ]5 ∨ 4[ ⊥  c ] ∧ [ c  > ]5
∧ ∨
c ⇒ [ ⊥  c ] ∧ [ ⊥  c ].

[ 137 ]
Which axiom, it should be noted, is accepted by many constructive analysts.
[ 138 ]
Which version is an easy consequence of the “disjunction property” that holds, for example, in the free topos with natural
numbers object. See my Numerology in topoi. Theory Appl. Categ. 16 (2006) No. 19, p522–528

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the “aspect ratio” to optimize the quality of the printed lines, that is, choose to make the
oblique edges to be the nicest of Latex obliques, those with slope ±1.
I like word-trees to grow to the right. (Why do so many like them to grow away from the
light? Why do they like roots an top? Leaves on the bottom?) Combined with the choice of
aspect we have a bonus: we don’t need labels—slopes are all that anyone needs.
Then to make it fit, we handle the exponential growth of the number of edges by
exponentially shrinking their length.
Presto! It becomes a wonderful illustration (appearing as our frontispiece) of just what
the signed-binary sequences are doing. Because of its magical properties—most yet to be
described—I have fallen into the habit of calling it the Houdini diagram.
If we view the diagram as a subset of the plane and take its closure, a boundary line is
added to the right-hand of the tree. It’s no less than the standard interval I. If we follow
a path from the far-left node we converge to the point in I named by the signed-binary
expansion that uses the labels (the slopes) of the path. Indeed, as we travel that path the
height above (or below) the central horizontal at each vertex is just what the signed-binary
expansion describes at that point of path.
When writing signed-binary expansions we’ll suppress the 1s and use the symbols
“−, ◦, +”
Given an object C with separated elements ⊥ and > and self-maps whose values are denoted
∨ ∧
x, x satisfying the continuous coalgebra condition we seek a procedure that delivers for each
c ∈ C a signed-binary expansion.
In the last Section, Theorem 29.1 said that a continuously ordered wedge is the union of
the three open halves: bottom, middle and top.
∧
When c is in the top open half we want to emit “+” and replace c with c, when in the
∨
bottom open half we want to emit “−” and replace c with c. And when c is in the middle
↔
open half we want to emit “◦” and replace c with its its mid-zoom c .
Whoops.
We don’t have a mid-zoom operation.
But the continuously ordered wedge C ∨ C does have a mid-zoom and that solves the
problem. For a moment let T be an arbitrary endofunctor on a category and F → T F a
final T -coalgebra. Given an arbitrary coalgebra g : C → T C we have another coalgebra
T g : T C → T 2 C. (Note that it’s an absolute tautology that g : C → T C is a map of
coalgebras.) If we can describe a coalgebra map T C → F then C → T C → F is the
unique coalgebra map from C to F. [139] In the case at hand the induced bottom-zoom
∨ ∨ ∧
function T C → T C sends hu, vi to hu, vi = h u, ui and the induced top-zoom function sends
∧ ∨ ∧
hu, vi to hu, vi = h v , vi. We define the mid-zoom to be the function that sends hu, vi to
↔ ∧ ∨
vi [140]
hu, vi = h u, vi.
∧ ∨ ∨ ∧
↔ ∧ ↔ ∨ [141]
The critical property of mid-zooming is: hu, vi = hu, vi and hu, vi = hu, vi.
[ 139 ] Has this fact ever been used before?
[ 140 ] The verification that the values of the mid-zoom all lie in the continuously ordered wedge uses again that the zoom
functions fix the endpoints. Indeed, for any pair g, h : C → C that fix ⊥ and > it is the case that if hu, vi is in the continuously
ordered wedge then so is hg(u), h(v)i. ∧ ∧ ∨ ∧ ∨ ∨
[ 141 ] Besides the evidence from the Houdini diagram we have a proof: hu,↔ ∧ ∨ ∨ ∨ ∨ ∧ ∧
vi = hu, vi = h v, vi = h v, vi = hu, vi.

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With the mid-zoom in hand we start again:

The triumvirate says that for any c in an interval coalgebra C
∨ ∧
∧ ∧ ∨ ∨
[ ⊥  c ] or [ c  > ] ∧ [ ⊥  c ] or [ c  > ].
Hence for any hu, vi ∈ C ∨ C:
∨ ∧
∧ ∧ ∨ ∨
[ h⊥, ⊥i  hu, vi ] or [ hu, vi  h>, >i ] ∧ [ h⊥, ⊥i  hu, vi ] or [ hu, vi  h>, >i ].
which translates to:
∨ ∧ ∨ ∧
∨ ∧ ∨ ∨ ∧ ∧ ∨ ∧
[ h⊥, ⊥i  hv, vi ] or [ hv, vi  h>, >i ] ∧ [ h⊥, ⊥i  hu, ui ] or [ hu, ui  h>, >i ].
∨
Note that the nth iteration of ⊥-zooming turns x into > iff it turns hx, yi into h>, >i for any
∨ ∧ ∨ ∧
y. Hence h⊥, ⊥i  hv, vi iff ⊥  v and dually, hu, ui  h>, >i iff u  >. When we specialize
∨ ∧ ∨ ∧
∧ ∧ ∧ ∧ ∨ ∨ ∨
x to u we obtain h⊥, ⊥i  hu, ui iff ⊥  u and dually hv, vi  h>, >i iff v  >. All of which
says that if hu, vi ∈ C ∨ C then the triumvirate of the open halves is equivalent to:
∧ ∨
[ ⊥  v ] or [ v  > ] ∧ [ ⊥  u ] or [ u  > ].
∨ ∧
Given an element c in a continuously ordered wedge let hu, vi = hc, c i and iterate
(forever) the non-deterministic parallel contrapuntal procedure:

∧ ∨
 
If ⊥  v If u  >
   
If ⊥  u and v  >
 emit “+”;  
 ||  emit “ ◦”;   emit “−”; 
 || 

 replace hu, vi with   replace hu, vi with 
  replace hu, vi with 
 
∧ ∨ ∧ ↔ ∧ ∨ ∨ ∨ ∧
hu, vi = h v , vi. hu, vi = h u, vi. hu, vi = h u, ui.

The Houdini diagram brought one important feature of the contrapuntal procedure to my
attention (13 years late). We’ll say that a signed-binary stream is an OK stream if it it does
not have a bad tail, to wit, an infinite stream of all +s or all −s but is not all +s or all −s
(that is, it is not one of the two edge streams, to wit, the unique streams for > and ⊥).
30.2 Lemma: Streams produced by the contrapuntal procedure are OK streams.
(This wonderful fact turns out to solve a number of irritating problems.) Suppose, first, that
an output stream were to end with the bad tail ◦+++· · ·. That first ◦ says that the procedure
determined that it was at a node where the second alternative is viable, that is, where the
rest of the output steam lies in the middle open half. Any initial segment of that infinite
stream does just that. But the bad tail in question converges to a point not in the middle
open half, hence that first ◦ would not have been possible. A similar argument holds for the
possibility of + − − − · · ·. That first + says that the first alternative is viable, that is, the
element being described lies in the top open half. But the bad tail in question describes an
element not in the top open half.[142]
[ 142 ] It is worth checking that even if we were to start with a stream procedure the output will not end with a bad tail.
∨ ∧
Suppose the input sequence is c = − + + + + · · · . Then h c, ci = h+ + + · · · , − − − · · ·i and only the second alternative holds
∨ ∧
hence the output is ◦ and—since both c and c are zooming fixed-points—all further further outputs will be ◦. Suppose—
∨ ∧
instead—that the input sequence is c = ◦ ++++ · · · . Then h c, ci = h+++ · · · , ◦◦◦ · · ·i and only the first alternative holds
hence the output is + and the pair is replaced, again, with h+++ · · · , −−− · · ·i and all further outputs will, again, be ◦.
See Section 43 (p??–??) for a way of making a much simpler finite automaton that removes bad tails.

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We can use even more. First, though, a definition. Note that in the Houdini diagram a
node can have 0,1 or 2 incoming edges. We need a name, in particular for the case of a single
incoming edge. We’ll call them monodes. A monode is named either by a finite word ending
in ◦ or it is a node on one of the two outer edges (the “one” excludes the root).
It turns out to be important that not only can we eliminate bad tails but we can eliminate
all streams with only finitely many monodes. We’ll call those with infinitely many monodes
good streams.
The following finite automaton converts any stream without a bad tail into a good stream
that names the same element of I. The states are S− , S◦ (the initial state) and S+ .

Next State Output

S− S◦ S+ S− S◦ S+
+ S− S+ S+ + ◦ +◦ +
◦ S◦ S◦ S◦ ◦ −◦ ◦ +◦
− S− S− S+ − − ◦
◦ “◦”

◦ “−◦” ◦ “+◦”



< <

− “−” + “+”
<

S−< S◦ S+
<
<
+ “◦” − “◦”



<
− +

The states may be interpreted as follows: in state S◦ the the input stream (so far) is
numerically equal to the present output stream. Whenever we leave S◦ a stammer occurs and
the machine moves to either S+ or S− ; in S+ the input stream is larger than the output; in
S− it’s smaller. Whenever we return to S◦ a stutter occurs: two output digits. The machine
is never more than one output digit behind the number of input pairs (that is, between every
pair of stammers there’s a stutter).[143]
Let α ∈ {−, ◦, +}∗ be a finite word of signed binary digits. The scope of the node
(α) is the regular open subinterval of I with endpoints α −−−− · · · and
defined by α, Sc (α),
α++++ · · ·. It is none other than the subset of I reachable by streams without bad tails that
are continuations of α. (Note that any regular open subinterval of I that shares an endpoint
with I must include that endpoint.) This is the critical use of the absence of bad tails. If all
streams are allowed we reach a closed subinterval from a given node, hence I saw no good
way of using the contrapuntal procedure—the very foundation of the finality of the standard
closed interval—as a tool for the construction of the standard interval.

[ 143 ] A non-stuttering machine (with 4 states, one of which, I, is strict initial) is available:
Next State Output
I S− S◦ S+ I S− S◦ S+
+ S+ S− S+ S+ + ◦ ◦ +
◦ S◦ S◦ S◦ S◦ ◦ − ◦ +
− S− S− S− S+ − − ◦ ◦

A stammer occurs at the very beginning, thereafter it is always exactly one output digit behind the number of input pairs;
alternatively, a machine that doesn’t stammer until it has to:
Next State Output
I S− S◦ S+ I S− S◦ S+
+ S+ S− S+ S+ + ◦ ◦ +
◦ I S◦ S◦ S◦ ◦ ◦ − ◦ +
− S− S− S− S+ − − ◦ ◦

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The set of nodes, N, is isomorphic to the set of natural numbers.[144] Given an interval
coalgebra C and an element c ∈ C we obtain the subset of nodes reachable by the contrapuntal
procedure with c as input. We wish to characterize those subsets.
In a topos with natural numbers object we can view the output, therefore, as a map
D : N → Ω. Given a finite word α of signed binary digits we may view D(α) as the “truth-
value” that the node named by α is in the subset (described by D). We’ll call D(α) the
domain of the node named by D, which function will be called the domain function.
The Lawvere test says that we need a condition on domain functions so that in Sh(X)
domain functions are in one-to-one correspondence with continuous maps f : X → I.
The only known way of eliminating the evil of bad tails is to use the plenitude of monodes.
We’ll use the conventions that λ names the empty word and that α ≺ β means that for
β = αγ for some non-empty γ that ends with a monode:
The output of the contrapuntal procedure can be described as a map D : N → Ω satisfying
the two conditions: D(λ) = True
W
D(α) ∧ D(β) = { D(δ) : α ≺ δ ∧ β ≺ δ }

(Keep in mind: D(α) = D(α) ∧ D(α).) We’ll call such functions Dedekind sutures.
In Sh(X) we can rewrite the conditions (Ω can be taken as the set of open subsets of X):
D(λ) = X
S
D(α) ∩ D(β) = { D(δ) : α ≺ δ ∧ β ≺ δ }

Given continuous f : X → [−1, +1] define D : N → Ω by D(α) = f -1 (Sc(α)). The

verification that such is a Dedekind suture is routine.
Given a Dedekind suture D : N → Ω define f : X T→
 [−1, +1] by taking f (x) to be the
unique element in f (x) to be the unique element in Sc(α) : x ∈ D(α) .
The fact that we do obtain an element, that is, the fact that the intersection of the good
scopes is non-empty, that fact is a critical use of the plenitude of monodes. Suppose their
intersection were empty. Replace each interval with its closure; the unique element in their
intersection is necessarily one of the new endpoints, hence necessarily a dyadic rational and
one not equal to ⊥ or >; but the only good streams converging to such are eventually all zero
and the dyadic rational is not a new endpoint; indeed it’s eventually the center of each of the
scopes.
For the continuity of f we use the easily verified facts that 1) the good-stream scopes form
a basis for the topology of the standard interval and that 2) f -1 (Sc(α)) = D(α) for all α.
If one starts with a continuous function, f, and proceeds to the construction of its domain
function D, it is easy to verify that the inverse images of the good-stream scopes of the
function constructed from D are, of course, the same as the inverse images of f. And in the
other direction, it is also routine that if one starts with a Dedekind suture, D, and constructs
a continuous function f : X → I then the domain function of f is D.[145]
[ 144 ] One choice of canonical names is the Kleene-regular set of words {−, +}∗ { }∗
◦ , that is, finite ◦-free words followed by
finite strings of ◦s. (Every dyadic rational strictly between −1 and +1 is described by a unique finite ◦-free word.) For a specific
isomorphism define N → N by sending (2n + 1)2m − 1 to the node reached by the word (f n)◦m where f : N → {+, −}∗ is
the unique function such that f 0 is the empty word, f (2n + 1) = (f n)+ and f (2n + 2) = (f n) − . Cf. [??] (p??).
[ 145 ] The “disjunction property” obviates the need for dependent choice. Given a space X and a continuous coalgebra C in

the category of sheaves Sh(X) and a partial section U → C we seek continuous f : U → I. (Dependent choice holds only
when X is totally disconnected.) For each x ∈ X we are prompted to pass to the category of “micro-sheaves” at x , to wit,

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Since we described the automata for zooming automata in the discrete case, we close
this section with the automata for the signed-binary-digit setting. The diagram below is for
>-zooming; negate the 6 signatures for ⊥-zooming.
Next State >-Zoom Output ⊥-Zoom Output
S− S◦ S+ S− S◦ S+ S− S◦ S+
+ S− S+ S+ + − + + + + +
◦ S− S◦ S+ ◦ − − ◦ ◦ ◦ + +
− S− S− S+ − − − − − − +
◦ “−”





< “−” <


<
echo all input S+ < + S◦ −
<
S− “−” for all input



In these machines at most one stammer occurs; restated, the output is never more than
one digit behind the input.[146]
It’s worth noting that no automaton, finite or not, can compute midpoints (or, as usually
pointed out, sums) in the context of streams of standard (unsigned) binary digits. If at any
point only heterogeneous pairs of standard digits (one 0, one 1) have been heard then we do
not know whether the result will be in the upper or lower open half of the unit interval and
if we are restricted to the digits 0 and 1 we can not specify its first digit. If, perchance, all
digit-pairs are heterogeneous we will never be able to compute the first digit.
There is a remarkably simple automaton, on the other hand, for midpointing signed binary
digits. See Section 43 (p??–??) for the scale-structure automata of signed binary expansions.

31. Appendix: Dedekind Cuts

If we wish to avoid the axiom of dependent choice, one approach is to use Dedekind cuts. As
previously noted, if experience with topoi is any guide we know in advance what I should
turn out to be in the category of sheaves over a topological space X, to wit, the sheaf of
continuous I-valued functions on X, that is, the sheaf whose stalks are germs of continuous
functions (as defined in the topos of sets) from X to I. There are a number of ways of define
a Dedekind cut—all of them equivalent in the category of sets—but not all give give the right
answer in a category of sheaves.
the result of identifying things when they agree when restricted to some neighborhood of x. In that category we are entitled to
view the partial section as global, a map from 1 to C . What we gain is the “disjunction property”: if the disjunction of two
(or, more to the point, three) sentences is true then one of them is already true. We may now repeat the above procedure to
obtain an unending stream of signed binary digits. Continuity is left as an exercise.
[ 146 ] There are four-state automata with strict initial states, I, that always stammer at the first input digit and never thereafter:

Next State >-Zoom Output ⊥-Zoom Output

I L M U I L M U I L M U
+ U L U U + + + + + − − +
◦ M L M U ◦ ◦ + + ◦ − − ◦
− L L L U − − + + − − − −

Six states are required for a mid-zoom machine and its unique stammer is also at the beginning:

Next State Output

I L2 L1 M U1 U2 I L2 L1 M U1 U2
+ U1 L2 M M U2 U2 + − − + + +
◦ M L2 L1 M U1 U2 ◦ − − ◦ + +
− U1 L2 L2 M M U2 − − − − + +

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When working with I rather than D it is convenient to define Dedekind cuts as subsets
◦
not of I but of its “interior,” I , the result of removing the two endpoints.[147]
◦
We say that L ⊆ I is a downdeal if:
` ∈ L ⇒ ∀`0 <` `0 ∈ L
and it is an open downdeal if, further:
` ∈ L ⇒ ∃`0 >` `0 ∈ L
We say that U is an updeal if
u ∈ U ⇒ ∀u0 >u u0 ∈ U
and it is an open updeal if, further
u ∈ U ⇒ ∃u0 <u u0 ∈ U
In the case of sheaves on X we may reinterpret L and U as a families of open subsets of X
◦
indexed by elements of I (that is, L gives the “extent to which ` ∈ L”). The conditions
then rewrite to: [ [
L` = L`0 Uu = Uu0
`0 >` u0 <u

Note that for any downdeal L, open or not, the largest open downdeal contained therein is
◦ n o
L = ` : ∃`0 ∈L `0 > `

and the largest open updeal contained in an updeal U is a

◦ n o
0 0
U = u : ∃ 0
u ∈U u < u

When working in sheaves these correspond to

◦ [ ◦ [
L` = L`0 Uu = Uu0
`0 >` u0 <u

For any lower semicontinuous map f : X → I we obtain an open downdeal by defin-

◦
ing L` = f -1 (`, >] ∩ I and for any upper semicontinuous g we obtain an open updeal
◦
with Uu = g -1 [⊥, u) ∩ I . Conversely, given an open downdeal, L, define f (x) to be
sup ` : x ∈ L` and given an open updeal, U, define g(x) = inf u : x ∈ Uu . It is easy to
check that these assignments establish a correspondence between upper/lower semicontinuous
functions and open up/down-deals.
In the classical setting there are several ways of defining I: as the set of open downdeals; as
the set of open updeals; as the set of pairs hL, U i where L is maximal among those downdeals
disjoint from U and U is maximal among those disjoint from L. In the more general setting,
none of these are guaranteed to satisfy the separation condition: on any space X and open
set V take L` to be constantly equal to V . The extent to which L is > is V and the extent to
which it is different from > is ¬V, to wit, its exterior (defined as the largest open set disjoint
from V, the interior of its complement). The extent to which L is not ⊥ is ¬¬V (the interior
◦
[ 147 ]
Bear in mind that I and I are isomorphic—as objects—to the natural numbers N. For a specific isomorphism define
◦ h(n)+1 h(n)−1
h : N → I to be the unique function such that h(0) = 0, h(2n + 1) = 2
and h(2n + 2) = 2
. Cf. [??] (p??).

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of the closure of V ). The union ¬V ∪ ¬¬V fails, in general, to be all of X. Note that the
maximal open updeal disjoint from L is U where Uu is constantly ¬V. The maximal open
downdeal disjoint from U is ¬V. If we take any “adjoint pair” (p??) of semicontinuous maps
hf, gi and x ∈ X such that f (x) < g(x) we may let r = f (x)|g(x) and apply a suitable power
of the dilatation at r to obtain an adjoint pair h(r/)n f, (r/)n gi that will recreate the same
sort of pathology. We need, in other words. a condition on hL, U i that will force f = g.
Given any function h : X → I there is a maximal lower semicontinuous f : X → I below
h and a minimal upper semicontinuous g above h. (To obtain f define a downdeal L by first
◦
taking L` to be the interior of h-1 (`, >] and then replacing L` with L` .) Of course a function
both upper and lower semicontinuous is plain continuous.
◦
A Dedekind cut on I is a pair of subsets hL, U i such that:

L is an open downdeal: ` ∈ L ⇒ [∀`0 <` `0 ∈ L] & [∃`0 >` `0 ∈ L]

U is an open updeal: u ∈ U ⇒ [∀u0 >u u0 ∈ U ] & [∃u0 <u u0 ∈ U ]
◦
L and U disjoint: q∈I ⇒ ¬[(q ∈ L) & (q ∈ U )]
L and U almost cover: ` < u ⇒ [` ∈ L] or [u ∈ U ]

(L and U each determines the other: given L then U = u : ∃u0 <u u0 6∈ L . The conditions

are, in fact, redundant: the 3rd and 4th conditions imply that L is a downdeal and U an updeal
(but not the openness condition).)
In the case of sheaves on X the conditions rewrite to:
S
Uu = u0 <u Uu0
S
L` = `0 >` L`0
Lq ∩ Uq = ∅
` < u ⇒ L` ∪ Uu = X
For any continuous f : X → I we obtain such a cut be defining L` = f -1 (`, >] and Uu =
f -1
[⊥, u). All Dedekind cuts so arise: given the L` s and Uu s define f : X → I by f (x) =
inf u : x ∈ Uu and verify that f is continuous (the key observation is that the closure of
0
 T T
Uu is contained in Uu0 whenever u < u , hence x : f (x) ≤
= u = u0 >u Uu0 = u0 >u U u0 is
closed and, dually, x : f (x) ≥= u is open).
◦
We define I to be the set of such Dedekind cuts. The bottom cut is h∅, I i; the top cut is
◦
hI , ∅i.
Note that hL, U i =
6 ⊥ iff L is non-empty and, dually, hL, U i =
6 > iff U is non-empty. The
almost-cover condition for the case ⊥ < > is precisely the separation condition for the set of
Dedekind cuts.
Define ∨ ∨ ∨
hL, U i = h { ` : > ` ∈ L}, {u : > u ∈ Ui }
∧ ∧ ∧
hL, U i = h { ` : < ` ∈ L}, {u : < u ∈ Ui }
∨ ∧
The Dedekind-cut conditions for hL, U i and hL, U i are pretty routine. (For the almost-
cover condition, given ` < u use the scale structure on I: since (>|`) < (>|u) we know that
either (>|`) ∈ L or (>|u) ∈ U . In the first case = (>|`)∧ ∈ L and in the second case
≤
< (>|u)∧ ∈ U .)

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The disjointness condition for the case is precisely the continuous coalgebra condition
for these operations.
Given an interval coalgebra, C, satisfying the separation and continuous coalgebra condi-
tion, and given c ∈ C, we wish to construct a Dedekind cut hL(c), U (c)i. By a
zooming sequence is meant an element of the free monoid on two generators, >-zooming
and ⊥-zooming.
` ∈ L(c) iff there is a zooming sequence α such that `α = ⊥ and ¬¬[c α = >].
u ∈ U (c) iff there is a zooming sequence α such that uα = > and ¬¬[c α = ⊥].
We need to verify the conditions for a Dedekind cut.
Before doing so let us pause to collect a few easily verified observations in the intuitionistic
setting. For any function, f, it is, of course, trivial that (x = y) ⇒ (f x = f y). Because
negation is contravariant we also have (f x 6= f y) ⇒ (x 6= y) and ¬¬(x = y) ⇒ ¬¬(f x = f y).
We will apply these trivial observations to the case when f is a zooming sequence and
incorporate the fact that > and ⊥ are fixed-points. Hence
x=> ⇒ xα = >
x=⊥ ⇒ xα = ⊥
xα 6= > ⇒ x 6= >
xα 6= ⊥ ⇒ x 6= ⊥
¬¬[x = ⊥] ⇒ ¬¬[xα = ⊥]
¬¬[x = ⊥] ⇒ ¬¬[xα = ⊥]
[148]
We will also use these trivial consequences of the apartness of > and ⊥:
¬¬[x = ⊥] ⇒ x 6= >
[??]
¬¬[x = >] ⇒ x 6= ⊥

And we will freely use all sorts of nice properties enjoyed by I (including the discrete coalgebra
condition).
Not so trivial is this critical lemma:
31.1 Lemma: For c ∈ C and u ∈ I the following conditions on c ∈ C and u ∈ I are
equivalent:
α : ∃α [¬¬(c α = ⊥) & (u α = >)]
 
β : ∃β ¬¬(c β = ⊥) & (u β 6= ⊥)
γ : ∃γ,v [(cγ 6= >) & (v < u) & (v γ = >)]
(Note that the γ-condition will tend to be much more computationally feasible than the other
two.) We need three implications:
α ⇒ γ:
Given α let v be the unique element in I such that v α = and γ the result of following
α with a ⊥-zooming (and use (¬¬(c α = ⊥) ⇒ ¬¬(cγ = ⊥) ⇒ (cγ 6= >).)
γ ⇒ β:
Given γ and v, we may assume that γ is the minimal zooming sequence for the task.
We know that it is non-empty since v < > and v γ = >. If γ ends with an >-zooming
then the sequence obtained by removing that final >-zooming would work as well. Thus
h i h i h i
[ 148 ] ¬¬[x = ⊥] ⇒ ¬[x 6= ⊥] ⇒ ¬[x 6= ⊥] ∧ [x 6= ⊥] ∨ [x 6= >] ⇒ [¬[x 6= ⊥] ∧ [x 6= ⊥] ∨ ¬[x 6= ⊥] ∧ [x 6= >] ⇒
h i
¬[x 6= ⊥] ∧ [x 6= >] ⇒ [x 6= >]

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from its minimality we may infer that γ ends with a ⊥-zooming. Let γ 0 be the zooming
0 0
sequence obtained by removing that final ⊥-zooming. Since v γ ∨ = > we know that v γ ≥ = ,
γ 0 0
γ∧ 0
γ∨
hence u > and u 6= ⊥. Since c 6= > the continuous zooming condition says that
0
¬¬[c γ ∧ = ⊥]. Thus we finish by defining β to be the result of following γ 0 with a >-zooming.
β ⇒ α:
Define α to be the result of following β with a sufficient number of ⊥-zoomings to insure
α
u = >.

Now for the Dedekind-cut conditions.

U (c) is an open updeal:
Suppose u ∈ U (c). Let γ, v be such that cγ 6= >, v < u and v γ = >. Then for any u0 > v
we have the same three conditions with u0 instead of u, hence u0 ∈ U (c) for all u0 > v.
L(c) and U (c) almost cover:
Given ` < u choose ` < k < v < u. [149] Let γ be a zooming sequence (say the shortest
one) such that k γ = ⊥ and v γ = >. The separation condition on C says that either cγ 6= > or
cγ 6= ⊥. In the first case we have u ∈ U (c) and, dually, in the second case ` ∈ L(c).
L(c) and U (c) disjoint:
We wish to reach a contradiction from the assumption that there is c ∈ A, q ∈ I and
zooming sequences σ, τ such that ¬¬(c σ = ⊥), q σ = >, ¬¬(c τ = >) and q τ = ⊥. We will
settle for a weaker condition: ¬¬(c σ = ⊥) implies c σ 6= > and ¬¬(c τ = >) implies c τ 6= ⊥.
That is, we will reach a contradiction just from c σ 6= >, q σ = >, c τ 6= ⊥ and q τ = ⊥.
If σ were empty then q = and it would not be possible for q τ = ⊥. Dually, τ is non-
>
∧ ∨
empty. If σ were to start with >-zooming we would know that q 6= ⊥ forcing q = > (the
discrete coalgebra condition holds in I) and thus q τ = >, contradicting q τ = ⊥. Hence σ
∨
starts with ⊥-zooming and, dually, τ with >-zooming. From c σ 6= > we may infer c 6= > and
∧
from c τ 6= ⊥ we infer c 6= ⊥. But the conjunctions of these two 6= s is precisely what the
continuous coalgebra condition says can not happen.
We must show that this assignment of Dedekind cuts preserves the coalgebra structure.
There is no difficulty in showing that hL(⊥), U (⊥)i and hL(>), U (>)i are what they should
∧ ∧ ∧
be. What we must show is hL(c), U (c)i = hL(c), U (c)i (the other equation, of course, is dual).
Restated: we must show
∧ ∧
` ∈ L(c) iff ≤
= ` ∈ L(c)
∧ ∧
u ∈ U (c) iff < u ∈ U (c)
∧α
The forward directions are immediate: if α is a zooming sequence such that ¬¬(c = >) and
∧α 0
` = ⊥ then if α0 is the result of following an >-zooming with α we have ¬¬(c α = >) and
0 ∧ ∧
` α = ⊥ (and if ` < then replace it with ). The same argument works when u ∈ U (c)
∧ α
(and since u = > we know that u > ).
= , ¬¬(c α = >) and ` α = ⊥. Then α is necessarily
For the reverse direction, suppose ` ≥
non-empty (that is, ` 6= ⊥) and it can not start with a ⊥-zooming (since ` ≥= implies
[ 149 ] For example, k0 = `|u, v = k|u.

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∨ ∧ α0
` = >). Let α0 be the rest of the sequence after the initial >-zooming. Then ¬¬(c = >)
∧ 0 ∧ ∧
and ` α = ⊥ forcing ` ∈ L(c)
Finally, suppose u > , ¬¬(cα = ⊥) and uα = >. If α is empty then the empty sequence
∧ ∧
also establishes u ∈ U (c). If α starts with a >-zooming we use the same sort of argument
∧ ∧
just above to establish u ∈ U (c). If α starts with a ⊥-zooming then for any u > it is the
∧
case that uα = >, hence we need to show that everything in I other than ⊥ is in U (a). But
∨
we may infer ¬¬(cα = ⊥) ⇒ (cα 6= ⊥) ⇒ (c 6= ⊥) and the continuous coalgebra condition
∧ ∧β ∧β
says that ¬¬(c = >). For β the empty sequence we thus have ¬¬(c = >) and u 6= ⊥ forcing
∧ ∧
u ∈ U (c).

32. Appendix: The Peneproximate Origins

I always disliked analysis. Algebra, geometry, topology, even formal logic, they captivated
me; analysis was different.
My attitude, alas, wasn’t improved when I was supposed to tell a class of Princeton
freshmen about numerical integration. I was expected to tell them that trapezoids were
better than Riemann and Simpson was better than trapezoids. I was not expected to prove
any of this.
I was appalled by the gap between applied mathematical experience and what we could
even imagine proving. How does one integrate over all continuous functions to arrive at the
expected error of a particular method?
Of course one can carve out finite dimensional vector spaces of continuous functions and
compute an expected error thereon. But all continuous functions? It’s easy to prove that there
is no measure—not even a finitely additive measure—on the set of all continuous functions
assuming at least that we ask for even a few of the most innocuous of invariance properties.
Yet experience said that there was, indeed, such a measure on the set of functions one actually
encounters.
But it wasn’t just a problem in mathematics: I learned from physicists that they succeed
in coming to verifiable conclusions by pretending to integrate over the set of all paths between
two points. Again it is not hard to prove that no such “Feynman integral” is possible once
one insists on a few invariance properties.
Even later I learned (from the work of David Mumford) about “Bayesian vision”: in
this case one wants to integrate over all possible “scenes” in order to deduce the most
probable interpretations of what is being seen. A scene is taken to be a function from, say,
a square to shades of gray. It would be a mistake to restrict to continuous functions—sharp
contour boundaries surely want to exist. Quite remarkable “robotic vision” machines had been
constructed for specific purposes by judiciously cutting down to appropriate finite-dimensional
vector spaces of scenes. But once again, there is no possible measure on sets of all scenes which
enjoy even the simplest of invariance conditions.
Thus three examples coming with quite disparate origins—math, science, engineering—
were shouting that we need a new approach to measure theory.
One line of hope arose from the observation that the non-existence proofs all require a
very classical foundation. There’s the enticing possibility that a more computationally realistic
setting—as offered, say, by “effective topoi”—could resolve the difficulties. A wonderful dream

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presents itself: the role of foundations in mathematics—and its applications—could undergo

a transformation similar to the last two centuries’ transformation of geometry.
Geometry moved from fixed rigidity to remarkable flexibility and—in the last century—
that liberal view became a critical tool in physics. We learned that there was a trade-off
between physics and geometry; we could still insist on classical (Euclidean) geometry but
only at the expense of a cumbersome physics. We no longer even view most questions on the
nature of geometry to be well put unless first the nature of physics be stipulated and—of
course—vice versa.
Could we now learn the same about foundations? Elementary topoi provide a general
setting for shifting foundations reminiscent of the role of Riemannian manifolds in geometry.
Might the trade-off between physics and geometry be replicated for physics and foundations?
Two hundred years ago there was only one geometry. It was more than taken for granted; it
was deemed to be certain knowledge.
Of course the geometry we now call Euclidean was certain; it may not be innate but it is
inevitable. I have no doubt that if we lived in a universe with a visibly non-zero curvature
we would get around to building our blackboards (or whatever we teach calculus with) with
zero curvature.[150]
It must be deemed remarkable that we learned to think—and make correct predictions—in
non-Euclidean geometry. We learned to imagine living in a 3-sphere, in spaces of higher genus,
even in projective space. The representation theorems for Riemannian manifolds (long before
they were all proved) played a critical role in that process; and so it is with foundations. Bill
Lawvere taught us that with a few topoi on hand for comparison we can learn to shift our
foundations between what’s called classical and what’s called (alas) intuitionistic. Again, the
representation theorems play a critical role: in a fully classical setting a category of sheaves
on a space can support a fully intuitionistic logic—change the topology and you can revert
to the classical.
Coming back to earth: I must confess that the perfectly obvious idea that one should
first establish ordinary integration in the right way on something as simple as the closed
interval, that simple idea took longer than it should have (it had to await a day’s boat trip
in Alaska, of all places). For some years I preached this doctrine to the category/topos crowd
and some trace of those preachings can be found scattered in the literature.[151] In September
1999 at an invited talk at the annual ctcs meeting (held that year in Edinburgh) I even
characterized the mean value of real-valued continuous functions on the closed interval as an
order-preserving linear operation that did the right thing to constants and had the property
that the mean value on the entire interval equaled the midpoint of the mean-values on the
two half intervals. I described it with a diagram that used (twice) the canonical equivalence
between I and I ∨ I.
But one equivalence, even used twice, doesn’t bring forth the general notion: it doesn’t
prompt one to invent ordered wedges; without ordered wedges one doesn’t define zoom
operators nor discover the theorem on the existence of standard models (Theorem 10.5,
p??). One doesn’t learn how remarkably algebraic real analysis can become.
What I needed was someone to kick me into coalgebra mode. Three months later two guys
did just that and on the 22nd day of December I wrote to the category list:
[ 150 ] In this neighborhood, of course, Euclidean geometry is the natural geometry from the very beginning. When I was a kid
a friend had measured the distance around a giant tree. We estimated the tree’s width by solving the same problem on a little
fruit-juice glass. We never questioned that the same ratio would hold for giant trees and little fruit-juice glasses.
[ 151 ] e.g., Abbas Edalat and Martı́n Hötzel Escardó, Integration in real PCF, LICS 1996, Part I (New Brunswick, NJ).

Information and Computation 160 (2000), no. 1–2, p128–166.

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There’s a nice paper by Duško Pavlović and Vaughan Pratt. It’s entitled
On Coalgebra of Real Numbers [152] and it has turned me on.

A solution, alas, for the three motivating problems still awaits; but, at least, now I like
analysis.

33. Addendum: A Few Latex Macros

I
\CI: \scalebox{1.14}{\ensuremath{\tt I}}
Requires graphics package. In its absence use the ossia, {\ensuremath{\tt I}} .
There are those who insist that this portrays a copulation of > and ⊥. (Indeed,
they go on to say that about the entire subject.)

⊥ >

\bt: \scalebox{.83}{\ensuremath{\bot}}
\tp: \scalebox{.8}{\ensuremath{\top}}
Ossias: {\ensuremath{\bot}} and {\ensuremath{\top}}
Z
R

\Fint: \;{\rotatebox{103}{\scalebox{.55}{$\int$}}}\hspace{-4.23mm}\int
The main “\int” is delayed until the end so that as a macro it accepts sub- and
superscripts. A less than satisfactory ossia: \;-\hspace{-4.5mm}\int
[1] [12] [123]

\fna[1]: {\ensuremath{^[\footnote{\hs{-6.5}$^[\hs{3.75}^]$\hs4#1}^]}}
\fnb[1]: {\ensuremath{^[\footnote{\hs{-10.25}$^[\hs{7.75}^]$\hs4#1}^]}}
\fnc[1]: {\ensuremath{^[\footnote{\hs{-14}$^[\hs{11.75}^]$\hs4#1}^]}}
(\hs{x} means \hspace{xpt}.) Its raison d’être is for use at the end of math display lines
(and should, in that case, usually be preceded with \;\;). Do not use for an asterisk at the
end of the title line. Other delimiters, of course, could be used: {1} h2i |3| b4e d5c
Ossia: \fn[1]: {\ensuremath{^[\footnote{\,#1}^]}}
∧ ∨ ↔
x x x

\tz[1]: \stackrel{\wedge}{#1} \bz[1]: \stackrel{\vee}{#1}

\mz[1]: \stackrel{\hspace{.1}\scalebox{.7}{$\leftrightarrow$}}{#1}
Use \hat x and \check x in subscripts: x̂ x̌.
. .
u (x + y)

\dt[1]: \stackrel{\mbox{\bf.}}{#1}
\hdot: \begin{picture}(0,8)\put(-1,8.1){\bf.}\end{picture}
Use \dot x in subscripts: ẋ. In footnotes use \put(-1,6) instead of \put(-2,8.1).
[ 152 ] Later published as The continuum as a final coalgebra CMCS’99 Coalgebraic methods in computer science (Amsterdam,

1999). Theoret. Comput. Sci. 280 (2002), no. 1-2, p105–122.

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∧ ∨
| | |
\mtz: scalebox{.7}[.9]{\begin{picture}(9,15)\put(2.4,0){$|$}}
\put(1,9){\scalebox{.8}{$\wedge$}}\end{picture}}
\mbz: \scalebox{.7}[.9]{\begin{picture}(9,15)\put(2.4,0){$|$}
\put(.8,9.1){\scalebox{.8}{$\vee$}}\end{picture}}
In footnotes:
\ftz: \scalebox{.7}[.9]{\begin{picture}(9,11)\put(2.4,0){$|$}
\put(1.05,6){\scalebox{.9}{$\wedge$}}\end{picture}}}
\fbz: \scalebox{.7}[.9]{\begin{picture}(9,11)\put(2.4,0){$|$}
\put(1.05,6.5){\scalebox{.9}{$\vee$}}\end{picture}}}
Ossias: \!\stackrel{\wedge}{|}\! and \!\stackrel{\vee}{|}\!
\md: \mbox{\huge${\mbox{\Large$|$}}$} In footnotes use large not Large.
\wmd: \;\md\; Used in formulas such as ((⊥|>)|(⊥| )) | (⊥|x)
◦
I I
\IC: \ensuremath{\mathbb I}
\ic: \begin{picture}(9,12)\put(0,0){$\IC$}\put(.2,9)
{\scalebox{.7}{$\circ$}}\end{picture} In footnotes use:
\ics \begin{picture}(6,9)\put(0,0){$\IC$}\put(.15,6.5)
\scalebox{.7}{$\circ$}}\end{picture}}
x≤
=y ≥y
x= x<>y x> <y
\lq \begin{picture}(17,0)\put(3.65,0){$\leq$}\put(4,-1.9)
{$\color{white}\rule{8.5pt}{5.8pt}$}\put(4,0){$=$}\end{picture}}
For \gq replace (3.65,0) with (4,0) and \leq with \geq
\lg: \begin{picture}(13,0)\put(1.7,2.9){$<$}
\put(2,-2.4){$>$}\end{picture} For \gl swap > and <

x-1 y -m z -2
\i: \def\i{\inv}\newcommand{\inv}[1]
{^{{\scalebox{1.3}[.76]{-}}\hs{-.5}#1}}
Use x\i1y\i mz\i2 instead of x^{-1}y^{-m}z^{-2} (x−1 y −m z −2 )
&

\Par: \begin{picture}(14,0)\put(2,7.5){\scalebox{-.9}{\&}}\end{picture}
Ossia: complain to Jean-Yves.
•
x −◦ y x ◦−
−◦ y x ◦−
− ◦y
\loli: \begin{picture}(20,0)\put(3,0){$-$}\put(4,0)
{$-$}\put(11.4,0){$\circ$}\end{picture}
In footnotes use \put(10,0) instead of \put(11.4,0) : −
−◦ .
\bimp: \begin{picture}(23,0)\put(2,0){$\circ$}
\put(3,0){$\loli$}\end{picture}
\abs: \stackrel{^\bullet}{\bimp}
◦ +−◦◦
\z: \scalebox{1.3}{\begin{picture}(6,0)
\put(0,-.6){$\circ$}\end{picture}}
For signed-binary and “symmetric ternary” expansions

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<



<


\lloop[1]: \scalebox{1.5}{\rotatebox{#1}{\begin{picture}(0,0)\put(0,0)
{\circle{16}}\put(-8.32,0){$\color{white}\rule{8.32pt}{8.3pt}$}
\put(-1.3,6.9){\rotatebox{-13}{\scalebox{.5}{$<$}}}\end{picture}}}
For the curlyheaded variation (as on page ??) replace last line with:
\put(-2.5,6.9){\rotatebox{-9}{\scalebox{.5}{$\prec$}}}\end{picture}}}
\rloop[1]: \reflectbox{\lloop{#1}}
[[w]] ((x)) hhyii bzc
\banana[1]: \,[\![#1]\!\!\,]
\pr[1]: \,(\!(#1\,)\!\!\,)
\Ang[1]: \langle\!\langle#1\rangle\!\rangle
\tr[1]: \lfloor#1\rfloor (and don’t forget \|v\| for kvk )

\fq: begin{picture}(9,0)\put(0,0){$\ct$}\put(0,3.75)
{\color{white}\rule{9pt}{5pt}}\end{picture}} For \tq replace 3.75 with
−2.85. Only with great reluctance did I forgo using the standard calendar lunar symbols
for first- and third-quarter. Ossias: \mbox{\small Q}_1 and \mbox{\small Q}_3

\venturi: \scalebox{.8}{\begin{picture}(17,7)\qbezier(5,5.5)(6,4)(12,4)
\qbezier(5,.5)(6,2)(12,2)\end{picture}}

34. Addendum: Heyting Scales [2009–04–05]

In [??] (p??) we pointed out that the support operator of a chromatic scale may be defined
using the Heyting structure (which was defined using the chromatic structure). But that does
not establish that any scale with a Heyting structure is, in fact, a chromatic scale. Hence this
addendum.
In fact, we use less than the full Heyting structure. For the chromatic structure it suffices
that a scale be a “negated scale” and that is precisely what allows us to show that an
order-complete scale is chromatic (see below).
Let’s establish some definitions.
The quickest definition of a Heyting semi-lattice (for a category theorist) is a poset
which when viewed as a category is exponential (sometimes “cartesian closed”). That is, a
meet semi-lattice with top and a binary operation whose values are denoted y → z charac-
terized by the “adjointness condition:”
x ≤
= y → z iff x ∧ y ≤
= z
Put another way, y → z is the largest element whose meet with y is bounded
by z. [153]
[ 153 ] The adjointness condition is equivalent to the three equations:
>→z = z
(y ∧ z) → z = >
x ∧ (y → z) = x ∧ ((x ∧ y) → z)

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A Heyting algebra is a lattice with bottom that is a Heyting semi-lattice. Any linearly
ordered set with top and bottom is such:

> if u ≤=v
u→v =
v if v < u

Clearly then, in any linear chromatic scale u → v is constructable as (u −◦ v) ∨ v. The

linear representation theorem for chromatic scales tells us that the adjointness condition
that characterizes the Heyting arrow holds for all chromatic scales (as do all universal Horn
sentences).
It should be noted, though, that the Heyting algebras that so appear are rather special.
They all satisfy the “equation of linearity” (u → v) ∨ (v → u) = >. There are no further
equations, or for that matter, universally quantified first-order properties: any countable
Heyting algebra that satisfies the equation of linearity may be faithfully represented in a
power of I (it’s not hard to show that sdi Heyting algebras enjoy the disjunction property).
The lemma that any scale with a Heyting structure is a chromatic scale uses less than
the entire Heyting structure: we need only the arrow operation when targeted at the bottom,
(x → ⊥). So:
Define a negated semi-lattice to be a meet semi-lattice with top and bottom and a unary
operation, whose values are denoted ¬y, that delivers the largest element disjoint from y.
That is, it satisfies the adjointness condition:
[ 154 ]
x ≤
= ¬y iff x ∧ y = ⊥

Note that the characterization of ¬x as the largest element disjoint from x easily implies that
negation is contravariant, hence double negation is covariant. Double negation is inflationary:
x≤= ¬¬x (because x ∧ ¬x = ⊥). The contravariance of negation then says ¬¬¬x ≤ = ¬x. But
a special case of x ≤
= ¬¬x is ¬x ≤ = ¬¬¬x (it’s the case obtained by replacing x with ¬x).
Thus ¬x = ¬¬¬x. In particular double-negation is a closure operation (inflationary and
idempotent)
A consequence is:
34.1 Lemma: In negated semi-lattices x ∧ y = ⊥ iff x ∧ ¬¬y = ⊥.
Because x ∧ y = ⊥ iff x ≤
= ¬y iff x ≤
= ¬(¬¬y) iff x ∧ ¬¬y = ⊥.
An important equation for us is the Lawvere-Tierney condition on a closure operation:
34.2 Lemma:
¬¬(x ∧ y) = ¬¬x ∧ ¬¬y
The verification of the equations is as follows: for the 1st equation note that t ≤ > → z iff t ∧ > ≤ z, that is, t ≤ > → z
iff t ≤ z; for the 2nd equation note that > ∧ (y ∧ z) ≤ z hence > ≤ ((y ∧ z) → z); for the 3rd equation note that t ≤ x ∧ (y → z)
iff t ≤ x and t ∧ y ≤ z whereas t ≤ x ∧ ((x ∧ y) → z) iff t ≤ x and t ∧ (x ∧ y) ≤ z and the two conditions are clearly the
same.
For the derivation of the adjointness condition from the equations assume first that x ≤ (y → z). Then x ∧ y ≤
y ∧ (y → z) = y ∧ ((y ∧ >) → z) = y ∧ (> → z) = y ∧ z ≤ z. Second, assume x ∧ y ≤ z or, as it will appear below,
x ∧ y ∧ z = x ∧ y. Then x = x ∧ > = x ∧ ((y ∧ z) → z) = x ∧ ((x ∧ y ∧ z) → z) = x ∧ ((x ∧ y) → z) = x ∧ (y → z) ≤ y → z.
For the independence of the three equations: taking y → z as > satisfies just the 2nd and 3rd equations; taking it as ⊥
satisfies just the 1st and 3rd ; taking it as y −−◦ z in any non-trivial scale satisfies just the 1st and 2nd
.
[ 154 ] Its equational characterization is given by:
¬⊥ = >
¬> = ⊥
x ∧ ¬y = x ∧ ¬(x ∧ y)
Both the adjointness condition and the equations are obtained, of course, just by replacing the variable z with ⊥ in the equations
for Heyting semi-lattices. The equivalence of the two definitions and the independence examples are easily obtained by following
through with that replacement.

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Because the covariance of double negations easily implies (indeed, is equivalent with)
¬¬(x ∧ y) ≤
= ¬¬x ∧ ¬¬y. For the other direction, ¬¬x ∧ ¬¬y ≤= ¬¬(x ∧ y), it suffices to show
¬¬x ∧ ¬¬y ∧ ¬(x ∧ y) = ⊥. The last lemma (used twice) says that this last equation is
equivalent to the obvious equation (x ∧ y) ∧ ¬(x ∧ y) = ⊥.

Define a negated scale to be a scale with a unary operator satisfying the equations for
a negated semi-lattice.

34.3 Lemma: In a negated scale, ¬x is an extreme point.

It suffices to show (¬x)∨ ≤

= ¬x and for that it suffices to show (¬x)∨ ∧ x = ⊥. So: (¬x)∨ ∧
x ≤= (¬x) ∧ x = (¬x ∧ x)∨ = ⊥∨ = ⊥. Recall that the >- and ⊥-zooming operations have
∨ ∨

the same fixed points. In particular, (¬x)∧ = ¬x.

We have now established:

34.4 Lemma: Double negation satisfies the four defining equations for the support operation:

¬¬⊥ = ⊥
(¬¬x)∧ = ¬¬x
x ∧ ¬¬x = x
¬¬(x ∧ y) = ¬¬x ∧ ¬¬y

.
In [??] (p??) we identified x not as the double negation but as (¬x). Since ¬x is an
extreme point we know that it is complemented and its complement (we’re in a distributive
lattice recall) is clearly its maximal disjoint element.

The scale structure allows us to construct the Heyting arrow operation starting with
∧.
negation: u → v = ¬(u |v) ∨ v. It’s easy to find negated lattices that aren’t Heyting algebras,
indeed aren’t even distributive lattices: take any lattice with top and formally adjoin a bottom
element (even if it already has one); the result is a negated lattice. (A semi-lattice with
bottom, on the other hand, is a Heyting semi-lattice iff every principal filter is a negated
semi-lattice.)

We close with

34.5 Lemma: All order-complete/injective scales are negated scales, hence chromatic.

We saw in Section 24 (p??–??) that the scales in question are of the form C(X) where X is
extremely disconnected. Given f ∈ C(X) let C ⊆X be the closure of x ∈ X : f (x) > ⊥ .
Since C is a “clopen” it has a continuous characteristic function. It is easy to see that χ. C
works as ¬f. [155]

[ 155 ] For an example of a metrically complete chromatic scale that is not order-complete (hence not injective) let Π I = IJ be
J
an uncountable cartesian power of I and S ⊂ IJ the subscale of elements that with the exception of a countable subset of J
J
are equal to an element in the image of the diagonal map (to wit, the unique scale homomorphism from I to I ). The chromatic
structure on S is clear as is the fact that any order-complete scale that contains it also contains a copy of IJ. (Chromatic
complete scales thus need not be order-complete but it is fairly easy to show that they do have countable sups and infs.)

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35. Addendum: Wilson Space [2009–04–25] In the 1999 category-list post in which I
first described the final-coalgebra characterization of I the ordered-wedge functor was defined
as follows:

In the category of posets with top and bottom consider the binary functor, X ∨ Y, obtained
by starting with the disjoint union X;Y, with everything in X ordered below Y, [156] and then
identifying the top of X with the bottom of Y.

That posting had a ps:

Just for comparison, consider the category of posets and the functor that sends X to X;1;X. The
open interval is an invariant object for this functor but it is not the final co-algebra. For that we
need—as we called it in Cats & Alligators—Wilson space. Actually, not the space but the linearly
ordered set, most easily defined as the lexicographically ordered subset of sequences with values
in {−1, 0, 1} consisting of all those sequences such that a(n) = 0 ⇒ a(n + 1) = 0 for all n (take a
finite word on {−1, 1} and pad it out to an infinite sequence by tacking on 0s).

Actually, in Cats & Alligators [157] Wilson space was not viewed as a poset but a topological
space. So let’s work in the category of spaces (we’ll come back to the poset view later). We
topologize X; 1; X by taking it as Scone(X + X) the scone of the disjoint union of two
copies of X. The scone of a space is the space that results when a new point is adjoined whose
only neighborhood is the (resulting) entire space. (Restated: a subset of Scone(X ) is open iff
it is entire or an open subset of X.) The new point is called the focal point (all sequences
converge to it). Scone(X ) classifies continuous partial maps with open domains: continuous
maps Y → Scone(X ) are in natural correspondence with continuous maps U → X where U
is an open subset of Y. The final coalgebra of Scone(−) is the space of “extended natural
numbers,” that is, the natural numbers plus a point at infinity, topologized by taking as its
only open nonempty subsets the infinite updeals (there’s only one nonempty finite updeal,
to wit, the one-element set {∞}). The focal point is 0. Given a space X and a continuous
partial map with open domain f : X → X, the induced map from X to the extended natural
numbers sends x ∈ X to sup n : f n (x)↓ (using the computer-science convention that ↓
means “the expression to the left is actually defined”). The final coalgebra, of course, needs
a coalgebra structure; it is given by the predecessor map (where it is understood that ∞ is
a fixed-point and that the predecessor of 0 is undefined). The map just described from X to
the extended natural numbers is then the unique co-homomorphism between coalgebras.
I find it remarkable that the final coalgebra for the functor Scone(X+X ) is the same space
that we defined for an entirely different reason in Cats & Alligators. [158]
[ 156 ] I was borrowing the computer-science use of the semi-colon for joining a pair of imperatives. Am I right in believing that

this first appeared in Kemeny’s BASIC ?

[ 157 ] Freyd, Peter and Scedrov, Andre, Categories, Allegories, North-Holland Publishing Co., Amsterdam, 1990
[ 158 ] It arose in the proof of the “geometric representation theorem” for intuitionistic logic. After establishing a completeness

theorem for intuitionistic logic in the semantics arising in set-valued functor categories we wish to establish a completeness
theorem for the category of sheaves on the reals. If T : B → A has the property (reminiscent of covering maps in topology)
that for any f ∈ A there is not only g ∈ B such that T (g) = f but for every B ∈ B such that T (B) = Dom(f ) there is
g ∈ B such that Dom(g) = B and T (g) = f then the functor induced by composition S T : S A → S B faithfully preserves
the semantics of first-order logic of the two categories. For any A let P be the “path tree,” to wit the partially ordered set
of non-empty finite sequences of composable maps in A ordered by prolongation on the right. Any partially ordered set may
be viewed as a category. and the obvious “forgetful functor” from P to A is an example of a functor that induces a functor
between functor categories that faithfully preserves the semantics of first-order logic. Each connected component of P is a rooted
tree and S P is a cartesian product of functor categories based on rooted trees. Any countable tree may be covered with an
ever-bifurcating tree.
As can any category of set-valued functors on a poset, the category of set-valued functors from an ever-bifurcating tree may
be viewed as the category of sheaves on a space, to wit, the ever-bifurcating tree where the open sets are all the ever-bifurcating
subtrees. When any space is made sober (see the first sentence of [??], p??) the category of sheaves remains the same. Wilson

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I’ll call a coalgebra on the functor whose values are denoted Scone(X +X ) a
partial-interval coalgebra and continue to call its final coalgebra Wilson space and
denote it as W. A partial-interval structure on X is given by a pair of continuous partial
∧ ∨
self-maps with disjoint open domains. I’ll denote the values of the partial maps as x and x
and call them partial >- and partial ⊥-zooming.
We construct W as the disjoint union {−, +}? ∪ {−, +},N to wit, the finite and infinite
words on a two-element alphabet {−, +}. The word “path” has been used to encompass both
finite paths (words) and infinite paths (sequences). It has two orderings of interest to us.
Besides the total ordering from 1999 there’s a partial ordering defined by prolongation on the
right. With this ordering the finite paths form a tree with the empty word as root. (I find
it easiest to view the tree not as going up or down but sideways. The root is on the left.)
For each finite word we obtain a subtree with that finite path as root. We topologize W by
taking all such rooted subtrees as basic open subsets. It may be noticed that the subspace of
infinite paths is none other than Cantor space.
∧
The partial-interval structure on W is such that p↓ iff the head of p is + in which case
∧
p is its tail (the head is the first element in the path, the tail is what’s left after the head
∨
is removed). p is defined dually. Note that neither zooming is defined on the empty word (it
doesn’t have a head).
Given a coalgebra structure on a space X we construct
f : X → W by defining f (x) ∈ W as the path, finite or infinite, obtained by iterating
the parallel procedure:
∧ ∨
" # " #
If x↓ then emit “ + ” If x↓ then emit “ − ”
∧ || ∨
and replace x with x and replace x with x
It is transparent that such defines a coalgebra homomorphism. For its continuity note that
the inverse image of a basic open set rooted at a given finite path is the domain of the partial
map determined by composing the sequence of >- and ⊥-zoomings corresponding to the +s
and −s in the given path.
In Cats & Alligators we needed an open continuous map from I to W (a “Freyd curve”).
We can construct is the unique coalgebra map for a particular partial interval structure
defined as follows. The domain of >-zooming will be the strictly positive elements in I. The
map from (0, 1] to [−1, 1] will be piecewise affine with infinitely many pieces. {2/3n }n∈N is
∧
the sequence of critical points. The critical values alternate between +1 and −1 with 1= 1.
[
As previously ⊥-zooming is defined as −(−x).
The resulting I → W can be (and originally was) defined using “symmetric ternary
expansions.” Not signed-binary expansions. Every element in I may be described as
∞
X an [ 159 ]
2
n=1
3n
space is none other than the ever-bifurcating tree made sober.
In order to move the semantics to a space more familiar than W we need an open continuous map from a familiar space to W.
If X → Y is open, continuous and onto then the induced map Sh(Y ) → Sh(X) faithfully preserves the first-order semantics.
We want an open continuous map onto I → W. (All sorts of familiar spaces have open continuous maps onto I. As just one
example to get started: the sine function from R to I.) Wonderfully enough the function we described in Cats & Alligators,
the one that had earned the name “The Freyd Curve” in the 70s, is the induced map I → Scone(W) arising from a particular
coalgebra structure on I.
[ 159 ] In Cats & Alligators the 2 was omitted and we worked in the interval [− 1 , + 1 ] (which, by the way, made the forthcoming
2 2
characterization of the elements with non-unique expansions a tiny bit harder).

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where an ∈ {−1, 0, +1}. As in Section 30 (p??–??) I’ll suppress the 1s and use the symbols
“−◦+.” The non-integers in I of the form (2m + 1)/3n have two such expansions, all other
elements have just one.
The induced function {−1, 0, +1}N → W can be described with a four-state automaton.
E is the initial state (actually, all choices of initial state work):
Next State Output
E E0 N N0 E E0 N N0
+ E N N E + + + − −
◦ E0 E N 0 N ◦
− E N N E − − − + +
The ◦ input toggles the pairs E, E0 and N, N0. The two non-◦ inputs have the same next-state
behavior: E and N are stationary for non-◦ inputs and they are the only non-◦-input targets.
In the diagram below, the vertical gray (double) arrows show the next-state behavior for the
◦ input, the horizontal and circular arrows show the next-state behavior for non-◦ input:
l
E ← N0
≺

l l
E0 → N ≺l

As for output: when z is the input there is no output (a stammer); non-◦ input is echoed in
the two left-hand states, E, E0 and negated in the two right-hand states, N, N0.
The first task is to show that the resulting function is defined not just on sequences but on
elements of the standard interval, that is, the same output is engendered by sequences that
name the same interval element. We need to consider the output of two machines starting in
the same place but one being fed the sequence +−−−− · · · and the other ◦ ++++ · · · .
We’ll do better with a single machine but with two demons jumping from state to state each
according to the commands issued by its appointed sequence. We start them at the same
state. For each of those four possible states the jump commanded by the input + produces
the same output as that commanded by ◦+ but the demons will land in different states. One
will be in E the other N and there they’ll henceforth remain. One of the demons will echo
the input, the other will negate it, which is just what is needed for the output engendered by
a constant sequence of +s for one demon to be the same as that engendered by a constant
sequence of −s for the other.
The “Freyd curve” in Cats & Alligators was different but can also be described with a
four-state automaton:
Next State Output
E0 E N N 0 E E N N0
0

+ E N E N + + + − −
◦ E E 0 N0 N ◦
− E N E N − − − + +

Any one of the states may be taken as initial. (To get the function described in Cats &
Alligators [160] start at E) The next state is always an adjacent state. The ◦ input toggles
[ 160 ]It was described there as the function that sends 2 ∞ -n ∈ I to {(−1)ni−1 a }L L
P
n=1 an 3 ni i=1 ∈ W where {ani }i=1 is the
∞
result of removing all 0s from {an }n=1 (and n0 is understood to be 0).

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the pairs E0, E and N, N0. The non-◦ inputs have the same next-state behavior: they both
always target one of the two middle positions which fact together with the fact that all next
states are adjacent determines their action. In the diagram below the gray (double) arrows
show the next-state behavior for the ◦ input.

E0 ↔ ↔ 0
→E ↔N ←N

As for output: when ◦ is the input there is no output (a stammer); non-◦ input is echoed
in the first two states, E0, E and negated in the last two, N, N0.
That the resulting function is defined not just on sequences but on elements of the interval
is obtained by an argument almost identical to that given for the first machine. We need,
again, to consider the output of two demons starting at the same state, one behaving according
to the sequence +−−−− · · · and the other ◦ ++++ · · · . On each of the four possible states
the jump commanded by the input + produces the same output as that commanded by ◦+
but the demons will land in different states. One will be in E the other N and they’ll spend
the rest of eternity trading places: at each subsequent input one of the demons will echo,
the other will negate, which is just what is needed for the output engendered by a constant
sequence of +s for one demon to be the same as that engendered by a constant sequence of
−s for the other.
For the continuity let’s consider a more general setting. We are considering functions given
by procedures that take finite paths of signals to finite paths in W. If we take a very general
view of procedure we are led to view functions of the form m : A? → B ? where A and B
are finite sets, which functions are covariant with respect to the prolongation ordering, that
is, if w, w0 ∈ A and if w0 is a prolongation of w then we require m(w0 ) to be a prolongation
of m(w) (which includes, of course, the possibility that m(w0 ) = m(w)). Such a function
induces a function m̃ : AN → B ? ∪ B N . In the classical case AN and B N are understood to
name elements in an interval (and, traditionally, A = B = {0, 1, .., b − 1}). In that case we
need two conditions: first, for any infinite input path the induced output path is also infinite;
second, if a pair of infinite input paths name the same element in the interval then so do
the pair of infinite output paths. The resulting function on the interval is then automatically
continuous. One would like, of course, that all continuous maps are so obtainable. Alas, even
something as simple as multiplying by 3/4 on the unit interval can not be so obtained when
using ordinary binary expansions (if the input is the unique binary expansion for 2/3 then
no finite initial subpath has enough information to determine even the head of the intended
output path). Signed-binary expansions were invented to take care of this problem. If the
interval in question is the standard interval I and if A = B = {−1, 0, +1} then for any
continuous f : I → I is of the form m̃ for some (perhaps many) m : A → B.
In our setting, the target is W, hence the 1st condition above is irrelevant. If we specialize
to the case that A = {−1, 0, +1} and, as above, paths in {−1, 0, +1}N name elements in I
via symmetric ternary expansions (not—it must be emphasized—signed binary expansions)
then given any m : {−1, 0, +1}? → {−, +}? covariant with respect to prolongation we obtain
m̃ : {−1, 0, +1}N → W. To obtain maps from I to W we still need the 2nd condition: two
infinite paths naming the same element in I must be sent to the same element in W by m̃.
Then continuity is automatic. For a proof, let w be a finite path in W and x ∈ I a point
sent by m̃ to a prolongation of w. We seek a neighborhood of x all of which is sent by m̃ to
prolongations of w. If x is named by a unique path in A? let k be such that its initial word
of length k is sent by m to a prolongation of w. Then, of course, the closed interval of all
prolongations of that initial k-word are still sent to prolongations of w and x is an interior

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point of that closed interval. If, perchance, x is named by two infinite paths, we need only
rephrase the argument. Let k by such that the initial word of length k of any infinite path
that name x is sent to prolongations of w. The interval of all prolongations of those initial
k-words are still sent to prolongations of w and x is an interior point of that closed interval.
(We obtain a closed interval for each of the two names and x is an endpoint of each, the top
end of one and the bottom end of the other, hence an interior point of their union.)
Finally, to show that it is an open continuous map, it suffices to find an open basis for
I each member of which is mapped onto an updeal in W. First, for each w ∈ {−1, 0, +1}?
note that the closed interval with w, −1, −1, . . . as lower bound and w, +1, +1, . . . as upper
has a basic open set in W as image. But note that that remains true for the open interval
with those same endpoints (w, +1, +1, . . . and w, 0, 0, +1, +1, +1, . . . are mapped to the same
element in W.) So, whenever an endpoint is other than ± 21 we delete it from the interval.
The family of all such intervals is a basis.
The 1999 definition above of Wilson space was the most efficient way of communicating its
total ordering (which ordering turns out to be implicit in its final coalgebra definition).[161]
The total ordering on W has endpoints and a canonical total interval coalgebra structure:
>-zooming sends every path, finite or infinite, with + as head to the rest of the word (its
“tail”) and sends all other words to the infinite path that’s all minuses. ⊥-zooming is, of
course, defined dually. The induced map W → I may be described as the function that reads
each path, finite or infinite, as a signed-binary expansion. Note that every element of I has a
signed-binary expansion with no 0s, hence W → I is onto. Every element of I not an interior
dyadic rational has a unique such expansion and every interior dyadic rational has exactly
three expansions coming from W.
We thus obtain an alternate characterization of the total ordering starting with a closed
interval viewed as an ordered set and replacing each element of a countable dense subset of
interior points with a three-element totally ordered set.
I has a canonical partial-coalgebra structure: partial >-zooming is obtained by cutting
the total >-zooming operation down to a partial map, to wit, the one that’s defined only
on elements strictly larger than . We treat, of course, ⊥-zooming in the dual fashion. The
induced map I → W followed by the previously induced map W → I is the identity map.
The composition of the two induced maps in the other order is most easily described in terms
of the last paragraph: it is the endomorphism on W obtained by collapsing to a point each of
the inserted three-element sets (collapse every pair of elements with nothing between them).

36. Addendum: Vector Fields [2009–06–21]

Theorem 10.5 on the existence of standard models (p??) tells us that many things can be
reduced to purely equational logic. For each natural number n consider the following Lipschitz
extension of the theory of harmonic scales: n n-ary operators with equations
_ _
(fi hx1 , x2 , . . . , xn i)2 = x2i
i i

For each i = 1, 2, . . . , n :

fi ht2 x1 , t2 x2 , . . . , t2 xn i = t2 fi hx1 , x2 , . . . , xn i
[ 161 ] Note that the 1999 ordering is not the standard lexicographic ordering (which would take a finite word as prior to any of

its prolongations) I said that there are two ordering of interest; the standard lexicographic ordering is not one of them.

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And X
xi fi hx1 , x2 , . . . , xn i = 0
i

If we view the f s as describing a self-map on n-space the first equation says that they
describe a self-map that preserves the `∞ -norm
The next n equations say that f s describe a self-map that’s an `2 -isometry on each ray.
The last equation says that the f s describe a self-map whose values are always orthogonal
—in the standard `2 sense—to its arguments. (The summation is easily avoidable: pad out
the two n-vectors with 0s to obtain vectors of dimension a power of two and replace adding
with a binary tree of midpointing.)
Left out are equations to make the f s Lipschitz. Add them at will (with Lipschitz constant
at least one.)
These equations are consistent iff n is even.
When n is even we have an easy model: define

f hx1 , x2 , . . . , xn i = hx2 , −x1 , x4 , −x3 , . . . , xn , −xn−1 i

For any n we may use a model of these equations to define a vector field on the
(n−1)-sphere: for vectors of unit `2 -norm simply define

f hx1 , x2 , . . . , xn i
ghx1 , x2 , . . . , xn i =
kf hx1 , x2 , . . . , xn ik2

Because spheres of even dimension have no non-vanishing vector fields we know that these
equations are inconsistent for any odd n. It is a sobering thought that beginning with the term
> and using only a sequence of substitutions, each of which uses one of the defining equations
with odd n, we can ultimately reach ⊥. It is even more sobering to realize that Adams’s
theorem on parallelizable spheres [162] —which used just about all of algebraic topology then
known [163] —was equivalent to a theorem about the consistency of certain finite families of
equations.[164]
[ 162 ] “Vector fields on spheres,” Bull. Amer. Math. Soc. 68 1962.
[ 163 ] In his first publication on the subject (ibid.) the parallelizability result appears as a special case of a (slightly later)
more general result that used more than all of algebraic topology then known. In particular it used the Adams operations the
construction of which required, conceptually, a quantification over functors; it was the first clearly important construction that
did so. (Mac Lane never agreed with me that this was the most important single event in the history of category theory following
its creation, indeed he omitted the Adams theorem from all of his histories of category theory: any actual application of category
theory was somehow not to be considered category theory).
[ 164 ] Anyone can easily verify that in each of the three columns below the vectors are pairwise orthogonal.

h+x0 + x1 i h+x0 + x1 + x2 + x3 i h+x0 + x1 + x2 + x3 + x4 + x5 + x6 + x7 i

h+x1 − x0 i h+x1 − x0 + x3 − x2 i h+x1 − x0 + x3 − x2 + x5 − x4 − x7 + x6 i
h+x2 − x3 − x0 + x1 i h+x2 − x3 − x0 + x1 + x6 + x7 − x4 − x5 i
h+x3 + x2 − x1 − x0 i h+x3 + x2 − x1 − x0 + x7 − x6 + x5 − x4 i
h+x4 − x5 − x6 − x7 − x0 + x1 + x2 + x3 i
h+x5 + x4 − x7 + x6 − x1 − x0 − x3 + x2 i
h+x6 + x7 + x4 − x5 − x2 + x3 − x0 − x1 i
h+x7 − x6 + x5 + x4 − x3 − x2 + x1 − x0 i

It is beguilingly easy to believe that the middle column is a natural generalization of the left, and the right column of the
middle. (If we erase everything except the indices we would be staring at tables for “nim-sum.” The problem is just a matter of
adjusting the signatures.) And clearly we should be able to obtain such a column of length 16. Adams became history’s most
conspicuous Fields non-Medalist by showing that if the top vector is hx0 , x1 , . . . , xn−1 i with n−1 others given by continuous
non-vanishing functions of the top and if the n vectors are pairwise orthogonal then n = 1, 2, 4 or 8.

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37. Addendum: Extending the Reach of the Theorem on Standard Models

[2009–07–23]

We wish to obtain existence proofs for structures on spaces more general than standard
n-cubes. We can describe any closed subset, C, of an n-cube, of course, as the kernel of a
Lipschitz function, M : In → I, defined on the n-cube (e.g., supy∈C x ◦−
−◦ y).
Suppose now that we wish to prove the existence of a fixed-point–free Lipschitz
function f : C → C. Given any Lipschitz f, Kirszbraun’s theorem delivers a Lipschitz function
f˜ : In → In such that f˜(x) = f (x) whenever it can, that is, whenever x ∈ C. [165] Thus we
may formalize the notion of a self-map on C with an operator f : In → In satisfying the
Horn sentence M x = > ⇒ M (f x) = >.
Because C is compact we know that if f is fixed-point–free then there’s a q ∈ I? such
that M x = > ⇒ f x ◦− −◦ x ≤
= q (recall that I? are the elements below >). But even if we
prove the consistency of such a condition the theorem on standard models does not deliver a
fixed-point–free map on C. The problem is the Horn condition: it’s not an equation.[166]
Fortunately the Lipschitz condition on M allows us to find an equation equivalent to the
Horn sentence:  
Mx ≤ = (>|)m+p M (f˜x)
where m and p are integers that stipulate the Lipschitz condition on M and f :

= (>|)m (M x ◦− = (>|)p (f x ◦−



(x ◦−
−◦ y) ≤ −◦ M y) and (x ◦− −◦ y) ≤ −◦ f y)

(Recall that any semiquation is equivalent to an equation).

= (>|)k (M f x) implies the Horn sentence for any integer k. For the
It is clear that M (x) ≤
necessity, given x ∈ In let x0 ∈  C be such that x ◦− −◦ x0 is maximal. Then
−◦ x0 ≤
x ◦− = (>|)p f˜x0 ◦−
−◦ f˜x ≤= (>|)m+p M (f˜x0 ) ◦−
−◦ M (f˜x) .
Thus if we replace the Horn sentence with this equational condition the consistency remains
the same as does the existence of a standard model (Theorem 10.5, p??). (Alternatively, we
could add an n-tuple of constants for each element of C and replace the Horn condition with
the family of all conditions of the form M (f ha1 , a2 , . . . , an i) = >. ) [167]
The slogan: consistent equational conditions have solutions.

38. Addendum: Boolean-Algebra Scales [2009–10–30]

One of the better-known constructions in mathematics is that for group algebras: given a
ring R and a group G, we construct R[G] as the ring generated by R and the elements of G
[ 165 ] If one uses the standard Euclidean metric on In then Kirszbraun delivers an f˜ with the same Lipschitz constant as f.

Given Hilbert spaces H1 , H2 and a closed subset C ⊆ H1 Kirszbraun extends any nonexpansive map f : C → H2 to all of
H1 . Such easily generalizes from nonexpansive to Lipschitz but we’re not done, we need to land in the standard cube. We do so
by composing with the clearly nonexpansive map that sends each point in H2 to its nearest point in the cube. (Such works for
all convex closed sets, hence they all have nonexpansive retractions. It’s worth noting the converse: the set of fixed-points for
any nonexpansive map is necessarily convex.) Finally, since any norm on a finite dimensional real vector space is Lipschitz with
respect to any other we obtain the advertised f˜.
[ 166 ] The theorem on standard models uses a simple quotient of the free algebra of an equational theory. Equations true for

an algebra are true for quotient algebras. Not so for Horn sentences. Consider how easily the familiar Horn condition on rings
[x2 = 0] ⇒ [x = 0] (i.e., “no nilpotents”) fails to be preserved when passing to a quotient ring.
[ 167 ] There can be better ways for some special cases. For one example, we can effectively convert the n-cube into a (flat)

n-torus by adding m equations for each new m-ary operator f, to wit, one for each 1 ≤ = i ≤= m:
f hx1 , . . . , xi−1 , ⊥|y , xi+1 , . . . , xm i = f hx1 , . . . , xi−1 , >|y , xi+1 , . . . , xm i.

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subject to the relations that the multiplication induced on G by the ring structure coincides
with that given by the group structure on G. Given a (chromatic, harmonic) scale S or any
Lipschitz extension thereof and a Boolean algebra B we construct the Boolean-algebra
(chromatic, harmonic) scale S[B] as the (chromatic, harmonic) scale generated by S
and the elements of B subject to the relations that the lattice structure
induced by the scale structure on the elements of B is as given by the Boolean-algebra
structure (including, of course, that the top and bottom of B are the top and bottom of
S[B]). Note that if x and y are a complementary pair in B they remain so in S[B], hence all
elements in B are extreme points in S[B] and complementation in B coincides with dotting
in S[B].
The maps from S[B] to any scale T are in natural correspondence with the pairs of maps,
one a scale-homomorphism S → T, the other a Boolean-homomorphism B → B(T ).[168]
For the special case S = I we have that the functor I[−] from Boolean algebras to scales is
the left adjoint of B(−) from scales to Boolean algebras. (More generally, if S is connected
then S[−] is the left adjoint of B(−) from the category of extensions of S to the category of
Boolean algebras.)
When S is a connected scale the adjunction map from B to B(S[B]) is an isomorphism.
For a quick and dirty proof note that we may construct S[B] as the S-valued continuous
functions from the space of ultrafilters on B (where the topology on S is discrete). More
generally (and constructively) we gain a handle on S[B] as follows: we will say that a term is
“pre-canonical” if it is of the form (s1 ∧ e1 ) ∨ (s2 ∧ e2 ) ∨ · · · ∨ (sn ∧ en ) where s1 , s2 , . . . , sn ∈ S
and {e1 , e2 , . . . , en } is a partition of unity in B (pairwise disjoint and e1 ∨e2 ∨· · ·∨en = >); and
“canonical” if, further, none of the ei s equals ⊥ and the si s are distinct. We need show only
that every element of S[B] is described by a canonical term, unique up to the ordering of the
partition of unity.[169] For the existence of the term it suffices to show that elements named
by canonical terms are closed under the scale operations: clearly any s ∈ S is named by the
canonical term s ∧ > and any e ∈ B not in S (that is, other than > or ⊥) by the canonical
.
term (> ∧ e) ∨ (⊥∧ e); hence if the set of terms named by canonical terms are closed under the
operations, then that set is necessarily all of S[B]. Zooming and—in the case of chromatic
scales—the support operation are easy since each distributes with the lattice operations and
∧ ∧
fixes the extreme points: ((s1 ∧ e1 ) ∨ (s2 ∧ e2 ) ∨ · · · ∨ (sn ∧ en ))∧ = (s1 ∧ e1 ) ∨ (s2 ∧ e2 ) ∨
∧
· · · ∨ (sn ∧ en ) and, in the case of chromatic scales, (s1 ∧ e1 ) ∨ (s2 ∧ e2 ) ∨ · · · ∨ (sn ∧ en ) =
(s1 ∧ e1 ) ∨ (s2 ∧ e2 ) ∨ · · · ∨ (sn ∧ en ). The latter terms may be only pre-canonical but it is clear
that every pre-canonical term is equal to a canonical term. Dotting requires a little work:
we need a lemma on scales, . . to wit, .that if the es form a partition of unity then ((s1 ∧ e1 ) ∨
.
(s2 ∧e2 )∨· · ·∨(sn ∧en )) = (s1 ∧ e1 )∨(s2 ∧ e2 )∨· · ·∨(sn ∧ en ). The linear representation theorem
for (chromatic, harmonic) scales comes to the rescue. Since the lemma is given by a family
of universally quantified Horn sentences it suffices to check the equation on linear scales. But
we know that in a linear scale all but one of the ei s will be ⊥ and the one that is not will be
> and that is quite enough. For midpointing suppose that (s01 ∧ e01 ) ∨ (s02 ∧ e02 ) ∨ · · · ∨ (s0m ∧ e0m )
is another canonical term. Then:
_ n _m n _
_ m
0 0
(si |s0j ) ∧ e00i,j


(si ∧ ei ) (sj ∧ ej ) =
i=1 j=1 i=1 j=1

where e00i,j = ei ∧ e0j (the “joint refinement” of the two given partitions of unity). Again, the
easiest proof is simply to consider the equality in the case of a linear scale. The right-hand
[ 168 ] B was defined on p??.
[ 169 ] Note that if ei were ⊥ uniqueness would be lost: any si would do.

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term is pre-canonical and—as before—can quickly be transformed into a canonical term.

Given any Lipschitz extension this proof may be easily adapted, in particular for harmonic
scales add the equation obtained by replacing midpointing with multiplication in the previous
equation.
For the uniqueness of the canonical term note first that in a pre-canonical term for > it
must be the case that si = > whenever ei 6= ⊥ (just specialize—once again—to any linear
quotient). Hence if (s1 ∧ e1 ) ∨ (s2 ∧ e2 ) ∨ · · · ∨ (sn ∧ en ) = (s01 ∧ e01 ) ∨ (s02 ∧ e02 ) ∨ · · · ∨ (s0m ∧ e0m )
then we know that si ◦− −◦ s0j = > whenever e00i,j 6= ⊥ which, in turn, says that si = s0j whenever
ei ∧ e0j 6= ⊥. We may thus infer that for every 1 ≤ =i≤ = n there is 1 ≤ =j≤ = m such that si = s0j
and vice versa. Because the si s in a canonical term are distinct this forces n = m and the
existence of a permutation π such that si = s0π(i) . Since ei ∧ e0j = ⊥ whenever j 6= π(i) we
may infer that ei = e0π(i) . [170]
Scales of the form I[B] thus succeed in mixing Aristotle, Lukasiewicz and Girard:
∨ & ∧
for extreme points | = = ∨ and | = ⊗ = ∧ hence x −◦ y is the same as the standard
material implication; the support and co-support operators translate from Lukasiewicz and
∧
Girard to Aristotle (x is the assertion, for example, that x is more probable than not).

39. Addendum: On the Definition of I-Scales [2010–07–28]

Our definition of I-scales should be considered as the minimal definition needed to make it
both unique on the standard interval and co-congruent with the theory of scales. It has a
unique maximal equational consistent extension.
In Section 3 (p??–??) we used the fact that a particular scale (I) appears as a subscale of
every non-trivial minor scale to imply the existence of a unique maximal equational consistent
extension. In this case we use the fact that a particular scale (I) appears as a quotient scale
of every non-trivial I-scale: if any equation is consistent with the theory of I-scales then it
has a non-trivial model which, in turn, has a quotient scale isomorphic (as an I-scale) to I,
hence holds for I. Among such equations are:
[ 171 ]
x= (r|s)x = rx|sx >x =x (rs)x = r(sx)

It was never clear to the writer just how I-scales should be defined. The minimal definition
was the choice. But note that the every theorem herein holds equally well for the maximal
definition.

40. Addendum: Proofs for Section 0 [2011–06–02]

At the beginning of Section 4 (p??–??) on lattice structure we explained how a closed-interval

homomorphism from C(X) to I necessarily preserves order. Clearly it also sends functions
constantly equal to an element in I to that same element in I. Combine with the covariance
and we have that it sends functions constantly equal to any element in I to that same element
in I. From that we may conclude that if f : X → I is bounded above/below by x ∈ I than
it is sent to an element in I also bounded above/below by x. [172]
[ 170 ] Having found this construction for scales, I assume that it is ancient knowledge that the analog construction for Boolean

algebras of central idempotents works as well. The case Z2 [B] returns us, of course, to Boole’s original Laws of Thought.
[ 171 ] Note that in the left-hand terms , r|s, >, rs refer to the structure of I and in the right-hand terms and rx|sx refer
to the structure in the given I-scale. (The right-hand term r(sx) uses composition of unary operations.)
[ 172 ] We will not need it here, but if one now adds the fact that any closed-interval homomorphism necessarily preserves dotting
. .
(since u is uniquely characterized by u|u = ⊥|>) to obtain that if C(X) and C(Y ) are viewed as metric spaces under the
uniform norm then any closed-interval map between them is necessarily nonexpansive.

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For f ∈ C(X) define its variance, by var(f ) = sup(f ) − inf(f ). It follows that any
closed-interval endomorphism on C(X) conserves low-variance (that is, does not increase it).
A special case, note, is that a convex combination has variance at most equal to the greatest
of the variances of the functions of which it is a combination.
[173]
In the case X = I we may rewrite the 4th equation (using the 3rd equation) as:

Z Z
f (x) dx = f (⊥|x) | f (>|x) dx

R
Let T : C(I) → C(I) denote the operator such that (T f )(x) = f (⊥|x) | f (>|x). Note first,
that T is a closed-interval endomorphism.[174] It suffices to show that for any f ∈ C(I) the
sequence {inf(T n f )} is non-decreasing and {sup(T n f )} is non-increasing and, finally, that
their difference converges to 0. We’ve already noted that T as any convex combinator is
monotonic on both inf and sup. Hence it suffices to show that lim var(T n f ) = 0.
We need a description of T n f. It is the arithmetic mean of 2n functions each of which is
of the form
e1 | e2 | en | f
I → I → I → ···I → I → I
where each ei is either ⊥ or >. Each of the 2n functions is a contraction down to a subinterval
2-n of the length of I followed by f. It thus suffices to show that for any ε > 0 we can find
n such that when f is restricted to any of those subintervals its variance is bounded by ε.
Suppose there were no such n. Consider the binary tree of intervals (partially ordered by
containment) arising from all the T n s; throw away all those intervals on which f is of variance
bounded by ε; if the resulting tree were infinite then—as for any infinite rooted tree with finite
branching—König’s lemma would give us an infinite path, that is, an infinite chain of closed
intervals—each half the length of the previous—on which f is of variance greater than ε.
But f could not possibly be continuous at the point that lies in their intersection. [175]
This proof easily generalizes to Im
. In the case m = 2 replace the 4th equation with two
[ 173 ] The 4th equation re-occurred to me via an outrageously circuitous route (I say “re-occurred” because I had quite forgotten

the diagram that illustrated it in my 1999 lecture mentioned in Section 32, p??–??.) We could, of course, convert integration
on I to integration on the circle: given f : I → I we could throw away one of the endpoints and view the resulting half-open
interval as a circle torn apart. The problem is that we will have converted the integration of a continuous function on I to the
.
integration of a function on the circle with a jump-discontinuity. But clearly the function whose value at x is f (x) should
.
have the same mean-value, hence instead of integrating f (x) we could integrate f (x)|f (x); its conversion to a circle does
yield a continuous function. But then we note that we’re integrating a function symmetric on I and it would suffice to find the
mean-value on either half. Choose “ the .top
” half. What we’ve done is replace the mean value of f (x) with the midpoint of the
mean values of f (>|x) and f (>|x) . At a lecture at the University of Cambridge I started calling this the “puff-pastry”
method: roll out the dough; coat with butter; fold over; repeat until done. Perhaps it was because I was so happy with that
. . .
name that it took me so long to note that (>|x) could be replaced with (⊥| x) and that the mean value of f (⊥| x) would be
the same without the dot.
[ 174 ] What we are showing, therefore, is that the mean-value operator from C(I) to I is characterized as the closed-interval

homomorphism that is constant on the orbits of T.

[ 175 ] We used a contradiction to show that a particular procedure works (to wit, trying each n in sequence). What will be

viewed by some as a more constructive proof for the existence of such an n start by taking I to be the˘standard interval, define ¯
var[x,y] (f ) to be the variance of f on the closed interval with endpoints x and y, and let h(x) = sup y : var[x,y] (f ) ≤ ε/2 .
Denote hi ⊥ as ai , let a∞ = supi ai and (using the continuity of f at a∞ ) let m = min i : varhai , a∞ i ≤ ε/2 . Then
˘ ¯
necessarily am+1 ≥ a∞ = > (the unique fixed-point of h). There’s a unique dyadic rational of minimal denominator in the open
interval (ai , ai+1 ) for each i ≤ m. Let 2n be the largest of those minimal denominators. The number being sought can be
taken as n. (For any pair of subsets A, B with A ∩ B 6= ∅ note that varA∪B (f ) ≤ varA (f ) + varB (f ) any f, any domain.)
This proof has another advantage: it works on functions continuous except for a finite number of jump-discontinuities. Note
first that T does not increase the number of discontinuities, second, that T halves the sum of all the jumps. Given ε begin
by iterating T until each jump is less than ε/2 and then continue as before. (And with a little more effort we can allow any
collection of jump-discontinuities as long as the sum of the jumps is finite.)

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equations:
Z Z Z Z Z
f h⊥|x, yi dx | f h>|x, yi dx = f hx, yi dx = f hx, ⊥|yi dx | f hx, >|yi dx
R

R
For each m the 4th equation is replaced by m equations, one for each dimension: for each
i = 1, 2, . . . , m there will be integrands f hx1 , . . . , ⊥|xi , . . . , xm i and f hx1 , . . . , >|xi , . . . , xm i.
The operator T becomes a convex combination of 2m functions (and for T n it’s 2mn ) and the
binary tree becomes a tree in which each vertex has 2m ranches. [176]
For the case that X is a a compact group, G, we need a few definitions. Given f ∈ C(G) let
Rf be the set of all convex combinations of the right-translates of f [177] and Rf the closure
of Rf (using, of course, the uniform norm on C(G)). It seems to this writer that one needs
two great von-Neumann ideas. The first is that Rf contains a constant function. A similar
argument yields that the closure, Lf , of Lf , the set of convex combinations of left-translates
of f also has a constant function. We will not need a great von-Neumann idea to establish
(below) that any constant function in Rf is equal to any constant function in Lf which more
than suffices to establish the uniqueness of both.[178] The fact that the constant values of
those unique constant functions yields a closed-interval homomorphism is also easy: given
f1 , f2 ∈ C(G) we will let T1 , T2 : C(G) → C(G) each denote an operator that delivers a given
convex combination (to be chosen) of a given sequence of right-translates (to be chosen) of
its argument; choose T1 so that T1 f1 is within ε of the unique constant function in Rf1 and
choose T2 so that T2 (T1 f2 ) is within ε of the unique constant function in RT1 f2 ⊆ Rf2 ;
it is still the case that T2 (T1 f1 ) is within ε of the unique constant function in Rf1 ; thus
T2 (T1 (f1 |f2 )) = (T2 (T1 (f1 )) | (T2 (T1 (f2 )) is within ε of the mean of constant functions in Rf1
and Rf2 .
To find a constant function in Rf we need the second great von-Neumann idea, to wit, use
that Rf is compact. We can then find h ∈ Rf of minimal variance and will finish by showing
that var(h) > 0 would allow an easy construction of an element of smaller variance.
So we start with the compactness argument. For any continuous f : G → I the
induced function G → C(G) that sends α ∈ G to f α is continuous (using the uniform
norm in G). The proof requires a little work (it will use, for example, the compactness
of G [179] ). Given ε > 0 we need an open neighborhood U of 1 ∈ G such that kf − f α k < ε
all α ∈ U. If there were no such neighborhood then Moore-Smith convergence works well for
reaching a contradiction: from the (down-)directed set of neighborhoods of 1 we can construct
a net by sending U to y ∈ G such that |f (y) − f α (y)| ≥
= ε for some α ∈ U ; the compactness
of G allows us to find a sub-net such that the y s converge to a point z ∈ G such that for
every U there’s an α ∈ U with |f (z) − f (zα)| ≥
= ε which belies the continuity of f.
Thus the set of right-translates of f, being the image of a continuous map from a compact
space, is itself a compact subset of C(G). For any ε > 0 we may choose a finite sequence
[ 176 ] Without mentioning mean-values we could have stated that for any continuous f : I → I the sequence of functions

obtained by iterating T uniformly converges to a constant function. If the domain is a cube of dimension m and if Tj defines the
operator that applies the T -operator to the jth coordinate then any sequence of Tj s in which each Tj appears infinitely often
for each j from 1 through m likewise will yield a sequence of functions uniformly converging to a constant function on Im .
[ 177 ] For our setting we need convex combinations using dyadic rationals, but such barely affects the proof. Indeed, one may

simply understand the phrase “convex combination” in this section to mean one that uses only dyadic rationals.
[ 178 ] More than suffices because, of course, it also establishes that right-invariance of integration on compact groups is equivalent

to left-invariance.
[ 179 ] If we take I to be the standard interval, G to be the group of positive reals under multiplication and f ∈ C(G) to be

defined by f (x) = cos(πx) then the sequence {αn = 1 + 1/n}∞ n=1 converges to 1 in G but kf − f
αn k
∞ is constantly equal to
2 (because (−1)n = f (n) = −f αn (n)). One may make this example look more complicated by using, instead, the (isomorphic)
group of reals under addition. But the more complicated version does have its merits: it’s easy to see that the distances between
adjacent critical points of cos(πex ) are arbitrarily small.

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{f α1 , f α2 , . . . , f αn } which is ε/2-dense in the set of all right-translates (that is, every right-
translate is within distance bounded by ε/2 from one of the f α s).
The set of convex combinations of the f α s is 2ε -dense in Rf by the following argument:
given r1 f β1 + r2 f β2 + · · · + rm f βm replace each of the f β s with one of the 2ε -close f α s to obtain
a convex combination of the f α s (albeit, one with repeated f α s). This convex combination is
of distance bounded by 2ε from the arbitrary one. We can then finish, of course, by reordering
the summands and using the distributive law to reduce down to a sum of multiples of at most
n of the f α s.
But the set of all convex combinations (dyadic rational or not) of the f α s is itself compact
(it is the image of a map from the unit simplex in n-space, that is, the set of n-tuples of non-
negative reals that add to 1.) hence has also a finite 2ε -dense subset which, perforce, is ε-dense
in the set of all convex combinations of the right-translates of f. All of which establishes that
Rf is totally bounded. In any complete metric space the closure of a totally bounded set is,
of course, compact and Rf has an element of minimal variance.
We could now easily obtain the existence of a constant function in Rf by taking it as an
element of minimal variance. It is even easier to take it as an element of minimal maximum
value; we need only show that for any non-constant function h there is a convex combination
of h-translates with smaller maximum. Let U ⊆ G be the open set x : h(x) < max h .
Let α1 , α2 , . . . , αn be such that G = U α1-1 ∪ U α2-1 ∪ · · · ∪ U αn-1 . Then the maximum value of
the convex combination hα1 |(hα1 |(· · · |(hαn ) · · ·)) is less than max h (it suffices to find for any
x ∈ G an i such that hαi (x) < max h but xαi ∈ U is—obviously—equivalent to x ∈ U αi-1 ).
Finally the advertised easy proof of uniqueness. Suppose that Lf has a function constantly
.
valued a and Rf one constantly valued b 6= a. Replace f with f, if necessary, to guarantee
that a < b. We know that there is convex combination of left-translates of f whose maximum
value is strictly less than a|b and a convex combination of right-translates whose minimum
value is strictly greater than a|b. We introduce a symbolic arithmetic just for this argument.
Given an arbitrary g ∈ C(G) and the q s and αs that are used for the convex combination in
Lf with max less than a|b we understand the formal product (q1α1 + q2α2 + · · · + qmαm
)g to be
α1 α2 αm αi
q1 g + q2 g + · · · + qm g where the g s denotes left-translates.
Given the r s and β s that describe the convex combination in Rg with min greater
than a|b we understand the formal product g(β1 r1 + β2 r2 + · · · + βn rn )
to be g β1 r1 + g β2 r2 + · · · + g βn rn . Then it does not matter how we associate the formal
triple product
(q1α1 + q2α2 + · · · + qm
αm
)f (β1 r1 + β2 r2 + · · · + βn rn )
In each case we obtain the sum X
qi αif βj rj
i,j

where, of course, αif βj is the function that sends x to f (αi xβj ). Since
 
(q1α1 + q2α2 + · · · + qm
αm
)f (β1 r1 + β2 r2 + · · · + βn rn )

is a convex combination of functions all of whose maximum values are strictly less than a|b
and  
α1 α2 αm β1 β2 βn
(q1 + q2 + · · · + qm ) f ( r1 + r2 + · · · + rn )
is a convex combination of functions all of whose minimum values are strictly greater than
a|b we obtain the desired contradiction.

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41. Addendum: Extreme Points, Faces and Convex Sets [2011–08–22]

Given a binary operation  on a set S, to say that a subset S 0 is closed under  means that
x, y ∈ S 0 implies xy ∈ S 0. We’ll say here that S 0 is co-closed if the converse holds: xy ∈ S 0
implies x, y ∈ S 0 [180] and bi-closed if both closed and co-closed.
We defined a subset of a scale to be a face if it bi-closed under midpointing. In the theory
of convex sets the usual condition is that a subset F is a face if it is bi-closed under all
convex binary combinations, that is, for all x, y in the convex set and 0 < a < 1 it is the
case that ax + (1 − a)y ∈ F iff x, y ∈ F. [181] A convex set is usually understood to be a
subset of a real vector space and in that case we will show below that the two definitions
of face are—fortunately—equivalent. The proof requires the Archimedean condition (if K is
a non-Archimedean harmonic scale note that the Jacobson radical of any non-trivial closed
interval is a >-face but not co-closed under all convex binary operations).
No special conditions are required to show that bi-closure under midpointing implies
convexity: given a convex combination ax + by where x and y are in a midpoint-face F,
then closure under midpointing says that x|y ∈ F and co-closure under midpointing then
says that ax + by ∈ F because, easily enough, (ax + by) | (bx + ay) = x|y. [182] The problem
is the co-closure.
Given positive a, b such that a + b = 1 we do have—even in the non-Archimedean case—
that if ax + by ∈ F then either x ∈ F or y ∈ F, to wit, the one
with the larger coefficient:
since if 0 ≤
=a≤ = b and a + b = 1 then ax + by = 2ax + (1 − 2a)y |y and midpoint co-closure
yields that y ∈ F. But we have more: 2ax + (1 − 2a)y ∈ F. By iteration we obtain that if
0≤=a≤ = b and a + b = 1 then it the case that 2an x + (1 − 2an )y ∈ F for all n =≥ 0 such
n
that 2 a ≤ = 1. In the Archimedean case we have a largest such n and hence a case where the
coefficient of x is the larger coefficient and that yields x ∈ F.
In the one-element case—that is, in the case of an extreme point—we do not need the
Archimedean condition. If {e} is co-closed under midpointing and if 0 < a ≤ = b,
a + b = 1 and ax + by = e then we have just seen that y = e. But from ax + (1 − a)e = e we
infer that ax = ae hence that x = e.

42. Addendum: Scale Spectra Are Compact Normal [2013–04–02]

Near the end of the discussion of Theorem 8.6, in the TAC version, I wrote in reference to
Spec(S):

It is always a spatial locale: the points are the “prime” congruences, that is, those that are not
the intersection of two larger congruences. Translated to filters: F is a point if it has the property
that whenever x ∨ y ∈ F it is the case that either x ∈ F or y ∈ F. Put another way, of course, the
points of Spec(S) are the linearly ordered quotients of S. We will show that Spec(S) is compact
normal (but not always Hausdorff).

Alas, I forgot that last promise.

First, note that the definition of prime is not quite right. We need to exclude the entire
filter. (One way to do so is to replace the word “two” with “a finite number of.”)
[ 180 ] Note that the weaker condition, x  y ∈ S 0 implies either x ∈ S 0 or y ∈ S 0 , holds iff the compliment of S 0 is closed in

which case we could say that S 0 is open under .

[ 181 ] An easy inductive argument shows that the closure and co-closure of binary convex combinations each implies the same

for n-ary convex combinations.

[ 182 ] We need both midpoint closure and co-closure to obtain convex closure. I is midpoint-closed but not, of course, convex-

closed in I.

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Just as in the ancestral subject, the topology is given by a basis whose members are of
the form n o
Us = F ∈ Spec(S) : s 6∈ F
where s ∈ S. We obtain the same list of identities as in Section 23 (p??–??):
Us ∩ Ut = Us ∨ t
Us ∪ Ut = Us ∧ t
Us = Uŝ
U> = ∅
U⊥ = Spec(S)

The compactness argument for Spec(S) is also a repetition of that in Section 23, p??–??
(for Max(S)): given a subset S 0 ⊆ S the necessary and sufficient condition that the family of
sets Us : s ∈ S 0 be a cover of Spec(S) is that every prime zoom-invariant filter excludes

some element of S 0, or—put another way—that no prime zoom-invariant filter contains all
of S 0. Since every zoom-invariant filter is the intersection of the prime zoom-invariant filters
that contain it, the condition is equivalent to the zoom-invariant filter generated by S 0 being
entire, that is, the condition that ⊥ be in the set obtained by closing S 0 under finite meet
and zooming. But, of course, only a finite number of elements of S 0 can be involved in any
such demonstration and thus their corresponding basic open sets yield the finite subcover.
Normality translates to the condition on the locale of open sets of a space X, that for any
open sets V, W such that V ∪ W = X there exist open sets V 0, W 0 such that: [183]
V0 ⊇ V0
W0 ⊇ W0
V 0 ∩ W0 = ∅
V 0 ∪ W0 = X
V 0 ∪ W0 = X
Using compactness the general case reduces to the case where V and W are finite unions
of basic open sets. [184] In the case at hand we have a basis closed under finite unions. Hence
we may assume that there are elements s, t ∈ S such that V and W are of the form Us
and Ut . And it is clearly sufficient (and—left as an exercise—necessary) to find V 0, W 0 of the
form Us0 , Ut0 . The condition Us ∪ Ut = Spec(S) is equivalent to Us ∧ t = Spec(S) which is, in
turn, equivalent to the condition that a finite iteration of >-zooming pushes s ∧ t down to
⊥. Since zooming distributes with the lattice operations we may replace s, t with the result
of that finite iteration. Hence we may assume that s ∧ t = ⊥. The same maneuver applies
to s0, t0. The condition V ⊇ V 0 translates, of course, to V ∩ V 0 = V 0, which after sufficient
>-zooming is equivalent to s ∨ s0 = s0, that is, s ≤ = s0. Hence the normality of Spec(S) would
be a consequence of the condition that given s, t ∈ S such that s ∧ t = ⊥ there exist s0, t0 ∈ S
such that:
s0 ≤= s0
0
t ≤ = t0
s0 ∨ t0 = >0
s 0 ∧ t 0 = ⊥0
s 0 ∧ t 0 = ⊥0
[ 183 ] The standard definition, of course, is that any pair of disjoint closed sets has a pair of disjoint open neighborhoods. Take

V and W as the complements of the closed sets, V 0 (W 0 ) as the neighborhood of the complement of W (V ).
[ 184 ] The entire space is covered by the basic open sets that are contained either in V or in W. Chose a finite subcovering.

Replace each of V and W with the union of those basic sets that are contained therein. The union of these two replacements
is still X and any pair V ,0 W 0 that satisfies the five conditions for these replacements automatically satisfies them for the
originals.

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Using (for the last time) that Us = Uŝ we replace the last two equalities with
0∧t = ⊥
s[
∧ t0 = ⊥
s[

We thus finish with the observation that s0 = t −◦ s and t0 = s −◦ t do just what is needed:
∨ .∨
s = ⊥| s = t | s = t −◦ s = s0
≤
∨ .∨
t = ⊥| t = s | t = s −◦ t = t0
≤
s0 ∨ t0 = (t −◦ s) ∨ (s −◦ t) = >
∧ ∧
0∧t ≤
s[ = s0 | t = (t −◦ s) | t = t ∧ s = ⊥
∧ ∧
∧ t0 ≤
s[ = s | t0 = s | (s −◦ t) = s ∧ t = ⊥

The middle row uses, of course, the equation of linearity and the last two rows the construc-
tion of the meet operation (and the fact that x ∧ y = (x ∧ y)|(x ∧ y) ≤ = x|y).

43. Addendum: Signed-Binary Automata [2013–10–25]

Given terms “t1 |(t2 |(t3 |(· · ·” and “u1 |(u2 |(u3 |(· · ·” their midpoint is “v1 |(v2 |(v3 |(· · ·” where
vi = ti |ui . We can easily construct an instantaneous automaton that produces a stream of
symbols from the set {⊥, , , , >} where the quarter-moons are defined by = ⊥| and
= |>. The problem, then, is to remove those quarter-moons. Consider:

|(>|x) = > |( |x) |(>|x) = |( |x)

|( |x) = > |( |x) |( |x) = |( |x)
|( |x) = |(>|x) |( |x) = |(⊥|x)
|( |x) = |( |x) |( |x) = ⊥ |( |x)
[??]
|(⊥|x) = |( |x) |(⊥|x) = ⊥ |( |x)

[185]
The significance of these computations is that each term may be replaced with one
whose left-most symbol is not a quarter-moon and that allows us to build an automaton that
systematically removes all quarter-moons from the output streams. The remarkably simple
finite automaton that results for midpointing signed-binary expansions has just three states.
Its nine pairs of input digits behaviorally group themselves into five classes defined by the
sum of the two digits, hence the inputs will be denoted with −2, −1, 0, +1 + 2. The states
will be S− , S◦ (the initial state) and S+ .
[ 185 ] An easy corollary of Theorem 15.1 (p??) but for the fastidious (see Section 46, p??–?? for subscorings):

|(>|x) = (>| )|(>|x) = >|( |x)

|( |x) = ( |>)|( |x) = ( | )|(>|x) =
“ ” “ ”
(>|⊥)|(>| ) |(>|x) = >|(⊥| ) |(>|x) = (>| )|(>|x) = >|( |x)
|( |x) = ( |>)|( |x) = |(>|x)
|( |x) = ( |>)|( |x) = ( | )|(>|x) =
“ ” “ ”
(⊥|>)|(⊥| ) |(>|x) = ⊥|(>| ) |(>|x) = (⊥| )|(>|x) = (⊥|>)|( |x) = |( |x)
|(⊥|x) = (>| )|(⊥|x) = (>|⊥)|( |x) = |( |x)
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The signed-binary midpoint automaton:

Next State Output

S− S◦ S+ S− S◦ S+
+2 S◦ S◦ S◦ +2 ◦◦ + +◦
−2 “−”
+1 S− S+ S− +1 ◦ +◦ +
−0 “ ◦ ”
0 S◦ S◦ S◦ 0 ◦− ◦ ◦+
+2 “+”
−1 S+ S− S+ −1 − ◦


−2 S◦ S◦ S◦ −2 −◦ − ◦◦


<
+2 “◦◦” S◦ −2 “◦◦”

<

<
0 “◦−” 0 “◦+”
−2 “−◦” −1 +1 +2 “+◦”

S

<
<
<
<
S− −1 “−” <
+
+1 “◦”  < +1 “+”  −1 “◦”
The states may be interpreted as follows: in state S◦ the midpoint of the input streams (so
far) is equal to the present output. Whenever we leave S◦ a stammer occurs and the machine
moves to either S+ or S− ; in S+ the midpoint of the present input streams is larger; in S− it’s
smaller. Whenever we return to S◦ a stutter occurs: two output digits. The machine is never
more than one output digit behind the number of input pairs (that is, between every pair of
stammers there’s a stutter).[186]
It is worth noting that if both input streams are without bad tails then so is the output:
whenever returning to S◦ a “◦” is produced hence an output bad tail would require the
machine eventually to stay entirely in the two lower states, S+ and S− , or eventually to stay
entirely in the single state S◦ ; in the 1st case the output would not have any adjacent pairs
of +s or −s; in the 2nd case a bad tail would be produced only if both input streams become
eventually all +s or eventually all −s (and if neither has a bad stream then both converge to
> or both to ⊥).

[ 186 ] A non-stuttering machine (with six states, one of which, I, is strict initial) is available:
Next State Output
I S−2 S−1 S0 S+1 S+2 I S−2 S−1 S0 S+1 S+2
+2 S+2 S+2 S0 S+2 S0 S+2 +2 − ◦ ◦ + +
+1 S+1 S+1 S−1 S+1 S−1 S+1 +1 − ◦ ◦ + +
0 S0 S0 S−2 S0 S+2 S0 0 − ◦ ◦ ◦ +
−1 S−1 S−1 S+1 S−1 S+1 S−1 −1 − − ◦ ◦ +
−2 S−2 S−2 S0 S−2 S0 S−2 −2 − − ◦ ◦ +

A stammer occurs at the very beginning, thereafter it is always exactly one output digit behind the number of input pairs;
alternatively, a machine that doesn’t stammer until it has to:
Next State Output
I S−2 S−1 S0 S+1 S+2 I S−2 S−1 S0 S+1 S+2
+2 I S+2 S0 S+2 S0 S+2 +2 + − ◦ ◦ + +
+1 S+1 S+1 S−1 S+1 S−1 S+1 +1 − ◦ ◦ + +
0 I S0 S−2 S0 S+2 S0 0 ◦ − ◦ ◦ ◦ +
−1 S−1 S−1 S+1 S−1 S+1 S−1 −1 − − ◦ ◦ +
−2 I S−2 S0 S−2 S0 S−2 −2 − − − ◦ ◦ +

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The lattice operations require, curiously, an automaton larger than the midpoint opera-
tion. First—for reasons to become clear—consider an automaton with seven states denoted
L3 , L2 , L1 , M (the initial state),U1 , U2 , U3 . We imagine an “upper” stream of signed binary
digits and a “lower” one. The nine input pairs again group themselves into five classes for
the next-state behavior. It is determined by their difference: the upper digit minus the lower
digit. The signed-binary lattice automaton;
−2, −1, 0, +1, +2


<
U3

<
+1,+2 −1, 0, +1, +2

<
Next State


0 <
L3 L2 L1 M U1 U2 U3 −1 U1 <
U2 −2

<


<

<
−2
+2 L3 L2 M U2 U3 U3 U3 +2
+1


<
+1 L3 L3 L1 U1 U3 U3 U3 M< 0


0 L3 L3 L2 M U2 U3 U3 −1

<
−2
−1 L3 L3 L3 L1 U1 U3 U3  +2

<
<


<
−2 L3 L3 L3 L2 M U2 U3 L L2


<
+1
 1 0 +2

<
−2, −1
 −2, −1, 0, +1

<
<
<
L3

−2, −1, 0, +1, +2

Before considering the output let us interpret the states: M occurs when the streams
presently describe the same number; the U-states occur when the upper stream presently
describes a number larger than the lower; U1 when it is possible that the upper stream will
end up smaller;[187] U2 and U3 when it is known that the upper stream will henceforth always
describe a larger number; U2 when it is possible that the numbers will converge, that is, even
though the upper will always be larger the difference may go to zero; U3 when it is known
that the numbers will not converge, that is, the difference is bounded away from zero. For
the Ls just replace, obviously, the upper s with lower s.
For the “max” operation define the output so that it echos the upper digit whenever
moving to or from a U-state and echos the lower digit whenever moving to or from an
L-state. Note that the only times when the automaton stays in state M is when the upper
and lower digits are equal—in that case echo that unique digit. (There is no direct motion
between the L- and U-states.) For the “min” operation just reverse, obviously, these output
rules.
For the lattice operations one can conflate the outer pairs of states, that is, we can replace
every subscript 3 with the subscript 2 to obtain a five-state machine.
But if we wish for a machine that can tell us when the two streams are describing neces-
sarily unequal numbers we need at least six states. It is possible in states U2 and L2 that the
numbers are equal (e.g., in state U2 when all subsequent upper digits are −1 and lower digits
+1). The six states required for an “apartness” machine can be realized by conflating L3
and U3 . The seven-state machine will do both tasks: the lattice operations and the apartness
information.
[ 187 ] If we view the automaton as a Markov process with two absorbing states and if we take the digits as randomly equidis-

tributed then the odds are 13 to 1 that the machine will eventually reach U3 starting in U1 . Using, instead, the distribution
suggested by the “better-stream” automaton (see below), to wit, a 1/2 probability of ◦ and 1/4 for each of + and − the
odds are 23 to 1. The expected waiting times are left, of course, as an exercise.

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As with the midpoint machine, the lattice machine does not create bad tails: if it were
then at least one of the input streams would have to converge to a dyadic rational different
from either > or ⊥ but by assumption any such input would have to end with all ◦s. [188]
The contrapuntal procedure produces only OK streams and the set of OK streams is closed
under action by the zoom, midpoint and lattice machines. But ultimately there will be a need
for a finite automaton that converts any stream into an OK Stream. Actually we are handed
one, to wit, the result, essentially, of feeding a signed-binary stream into the contrapuntal
procedure. But it can be much simplified,

The OK-stream automaton:

I
− ◦ +
◦  “◦”
< <
<

◦ “−” ◦ “+”



<

<
<

S− S◦ S+
< <
− “−” + “+”



<

<
”
◦−

◦
“◦

+ −
◦“

− +
+
”
<

 <
<

<


+ “+” 
+ “◦” S◦−< S◦<+ − “◦”


< <
− “−”

All down-slopping arrows are stammers, all up-slopping arrows are stutters. The two states
S− , S+ occur only if all inputs have been the same non-zero digit, hence can not be producing
a bad tail. It is easy to check that starting from any other state, an input triple of just
+s or just −s will always produce an output that includes a ◦. That the output stream is
numerically equivalent to the input stream can be checked by noting that at each step the
present input word is numerically equivalent to the result of catenating the present output-
word with the word that appears as the subscript in the name of the state (where, of course,
the subscript of I is the empty word).

We defined a good stream to be one that described a path that hit an infinity of monodes
(on the Houdini diagram). The stream-docking automata of the last section does not always
produce good streams. (The stream + − + − + − · · · is an example of such) To convert any
stream into a good stream we could follow a OK stream machine with the machine on page
??.

[ 188 ] If we are confident that the input streams are without bad tails we can get by with just 5 states: the states with subscript

2 can be merged with the those with subscript 3. Such can yield quicker apartness results. The words −+k and ◦ k − are
numerically equal but when we know that there are no bad tails apartness from 0 can be inferred k+1 digits earlier with the
first than with the second.

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Better, use this good-stream automaton that does both:

I
− ◦ +
◦ “◦”


< <

<
◦ “−” ◦ “+”



<


<

<
S− S◦ S+
< <
− “−” + “+”



<
<
”

<
− ◦

<
“◦ “◦
+ ◦ + −
−”

◦ “◦
◦ “◦ −

++

<
<

− +
 <


”
+ “◦” S◦− < +< <
S◦< + − “◦”



<
◦”
<
“+
◦” “−

<
−
+ “◦−” − + − “◦+”
<

<

S◦−− S◦++

As for the earlier machine, all down-slopping arrows are stammers, all up-sloping arrows
are stutters. The only outputs that don’t include a ◦ are from just two states, S− , S+ , and
those two states occur only on the edges. All outputs, therefore, include a monode. The same
argument as previously yields that the input and output streams are numerically equal.

The array below describes a stutter-free version of the good stream automaton. The 28
states are named by words of signed binary digits of length of 0,1,2 and 3. (The initial state
is named by the empty word.) They appear in the leftmost column. The input digits appear
on the top row. For the 11 transient states (that is, those named by words of length less
than 3) each entry describes the next state. For the 17 recurring states named by words of
length 3, each entry describes, first, the output digit (between the quotation marks), second
the next state. Each entry has the property that the catenation of the output symbol with
the “goto-word” is numerically equal to the catenation of the name of the present state with
the input.

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The stutter-free good-stream automaton:

− ◦ +
− ◦ +
+ ◦+ +◦ ++
◦ ◦− ◦◦ ◦+
− −− −◦ ◦−
++ + ◦+ ++ ◦ +++
+◦ + ◦− +◦◦ + ◦+
◦+ ◦ ◦+ ◦+ ◦ ◦++
◦◦ ◦ ◦− ◦◦◦ ◦ ◦+
◦− ◦ −− ◦+ ◦ ◦++
−◦ − ◦− −◦◦ − ◦+
−− −−− −− ◦ − ◦−

− ◦ +
+++ “+ ” + ◦+ “+ ” ++ ◦ “+ ” +++
++ ◦ “+ ” + ◦− “+ ” +◦◦ “+ ” + ◦+
+ ◦+ “+ ” ◦ ◦+ “+ ” ◦+ ◦ “+ ” ◦++
+◦◦ “+ ” ◦ ◦− “+ ” ◦◦◦ “+ ” ◦ ◦+
+ ◦− “+ ” ◦ −− “+ ” ◦− ◦ “+ ” ◦ −+
◦++ “◦ ” + ◦+ “◦ ” ++ ◦ “+ ” ◦ ◦−
◦+ ◦ “◦ ” + ◦− “◦ ” +◦◦ “◦ ” + ◦+
◦ ◦+ “◦ ” ◦ ◦+ “◦ ” ◦+ ◦ “◦ ” ◦++
◦◦◦ “◦ ” ◦ ◦− “◦ ” ◦◦◦ “◦ ” ◦ ◦+
◦ ◦− “◦ ” ◦ −− “◦ ” ◦− ◦ “◦ ” ◦ ◦−
◦− ◦ “◦ ” − ◦− “◦ ” −◦◦ “◦ ” − ◦+
◦ −− “− ” ◦ ◦+ “◦ ” −− ◦ “◦ ” − ◦−
− ◦+ “− ” ◦ ◦+ “− ” ◦+ ◦ “− ” ◦++
−◦◦ “− ” ◦ ◦− “− ” ◦◦◦ “− ” ◦ ◦+
− ◦− “− ” ◦ −− “− ” ◦− ◦ “− ” ◦ ◦−
−− ◦ “− ” − ◦− “− ” −◦◦ “− ” − ◦+
−−− “− ” −−− “− ” −− ◦ “− ” − ◦−

But there’s a better automaton. By adding two more states we obtain the machine shown
on the next page, the output of which is asymptotically at least half monode. It suffices to
note that each “output word” (a word that’s enclosed in quotation marks) is at least half
monode.[189] Indeed, the output is not just asymptotically monodal, at least half of any even-
length initial segment is monodal.[190] (If any initial word is a catenation of output-words
we’re done. If not, then it’s a catenation of goto-words followed by an initial segment of
one of the output-words. But since each even-length initial segment of an output-word is
half monodal, we need worry only about a catenation of output-words followed by an initial
output-word segment of odd length. In all such initial segments the number of non-monodes
is at most one more than the number of monodes. The hypothesis—the assumption that we’re
looking at an initial segment of even length—forces at least one of the previous output-words
to be of length one. And any such word consists of a single monode.)

[ 189 ] To wit, the two single-digit edge-outputs. plus ◦ , ◦ −, ◦ +, + ◦ , − ◦ , ◦ −− ◦ , ◦ ++ ◦ .

[ 190 ] Hence for any initial segment of length n the number of monodes is at least the integer part of n/2.

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The better-stream automaton:[191]

I
J
J

− ◦ +
◦ “◦”


< <

<
◦ “−” ◦ “+”



<


<

<
S− S◦ S+
< <
− “−” + “+”



<
<
” ◦“

<
<
−
“◦ ◦+
◦

◦”
+ ” −

◦ “ ◦+
◦ “◦ −−

<
<

+◦”
− +



+ “◦” S◦−<< <
S◦<+ − “◦”



< <
”
“− ◦
+“
+◦

<
<

” −
+ “◦−” − + − “ ◦+”
<

S◦−− <
S◦++
< < <
<

+
“+
◦” “−
◦”
+ “◦ −” ◦ ◦ − “◦+”
−
<

<

S◦−−◦ S◦++◦ [??]

Can we do better with an even bigger machine? Not if the goal is always asymptotically
better than half monodal. A fairly simple argument shows that there can not be any pair of
adjacent ◦s in any stream of signed binary digits that converges to the number 1/3. [192]

[ 191 ] Honest, I didn’t notice its resemblance to the bejeweled—but no longer worn—symbol of power until several days after its

creation.
[ 192 ] Start with the fact that any stream starting with
◦◦ converges to an element of I between −1/4 and 1/4: if ◦◦ were
to appear in a stream converging to 1/3 let n be the number of digits prior to that ◦◦, m the integer (necessarily positive)
named by those digits and x the element in the interval from −1/4 to 1/4 named by the stream that remains after those first
n digits are removed; if x > 0 we could infer from 2n/3 = m + x that x is the “fractional part” of 2n/3, that is, it would have
to be 1/3 or 2/3; but if x < 0 then we could infer from 2n/3 + 1 = m + (1+ x) that 1+x is the fractional part of (2n + 3)/3,
that is, x would have to be −1/3 or −2/3. (Note that arbitrarily long initial segments of + ◦ − ◦ − ◦ − · · · are less than half
monodal; hence the word “asymptotically.”)

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The stutter-free better-stream automaton:

− ◦ +
+̇ +̇ +̇ + “+ ” +̇ +̇ ◦ + “+ ” . ◦
+̇ +̇ +̇ “+ ” +̇ +̇ +̇ +
+̇ +̇ +̇ ◦ “+ ” . ◦−
+̇ +̇ “+ ” . ◦◦
+̇ +̇ “+ ” +̇ +̇ ◦ +
+̇ +̇ ◦.+ “+ ” +̇ ◦. ◦+ “+ ” +̇ ◦ . ◦
. +̇ “+ ” +̇ ◦. ++
+̇ +̇ ◦ ◦ “+ ” +̇ ◦ ◦ − “+ ” +̇ ◦. ◦◦ “+ ” +̇ ◦. ◦+
− ◦ + +̇ +̇ ◦ − “+ ” . ◦ −−
+̇ “+ ” +̇ ◦ −̇ ◦ “+ ” +̇ ◦. ◦−
+̇ ◦. ++ “+ ” ◦. +̇ ◦ + “+ ” ◦. + +. ◦ “+ ” . ◦ ◦−
+̇
− ◦ +
+̇ ◦
◦+ . +̇ ◦ “+ ” ◦. +̇. ◦ − “+ ” ◦. +̇. ◦ ◦ “+ ” ◦. +̇ ◦ +
+ . ◦
+̇ +̇ +
+̇ ◦
◦ ◦− ◦◦ ◦+ . ◦.+ “+ ” ◦. ◦. ◦ + “+ ” ◦. ◦. +̇. ◦ “+ ” ◦. ◦. + +
+̇ ◦. ◦◦ “+ ” ◦. ◦ ◦ − “+ ” ◦. ◦. ◦ ◦ “+ ” ◦. ◦. ◦ +
− −̇ − −̇ ◦ ◦− +̇ ◦. ◦− “+ ” ◦. ◦ − − “+ ” ◦. ◦ −̇. ◦ “+ ” ◦. ◦ ◦ −
+̇ + +̇ ◦ + . ◦
+̇ +̇ +̇ +̇ +
+̇ ◦ −̇ ◦ “+ ” ◦ −̇. ◦ − “+ ” ◦ −̇ ◦ ◦ “+ ” ◦. −̇ ◦ +
+̇ ◦ . ◦−
+̇ . ◦◦
+̇ +̇ ◦ +
+̇ ◦ − − “+ ” . ◦ ◦+
−̇ “+ ” ◦ − −. ◦ “+ ” ◦. −̇ ◦ −
◦. + ◦. ◦ + ◦. +̇. ◦ ◦. + + ◦. + + ◦ ◦ −̇. ◦ − “◦ ”
◦◦ ◦ ◦− ◦. ◦ ◦ ◦. ◦ +
“+ ” . ◦◦
+̇ +̇ “+̇ ” ◦ ◦ −−
◦− ◦ −− ◦ −̇. ◦ ◦ ◦− ◦. +̇ ◦. + “◦ ” +̇ ◦. ◦+ “◦ ” +̇ ◦ . ◦
. +̇ “◦ ” +̇ ◦. ++
−̇ ◦ −̇ ◦ − −̇ ◦ ◦ −̇ ◦ + ◦. +̇ ◦ ◦ “◦ ” +̇ ◦ ◦ − “◦ ” +̇ ◦. ◦◦ “◦ ” +̇ ◦. ◦+
−̇ − −̇ −̇ − −̇ −̇ ◦ −̇ ◦ − ◦. +̇ ◦ − “◦ ” . ◦ −−
+̇ “◦ ” +̇ ◦ −̇ ◦ “◦ ” +̇ ◦. ◦−
+̇ +̇ ◦ + ◦. ◦. + + “◦ ” ◦. +̇ ◦ + “◦ ” ◦. + +. ◦ “◦ ” . ◦−
+̇ ◦
+̇ +̇ + +̇ +̇ +̇. ◦ +̇ +̇ +̇ +
◦. ◦. +̇ ◦ “◦ ” ◦. +̇. ◦ − “◦ ” ◦. +̇. ◦ ◦ “◦ ” ◦. +̇ ◦ +
+̇ +̇ ◦ +̇ +̇. ◦− +̇ +̇. ◦◦ +̇ +̇ ◦ +
+̇ ◦ +̇ ◦ ◦. ◦. ◦. + “◦ ” ◦. ◦. ◦ + “◦ ” ◦. ◦. +̇. ◦ “◦ ” ◦. ◦. + +
.+ . ◦+ +̇ ◦ . ◦
. +̇ +̇ ◦ . ++ ◦. ◦. ◦ ◦ “◦ ” ◦. ◦ ◦ − “◦ ” ◦. ◦. ◦ ◦ “◦ ” ◦. ◦. ◦ +
+̇ ◦ ◦ +̇ ◦ ◦ − +̇ ◦ . ◦◦ +̇ ◦ . ◦
.+
+̇ ◦ − ◦. ◦. ◦ − “◦ ” ◦. ◦ − − “◦ ” ◦. ◦ −̇. ◦ “◦ ” ◦. ◦ ◦ −
. ◦ −−
+̇ +̇ ◦ −̇ ◦ +̇ ◦ . ◦ −̇ ◦. ◦ −̇ ◦ “◦ ” ◦ −̇. ◦ − “◦ ” ◦ −̇ ◦ ◦ “◦ ” ◦. −̇ ◦ +
◦. + + ◦. +̇ ◦ + ◦. + +. ◦ . ◦ ◦−
+̇
◦. ◦ − − “◦ ” −̇ ◦
◦. +̇ ◦ ◦. +̇. ◦ − ◦. +̇. ◦ ◦ ◦. +̇ ◦ + . ◦+ “◦ ” ◦ −. − ◦ “◦ ” ◦ −̇ ◦ −
◦. ◦. + ◦. ◦. ◦ + ◦. ◦. +̇. ◦ ◦. ◦. + + ◦. −̇ ◦. + “◦ ” −̇ ◦. ◦+ “◦ ” −̇ ◦ . ◦
. +̇ “◦ ” −̇ ◦. ++
◦. ◦ ◦ ◦. ◦ ◦ − ◦. ◦. ◦ ◦ ◦. ◦. ◦ + ◦. −̇ ◦ ◦ “◦ ” −̇ ◦ ◦ − “◦ ” −̇ ◦. ◦◦ . “◦ ” −̇ ◦. ◦+
◦. ◦ − ◦. ◦ − − ◦. ◦ −̇. ◦ ◦. ◦ ◦ − ◦ −̇ ◦ − “◦ ” . ◦ −−
−̇ “◦ ” −̇ ◦ − . ◦ “◦ ” . ◦ ◦−
−̇
◦ −̇ ◦ ◦ −̇. ◦ − ◦ −̇ ◦ ◦ ◦. −̇ ◦ + ◦ −− ◦ “− ” ◦. ◦ + + “◦ ” −̇ −̇ ◦ ◦ “− ” ◦ +̇. ◦ +
−̇ ◦. ++ “− ” ◦. +̇ ◦ + “− ” ◦. + +. ◦ “− ” . ◦ ◦−
+̇
◦ −− −̇ ◦ . ◦+ ◦ −. − ◦ ◦ −̇ ◦ − −̇ ◦
−̇ ◦ −̇ ◦ . +̇ ◦ “− ” ◦. +̇. ◦ − “− ” ◦. +̇. ◦ ◦ “− ” ◦. +̇ ◦ +
.+ . ◦+ −̇ ◦ . ◦
. +̇ −̇ ◦ . ++ −̇ ◦
−̇ ◦ ◦ −̇ ◦ ◦ − −̇ ◦ . ◦.+ “− ” ◦. ◦. ◦ + “− ” ◦. ◦. +̇. ◦ “− ” ◦. ◦. + +
. ◦◦ −̇ ◦ . ◦+ −̇ ◦
−̇ ◦ − −̇ ◦ − − −̇ ◦ −̇ . ◦◦ “− ” ◦. ◦ ◦ − “− ” ◦. ◦. ◦ ◦ “− ” ◦. ◦. ◦ +
. ◦ −̇ ◦ ◦ −
−̇ ◦
−̇ −̇ ◦ −̇ −̇ ◦ − −̇ −̇ ◦ ◦ −̇ −̇ ◦ + . ◦− “− ” ◦. ◦ − − “− ” ◦. ◦ −̇. ◦ “− ” ◦. ◦ ◦ −
−̇ ◦ −̇ ◦ “− ” ◦ −̇. ◦ − “− ” ◦ −̇ ◦ ◦ “− ” ◦. −̇ ◦ +
−̇ −̇ − −̇ −̇ −̇ − −̇ −̇ −̇ ◦ −̇ −̇ ◦ −
−̇ ◦ − − “− ” −̇ ◦. ◦+ “− ” ◦ −. − ◦ “− ” ◦ −̇ ◦ −
−̇ −̇ ◦.+ “− ” −̇ ◦. ◦+ “− ” −̇ ◦ . ◦
. +̇ “− ” −̇ ◦. ++
−̇ −̇ ◦ ◦ “− ” −̇ ◦ ◦ − “− ” −̇ ◦. ◦◦ “− ” −̇ ◦. ◦+
−̇ −̇ ◦ − “− ” −̇ ◦ − − “− ” −̇ ◦ −̇. ◦ “− ” −̇ ◦ ◦ +
−̇ −̇ −̇ ◦ “− ” −̇ −̇ ◦ − “− ” −̇ −̇ ◦ ◦ “− ” −̇ −̇ ◦ +
−̇ −̇ −̇ − “− ” −̇ −̇ −̇ − “− ” −̇ −̇ −̇ ◦ “− ” −̇ −̇ ◦ −

The dots have no effect on the behavior of the machine; their purpose is only to ease a
(forthcoming) verification that the stutter-free machine’s output streams are the same as the
those of the diagrammed machine. The 67 states are named by words of signed binary digits
of length 0 through 4. (The initial state is named by the empty word.) They appear in the
leftmost columns. The input digits appear on the top row. For the 28 transient states (that
is, those named by words of length less than 4) each entry describes the next state. For the
39 recurring states (named by words of length 4) each entry describes, first, the output digit
(between the quotation marks), second the next state. Each entry has the property that the
catenation of the name of the present state with the input symbol is numerically equal to the
catenation of the output symbol with the name of the goto-state.
It’s another thing to show that it produces the same output stream as the diagrammed
machine. That’s what the dots are for. They identify those digits guaranteed to be output: if
the name of a present state begins with ẋ then the output—regardless of input—will be x;
the subsequent digits of the name marked with a dot will appear as an initial segment of the
next state—again, regardless of input). The rest of digits in the present-state name describe
the (subscript) of the state in the diagrammed machine that the stutter-free automaton is
emulating. And the rest of the digits in the goto-word describe the next state in the stutter-
free machine.[193]
[ 193 ] In case you wish to check some of these claims (including the implicit claim that every goto-state is, indeed, one of the

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There is yet one more improvement to make. As noted there may be no improvement in
the frequency of monodes but at least we can eliminate the 2-digit subwords +− and −+.
One way of course would be to follow the better stream-automaton with a fairly simple
automaton designed just for this purpose. But we can, in fact, get by with a fairly minimal
adjustment to obtain the stutter-free even-better-stream automaton:

− ◦ +
++++ “+ ” ++ ◦+ “+ ” +++ ◦ “+ ” ++++
+++ ◦ “+ ” ++ ◦− “+ ” ++ ◦ ◦ “+ ” ++ ◦+
++ ◦+ “+ ” + ◦ ◦+ “+ ” + ◦+ ◦ “+ ” + ◦++
++ ◦ ◦ “+ ” + ◦ ◦− “+ ” +◦◦◦ “+ ” + ◦ ◦+
− ◦ + ++ ◦− “+ ” + ◦ −− “+ ” + ◦− ◦ “+ ” + ◦ ◦−
+ ◦++ “+ ” ◦+ ◦+ “+ ” ◦++ ◦ “+ ” + ◦ ◦−
− ◦ +
+ ◦+ ◦ “+ ” ◦+ ◦− “+ ” ◦+ ◦ ◦ “+ ” ◦+ ◦+
+ ◦+ +◦ ++
+ ◦ ◦+ “+ ” ◦ ◦ ◦+ “+ ” ◦ ◦+ ◦ “+ ” ◦ ◦++
◦ ◦− ◦◦ ◦+ +◦◦◦ “+ ” ◦ ◦ ◦− “+ ” ◦◦◦◦ “+ ” ◦ ◦ ◦+
− −− −◦ ◦− + ◦ ◦− “+ ” ◦ ◦ −− “+ ” ◦ ◦− ◦ “+ ” ◦ ◦ ◦−
++ + ◦+ ++ ◦ +++
+ ◦− ◦ “+ ” ◦− ◦− “+ ” ◦− ◦ ◦ “+ ” ◦− ◦+
+◦ + ◦− +◦◦ + ◦+
+ ◦ −− “◦ ” + ◦ ◦+ “◦ ” + ◦+ ◦ “+ ” ◦− ◦−
◦+ ◦ ◦+ ◦+ ◦ ◦++ ◦++ ◦ “+ ” ◦− ◦− “◦ ” ++ ◦ ◦ “+ ” ◦ ◦ −−
◦◦ ◦ ◦− ◦◦◦ ◦ ◦+ ◦+ ◦+ “◦ ” + ◦ ◦+ “◦ ” + ◦+ ◦ “◦ ” + ◦++
◦− ◦ −− ◦− ◦ ◦ ◦− ◦+ ◦ ◦ “◦ ” + ◦ ◦− “◦ ” +◦◦◦ “◦ ” + ◦ ◦+
−◦ − ◦− −◦◦ − ◦+
−− −−− −− ◦ − ◦− ◦+ ◦− “◦ ” + ◦ −− “◦ ” + ◦− ◦ “◦ ” + ◦ ◦−
+++ ++ ◦+ +++ ◦ ++++ ◦ ◦++ “◦ ” ◦+ ◦+ “◦ ” ◦++ ◦ “◦ ” + ◦ ◦−
++ ◦ ++ ◦− ++ ◦ ◦ ++ ◦+ ◦ ◦+ ◦ “◦ ” ◦+ ◦− “◦ ” ◦+ ◦ ◦ “◦ ” ◦+ ◦+
+ ◦+ + ◦ ◦+ + ◦+ ◦ + ◦++ ◦ ◦ ◦+ “◦ ” ◦ ◦ ◦+ “◦ ” ◦ ◦+ ◦ “◦ ” ◦ ◦++
+◦◦ + ◦ ◦− +◦◦◦ + ◦ ◦+ ◦◦◦◦ “◦ ” ◦ ◦ ◦− “◦ ” ◦◦◦◦ “◦ ” ◦ ◦ ◦+
+ ◦− + ◦ −− + ◦− ◦ + ◦ ◦− ◦ ◦ ◦− “◦ ” ◦ ◦ −− “◦ ” ◦ ◦− ◦ “◦ ” ◦ ◦ ◦−
◦++ ◦+ ◦+ ◦++ ◦ + ◦ ◦− ◦ ◦− ◦ “◦ ” ◦− ◦− “◦ ” ◦− ◦ ◦ “◦ ” ◦− ◦+
◦+ ◦ ◦+ ◦− ◦+ ◦ ◦ ◦+ ◦+ ◦ ◦ −− “◦ ” − ◦ ◦+ “◦ ” ◦ −− ◦ “◦ ” ◦− ◦−
◦ ◦+ ◦ ◦ ◦+ ◦ ◦+ ◦ ◦ ◦++ ◦− ◦+ “◦ ” − ◦ ◦+ “◦ ” − ◦+ ◦ “◦ ” − ◦++
◦◦◦ ◦ ◦ ◦− ◦◦◦◦ ◦ ◦ ◦+ ◦− ◦ ◦ “◦ ” − ◦ ◦− “◦ ” −◦◦◦ “◦ ” − ◦ ◦+
◦ ◦− ◦ ◦ −− ◦ ◦− ◦ ◦ ◦ ◦− ◦− ◦− “◦ ” − ◦ −− “◦ ” − ◦− ◦ “◦ ” − ◦ ◦−
◦− ◦ ◦− ◦− ◦− ◦ ◦ ◦− ◦+ ◦ −− ◦ “− ” ◦ ◦++ “◦ ” −− ◦ ◦ “− ” ◦+ ◦+
− ◦++ “− ” ◦+ ◦+ “◦ ” ◦ ◦ “◦ ” ◦◦
◦ −− − ◦ ◦+ ◦ −− ◦ ◦− ◦− − ◦+ ◦ “− ” ◦+ ◦− “− ” ◦+ ◦ ◦ “− ” ◦+ ◦+
− ◦+ − ◦ ◦+ − ◦+ ◦ − ◦++
− ◦ ◦+ “− ” ◦ ◦ ◦+ “− ” ◦ ◦+ ◦ “− ” ◦ ◦++
−◦◦ − ◦ ◦− −◦◦◦ − ◦ ◦+
−◦◦◦ “− ” ◦ ◦ ◦− “− ” ◦◦◦◦ “− ” ◦ ◦ ◦+
− ◦− − ◦ −− − ◦− ◦ − ◦ ◦−
− ◦ ◦− “− ” ◦ ◦ −− “− ” ◦ ◦− ◦ “− ” ◦ ◦ ◦−
−− ◦ −− ◦ − −− ◦ ◦ −− ◦ +
− ◦− ◦ “− ” ◦− ◦− “− ” ◦− ◦ ◦ “− ” ◦− ◦+
−−− −−−− −−− ◦ −− ◦ −
− ◦ −− “− ” − ◦ ◦+ “− ” ◦ −− ◦ “− ” ◦− ◦−
−− ◦ + “− ” − ◦ ◦+ “− ” − ◦+ ◦ “− ” − ◦++
−− ◦ ◦ “− ” − ◦ ◦− “− ” −◦◦◦ “− ” − ◦ ◦+
−− ◦ − “− ” − ◦ −− “− ” − ◦− ◦ “− ” − ◦ ◦+
−−− ◦ “− ” −− ◦ − “− ” −− ◦ ◦ “− ” −− ◦ +
−−−− “− ” −−−− “− ” −−− ◦ “− ” −− ◦ −

The bold face has no effect on the behavior. Its purpose is only to mark the changes from
the previous array. In that array there are just two places where a +− is produced, the first
is immediately clear in the first of the three responses in the row:
. .
+̇ ◦ − − “+ ” −̇ ◦ ◦ + “+ ” ◦ −− ◦ “+ ” ◦ −̇ ◦ −

The second, not so immediately clear, starts also in state +̇ ◦ − − . Starting there the input
string ◦− produces the output + − . [194]
We’ll replace the row with the numerically equivalent:

+ ◦ −− “◦ ” + ◦ ◦+ “◦ ” + ◦+ ◦ “+ ” ◦− ◦−
states) you might want to use http://www.math.upenn.edu/~pjf/analysisCHECK.pdf
[ 194 ] We have now found all possibilities for the production of the output string +−, that is we have found all appearances of

“+” followed by a goto-state that starts (case 1) with −̇ or (case 2) with an undotted digit but which allows a “−” among its
output possibilities. Each case appears just once following “+” and the two appearances are in the same row.

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It strikes me as remarkable that any stream is numerically equal to one consisting of

repeated monopodes alternating with the words +, ++, − or −−. Put another way, all we
need are streams of extended Kleene form
n N N
o
{+∗ , −∗ } ◦, ◦+, +◦, ◦−, −◦ ∩ ◦, {++, +, −−, −}◦


These automata can be easily modified for use in the usual real-number context. The first
modification inserts a binary point [195] in the proper place in the output stream.[196] The
second modification is remarkably simple: add a ◦ as the first digit. We don’t need the word
“monode” (nor {+∗ , −∗ } in the Kleene expression); each even initial segment of the output
will be at most one digit away from being at least half ◦s.
We can simplify the automaton by adding not just one ◦ to the beginning but four ◦s.
We use 31 states to obtain:

The real-number even-better-stream automaton

− ◦ +
++ ◦ ◦ “+ ” + ◦ ◦− “+ ” +◦◦◦ “+ ” + ◦ ◦+
+ ◦++ “+ ” ◦+ ◦+ “+ ” ◦++ ◦ “+ ” + ◦ ◦−
+ ◦+ ◦ “+ ” ◦+ ◦− “+ ” ◦+ ◦ ◦ “+ ” ◦+ ◦+
+ ◦ ◦+ “+ ” ◦ ◦ ◦+ “+ ” ◦ ◦+ ◦ “+ ” ◦ ◦++
+◦◦◦ “+ ” ◦ ◦ ◦− “+ ” ◦◦◦◦ “+ ” ◦ ◦ ◦+
+ ◦ ◦− “+ ” ◦ ◦ −− “+ ” ◦ ◦− ◦ “+ ” ◦ ◦ ◦−
+ ◦− ◦ “+ ” ◦− ◦− “+ ” ◦− ◦ ◦ “+ ” ◦− ◦+
+ ◦ −− “◦ ” + ◦ ◦+ “◦ ” + ◦+ ◦ “+ ” ◦− ◦−
◦++ ◦ “+ ” ◦− ◦− “◦ ” ++ ◦ ◦ “+ ” ◦ ◦ −−
◦+ ◦+ “◦ ” + ◦ ◦+ “◦ ” + ◦+ ◦ “◦ ” + ◦++
◦+ ◦ ◦ “◦ ” + ◦ ◦− “◦ ” +◦◦◦ “◦ ” + ◦ ◦+
◦+ ◦− “◦ ” + ◦ −− “◦ ” + ◦− ◦ “◦ ” + ◦ ◦−
◦ ◦++ “◦ ” ◦+ ◦+ “◦ ” ◦++ ◦ “◦ ” + ◦ ◦−
◦ ◦+ ◦ “◦ ” ◦+ ◦− “◦ ” ◦+ ◦ ◦ “◦ ” ◦+ ◦+
◦ ◦ ◦+ “◦ ” ◦ ◦ ◦+ “◦ ” ◦ ◦+ ◦ “◦ ” ◦ ◦++
INITIAL STATE → ◦◦◦◦ “◦ ” ◦ ◦ ◦− “◦ ” ◦◦◦◦ “◦ ” ◦ ◦ ◦+
◦ ◦ ◦− “◦ ” ◦ ◦ −− “◦ ” ◦ ◦− ◦ “◦ ” ◦ ◦ ◦−
◦ ◦− ◦ “◦ ” ◦− ◦− “◦ ” ◦− ◦ ◦ “◦ ” ◦− ◦+
◦ ◦ −− “◦ ” − ◦ ◦+ “◦ ” ◦ −− ◦ “◦ ” ◦− ◦−
◦− ◦+ “◦ ” − ◦ ◦+ “◦ ” − ◦+ ◦ “◦ ” − ◦++
◦− ◦ ◦ “◦ ” − ◦ ◦− “◦ ” −◦◦◦ “◦ ” − ◦ ◦+
◦− ◦− “◦ ” − ◦ −− “◦ ” − ◦− ◦ “◦ ” − ◦ ◦−
◦ −− ◦ “− ” ◦ ◦++ “◦ ” −− ◦ ◦ “− ” ◦+ ◦+
− ◦++ “− ” ◦+ ◦+ “◦ ” − ◦− ◦ “◦ ” − ◦ ◦−
− ◦+ ◦ “− ” ◦+ ◦− “− ” ◦+ ◦ ◦ “− ” ◦+ ◦+
− ◦ ◦+ “− ” ◦ ◦ ◦+ “− ” ◦ ◦+ ◦ “− ” ◦ ◦++
−◦◦◦ “− ” ◦ ◦ ◦− “− ” ◦◦◦◦ “− ” ◦ ◦ ◦+
− ◦ ◦− “− ” ◦ ◦ −− “− ” ◦ ◦− ◦ “− ” ◦ ◦ ◦−
− ◦− ◦ “− ” ◦− ◦− “− ” ◦− ◦ ◦ “− ” ◦− ◦+
− ◦ −− “− ” − ◦ ◦+ “− ” ◦ −− ◦ “− ” ◦− ◦−
−− ◦ ◦ “− ” − ◦ ◦− “− ” −◦◦◦ “− ” − ◦ ◦+

Again, each entry has the property that the catenation of the name of the present state
with the input symbol is numerically equal to the catenation of the output symbol with the
name of the goto-state.
The subwords “+−” and “−+” do not appear and no subword of length four has fewer
than two ◦s (hence any initial segment is more than half ◦).
[ 195 ] in lieu of a decimal point
[ 196 ] There are two ways of defining proper place, one for midpointing, one for addition.

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It should be noted that there is a down side in using these automata: they can throw
away information. When we know that a stream can not have a bad tail a long word of the
form − + + + + · · · would be replaced by one that starts with ◦◦◦◦◦ · · · and that delays
the information that the stream will not end positively. (If the stream is the output of the
contrapuntal procedure then we’re throwing away even more: the first word is produced only
when the stream is destined to end negatively.) In [??] (p??) we pointed out that −+k can
be better than ◦k −.
We find ourselves in a situation quite familiar to those born early enough to remember
“table of values” for functions such as logs and sines. They were not values, of course, but
approximations of values. If the last digit is a 5 and you wish to drop that last digit by further
“rounding” you do not know whether to round up or down. Thus many tables used different
5s to indicate whether the 5, itself, was the result of rounding up or down.
• and ◦ .
For us, perhaps, ◦ •

44. Addendum: The Rimsky Scale and Dedekind Incisions [2015–1–13]

(draft still in progress)

I noted in Section 31 (p??–??) that a particular interpretation of Dedekind’s definition
of cuts—when interpreted in the category of sheaves on a space—delivers continuous I-
valued functions on that space. But there are other interpretations—equivalent in a Boolean
topos—that give you other functions. Dedekind didn’t need both the upper and lower cuts:
eiher one would suffice. But in the category of sheaves on a space it’s the sheaf of upper-
semicontinuous functions that’s constructed with open updeals, and—of course—it’s the sheaf
of lower-semicontinuous functions that’s constructed with open downdeals. More interesting:
consider a disjoint pair of open sets, one an open updeal, the other an open downdeal, neither
of which can be enlarged without overlapping the other. We obtain, obviously, a pair, hl, ui,
of semicontinuous fucntions, one lower-semicontinuous, one upper-semicontinuous. Not only
is T X ≤= ux for all x but l is the highest lower-semicontinuous function so related to u which,
in turn, is the lowest upper-semicontinuous function so reatled to l. I’ll call such an adjoint
pair of semicontinuous functions.
Now for something that doesn’t seem to have anything to do with categories.
A pre-continuous function is a continuous bounded partial function with a
maximal domain, that is, a domain that cannot be enlarged in a way that maintains conti-
nuity. Necessarily such a domain must be dense and with no “removable singularities.” Let
pcf(X) be the real Banach algebra whose elements are pre-continuous real-valued functions
on X.
What? Banach algebra? We’ll prove an important lemma: The domain of any pre-
continuous real-valued function on a compact metric space is “second category,” that is,
contains a dense countable intersection of open sets.
In this and the next section we’ll be working with partial functions. We’ll use
= signs between partial functions only in the case of “Kleene equality,” that is, only when
they’re equal and have the same domain. (Hence the usual formula for the linearity of deriva-
tives is not a Kleene equality.) We’ll use the “venturi tube” for a “semi-Kleene equality”:
if the left side is defined then so is the right and the values are equal. (Thus in calcu-
lus f 0 + g 0 (f + g).0 ) [197] When working with sets of partial functions it’s often useful to
[ 197 ] Note that “Kleene semi-equality” (as opposed to “semi-Kleene equality”) would refer—instead—to an ordering relation

between a pair of partial functions with the same domain).

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describe their partial ordering by as “graph-inclusion.” We’re interested in the connected

components of the set of bounded continuous partial functions on dense domains (recall that
a poset is “connected” if for every pair of elements there’s a sequence of comparable elements
connecting them). In fact, each component has a maximal element and we’ll call such a pre-
continuous function, to wit, a continuous bounded partial function with a maximal domain,
that is, a domain that cannot be enlarged in a way that maintains continuity. Necessarily
such a domain must be dense and without any “removable singularities.”
We’ll easily show that much of what we’ll be doing with scales can be generalized to
real Banach algebras. A preview: let pcf(X) be the Banach algebra whose elements are
pre-continuous real-valued functions on X.
Yes, it is a Banach algebra. The norm of f ∈ pcf(X) is the uniform norm (note that
f was required to be bounded). f + g = h and f g = h mean that the equations hold on
the intersection of the domains. To construct the limit of a Cauchy sequence {fn } start by
working on the restriction of the f s to the intersection of their domains, construct the Cauchy
limit thereon and finish by removing all removable singularities.
If I’m the first to describe such a Banach algebra I’m not the first to live in one. I wrote in
Section 32 (p??) with regard to scene analysis: “It would be a mistake to restrict to continuous
functions—sharp contour boundaries surely want to exist.” But long before scene analysis
we’ve been idealizing physical objects as closed subsets of space—or maybe open subsets. We
tried to ignore the ambiguity.[198]
Define g ↓ and g by ↑

[ 199 ]
g ↓ (x) = g(x) ∧ lim inf g(y) and g ↑ (x) = g(x) ∨ lim sup g(y)
y→x y→x

Note that g ↓ is lower-semicontinuous and g ↑ is upper-semicontinuous,[200] indeed g ↓ is the

highest lower-semicontinuous function below g and g ↑ is the lowest upper-semicontinuous
function above g. One consequence is that the operations denoted by ↓ and ↑ are clearly
idempotent and covariant.
We say that h`, ui is a adjoint pair of semicontinuous functions if ` ↑ = u and u ↓ = `.
Note well: g ↓ and g ↑ needn’t be adjoint. A revealing example: let S ⊆ X and g = χ S .
Then g ↓ is the characteristic function of the interior of S and g ↑ of its closure. When S is
open g ↑ ↓ = g ↓ means that S is a regular open set.
For any g : X → I we obtain adjoint pairs hg ↑ ↓ , g ↑ ↓ ↑ i and hg ↓ ↑ ↓ , g ↓ ↑ i and—all together—up
to seven functions: g, g ↓ , g ↑ , g ↓ ↑ , g ↑ ↓ , g ↓ ↑ ↓ , g ↑ ↓ ↑ . [201]
For a proof it helps to move to a general setting.
34.6 Lemma: On an arbitrary poset, J, let ↓ and ↑ denote a pair of idempotent covariant
endofunctions such that x ↓ ≤
= x ↑ for all x. Then the semigroup generated by ↓ , ↑ has as at most
6 elements:
↓
, ↑ , ↓↑ , ↑↓ , ↓↑↓ , ↑↓↑
[ 198 ] When two objects come in contact are they still disjoint?
[ 199 ] Note that the standard definitions of lower- and upper-semicontinuity do not employ equal signs: “ g(x) ≤ lim inf y→x g(y) ”
and “ g(x) ≥ lim supy→x g(y) ” hence the need for ∧ and ∨ in the definitions of g ↓ and g ↓ .
[ 200 ] That is, (g ↓ ) ↓ = g ↓ and (g ↑ ) ↑ = g ↑ .
[ 201 ] For an an example where all the values are different take g to be the characteristic g ( )( )
g↑ [ ] [ ]
function of a subset of I consisting, first, of an open subinterval with one point removed, g↑ ↓ ( ) ( )
second, a subinterval with its rational elements removed, third, a single isolated point. g↑ ↓ ↑ [ ] [ ]
For the record, g ↓ ≤ g ↓ ↑ ↓ ≤ h ≤ g ↑ ↓ ↑ ≤ g ↑ for any g where h can be either g ↓ ↑ or g ↑.↓ g↓ ( )( )
g↓↑ [ ]
g↓↑↓ ( )
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each of which is an idempotent. Moreover

J ↓↑↓ = J ↑↓ and J ↑↓↑ = J ↓↑
and if J is viewed as a category
J ↑ ↓ is a reflective full subcategory of J ↓
J ↓ ↑ is a co-reflective full subcategory of J ↑
If we add the condition x ↓ ≤
= x ≤
= x ↑ then
J ↓ is a co-reflective full subcategory of J
J ↑ is a reflective full subcategory of J
Finally, if J is a complete lattice then so are J ↓, J ↑, J ↓ ↑, J ↑ ↓, J ↓ ↑ ↓, J ↑ ↓ ↑.
Because, for idempotency: ↓ ↑ = ↓ ↓ ↓ ↑ ≤ = ↓↑↓↑ ≤ = ↓ ↑ ↑ ↑ = ↓ ↑ (and use the dual argument for
↑↓
= ↑ ↓ ↑ ↓ ). For ↓ ↑ ↓ and ↑ ↓ ↑ , it’s easier: ↓ ↑ ↓ ↓ ↑ ↓ = ↓ ↑ ↓ ↑ ↓ = ↓ ↑ ↓ . Given any word on ↓ and ↑
use, first, their idempotency to reduce to a word of alternating letters and, second, use the
idempotency of ↓ ↑ and ↑ ↓ to reduce to a word of at most three letters. [202]
To see that the set of elements of the form x ↓ ↑ ↓ is equal to the set of elements of the form
y ↑ ↓ : given x take y = x ↑ ; given y take x = y ↓ .
As always, reflective full subcategories are known (since their invention in 1958) by the
endofunctors called their “reflectors” and on a poset reflectors are precisely the increasing
covariant idempotents: the subset of all values (or, if preferred, of all fixed points) of any
increasing covariant idempotent f is a reflective subposet: x ≤
= f (y) iff f (x) ≤
= f (y).
When restricted to J ↓ the function described as ↑↓
is increasing: x ↓ = x ↓ ↓ ≤
= x↑ ↓.
Given any continuous g : S → R where S is a dense subset of a space X we obtain a
adjoint pair on X hg ↓, g ↑ i.
For x ∈ S, of course, g ↓ (x) = g(x) = g ↑ (x). For any x ∈ X there’s no problem in seeing
that g ↓ ↑ (x) ≤
= g ↑ (x). In order to establish that g ↓ ↑ (x) ≥
= g ↑ (x) we need to show that for any
ε > 0 and any neighborhood of x there exists y ∈ S such that g ↓ (y) ≥ = g ↑ (x) − ε. But we
know that there exists such a y in S in that neighborhood such that g(y) = ≥ g ↑ (x) − ε, But,
as just observed, g ↓ (y) = g(y). The dual proof yields g ↑ ↓ (x) = g ↓ (x).
Note that if h`, ui is an adjoint pair then it’s equal to hg ↓ , g ↑ i for any ` ≤
=g≤
= u.
Theorem: For any adjoint pair h`, ui the equalizer of ` and u is a dense subset, indeed,
the intersection of the graphs of ` and u is the graph of a pre-continuous function and all
pre-continuous functions so arise. Moreover the domain of any pre-continuous function is
substantial [203] that is, it contains a dense Gδ .
substantial,
We’ll view the equalizer of ` and u as the intersection of the sets for all positive ε :
[ 204 ]
Sε = { x ∈ X : |u(x) − `(x)| < ε }
We need that Sε is open and dense.
For any open real interval (a, b) let h`, ui-1 (a, b) ⊆ X denote

{ x ∈ X : a < l(x) and u(x) < b }

[ 202 ] Indeed, this sentence shows any idempotent semigroup with two generators has at most 6 elements.
[ 203 ] Sometimes “comeagre,” sometimes “residual,” sometimes “second category.”
[ 204 ] Yes, the absolute-value bars aren’t needed.

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Because lower-semicontinuous maps are precisely the functions whose inverse images carry
open updeals to open sets in X and dually for upper-semicontinuous maps we have that that
hl, ui-1 (a, b) is open. Hence so is
[
Sε = hl, ui-1 (x, x + ε)
x

For the density of Sε suppose that U ⊆ X is open and disjoint from Sε . We need to
show that U is empty. The characteristic map χ U is lower-semicontinuous, therefore so is
` + εχ U .[205] Since ` is the highest lower-semicontinuous map below u it must be that U is
empty.

45. Addendum: Lebesgue Integration and Measure, Rethought [2015–5–18]

In this section the interval I will be understood to be the unit interval, [0, 1].
Theories of integration and of measure are, of course, intimately related but they differ in
their motivations.
We start with the first, the theory of integration. In Section 40 (p??–??) we described a
covariant mean-value function C(In ) → I that preserves top, bottom and midpointing. We’ll
denote its values here as kf k1 and use it to establish a metric space structure on C(In ).[206]
Our goal is to extend this function to a larger scale of integrable functions denoted as L(In ).
The simplest answer was given by Peter Lax: take L(In ) to be the L1 - metric-space completion
of C(In ). [207] That simple answer was likely given by many others. Lax’s great contribution
was to give us a description of the elements of L(In ) better than “equivalence-classes-of-L1 -
Cauchy–sequences”: he gave us what he labeled “realizations.” And since we’re looking at
functions that are I-valued (rather than R-valued) it’s even easier to describe them. Hold on.
We first need just a little from the other view. Given an open set U ⊆ In define µ(U ) to
be the supremum of the mean values of all continuous h : In → I with support contained in U :

khk1 : h ∈ C(In ), h ⊆ U

µ(U ) = sup
Three important lemmas:
45.1 Lemma: If U1 ⊆ U2 ⊆ · · · then µ(U1 ∪ U2 ∪ · · ·) = supi µ(Ui )
It is clear that µ is covariant. For any h and ε where h ∈ C(In ) has its support in the
union of the Ui s and ε is positive it suffices to show that khk1 ≤
= µ(Un ) + ε for large n.
Since the compact set { x : h(x) = ≥ ε } is contained in the union of the Ui s we can find
n such that it is contained in Un . Let bh − εc be the function such that bh − εc(x) =
max{0, h(x)−ε}. Then the support of bh−εc is contained in Un and khk1 ≤ = kbh−εc+εk1 ≤ =
kbh − εck1 + kεk1 ≤ = µ(Un ) + ε.
45.2 Lemma: µ(U0 ∪ U1 )) ≤
= µ(U0 ) + µ(U1 )
[ 205 ] The easiest proof uses the view that lower-semicontinuity is equivalent to continuity where the target’s only open sets are

the open downdeals but there’s an approach here that avoids using that equivalence: replace εχ U with a positive multiple of
disthx, X \ U i. It’s easy to see that the sum of a continuous and a semicontinuous is a semicontinuous.
[ 206 ] As usual, for f, g ∈ C(In ) we understand kf − gk
1 to be the mean value of the absolute difference of f, g. Note that
k k1 is quite different from the intrinsic metric (p??), to wit, the one usually denoted k k∞ .
[ 207 ] Rethinking the Lebesgue integral. Amer. Math. Monthly 116 (2009), no. 10, 863–881. Lax wrote with regard to the other

approach (that is, the theory of measure): “In our development Lebesgue measure is a secondary notion. A set S is measurable
if its characteristic function is one of the functions in L1 . Its measure is defined as the integral of the characteristic function. To
be sure, such an approach is anathema to probabilists; their object of desire is sigma-algebra of measurable sets.”

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For any continuous h with support in U0 ∪ U1 we need a pair of continuous maps h0 , h1

such that h0 + h1 = h where the support of hi is contained in Ui . We’ll use the fact that if
f is a bounded continuous map on an open set V and g is a continuous map on the entire
space with support in V then the map equal to f g on V can to be extended to a continuous
map on the entire space by setting it to 0 on V ,C the complement of V. Hence we prove the
lemma by taking
disthx, UiC i
hi (x) = h(x)
disthx, U0C i + disthx, U1C i

These two lemmas combine for the third:

S P
45.3 Lemma: µ ( i Ui ) ≤
= i µ(Ui )
We’ll say that a set S ⊆ Im is negligible if it is contained in open sets of arbitrarily
small µ-value and that it is pervasive if its complement is negligible. An easy—but necessary
to our purposes—simplification: there’s no need for the functions to be defined everywhere.
We’ll allow partial functions whose domains are pervasive.
We’ll use the simple fact that a partial function is a special case of a relation, and a
relation—as usual—can be named by a set of ordered pairs. We replace Lax’s realization
with our virtual map, to wit, a partial function with a pervasive domain whose graph is a
countable union of closed subsets, that is, an Fσ set. (Without loss of generality, all Fσ s will
be understood to be ascending unions of closed sets.)
We’ll be using the fact that when its target is compact a function is continuous iff its
graph is a closed set of ordered pairs. [208] Hence a partial I-valued function has an Fσ graph
iff it is the union of a sequence of continuous partial functions, each with a closed domain.
And it is a virtual map if, moreover, the union of those closed domains is a pervasive set.
Lax’s definition of “realization” was a function f for which there is an L1 -Cauchy sequence
of continuous functions that almost everywhere pointwise coverges to f . We’ll use virtual
maps instead. Our first task is to find such a sequence for any virtual map. And that’s easy.
Use Tietze [209] to extend each of the continuous partial maps on closed domains to an entire
continuous map. The resulting sequence is not only pointwise convergent on the domain of
the virtual map: it is pointwise eventually constant. Done.
Fσ sets are closed, of course, under finite intersection. An easy exercise is that there’s a
pervasive set on which two virtual maps agree iff their intersection is not only Fσ but still a
æ
virtual map, that is, it still has a pervasive domain. We’ll abbreviate all that as f = g.
æ
We take the =-classes of virtual maps as the elements of L(In ). [210]

The converse is harder:

[ 208 ] Reaping once again the advantage of I over R. Note that a “quasi-inverse” function on R (indeed, any entire function
that extends the partial function which sends x 6= 0 to x-1 ) has a closed subset as graph. Much of this material, though, does
generalize to R-valued functions. Just replace “Fσ ” with “σ-compact,” that is, a countable union of compact subsets.
[ 209 ] The lemma was proved by L.E.J. Brouwer and H. Lebesgue for Rn, by H. Tietze for arbitrary metric
spaces, and by P.S. Urysohn for normal spaces. I think it noteworthy that Tietze didn’t need the axiom of
choice. (Urysohn needed it when he generalized Tietze’s theorem to normal spaces.) For the record: Given a
closed subset A of a metric space X and a continuous f : A → [−1, +1] extend f as limn→∞ gn
where gn : X → [−1, +1] is defined inductively by taking g0 to be constantly 0 and (using disthx, ∅i = 1)
n−1 disthx, B i − disthx, B i
gn x = gn−1 x + 23n disthx, B− i + disthx, B+ i with B± = x ∈ A : ±(f x − gn−1 x) ≥ 2n−1 /3n .
˘ ¯
− +

Then |f x − gn x| ≤ (2/3)n for all x ∈ A. and since |gn x − gn−1 x| ≤ 2n−1 /3n for all x ∈ X the gn s uniformly converge to
a continuous map.
[ 210 ] The set of virtual maps is partially ordered by . A poset is “connected” if for every pair of elements there’s a sequence of
comparable elements connecting them. The elements of L(In ) could be taken to be the connected compontents of the -poset
of virtual maps. (In the category of posets—if you will— L(In ) is the reflection of the -poset of virtual maps into the full
subcategory of discrete posets.)

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45.4 Theorem: (Peter Lax) Any L1 -Cauchy sequence of continuous functions converges to
a virtual map.
Given an L1 -Cauchy sequence {fn } of continuous functions we follow Lax and replace the
sequence with a “rapidly converging” subsequence, to wit, one that satisfies the condition
= 1/4n. (Working in the metric completion we simply chose each fn to be within
kfn − fn+1 k1 ≤
1/22n+1 of the sequence’s limit.) [211]
It suffices to find an ascending chain of closed sets {An } such that the fn s converge
uniformly on Ai for each i and such that A1 ∪ A2 ∪ · · · is pervasive and we’ll do that by
first defining a sequence of (not—in general—descending) open sets {Un } and then defining
each An to be the complement of Un ∪ Un+1 ∪ Un+2 ∪ · · · . (The Ui s will be what Swe could
call “obstructions to uniform convergence”) We will guarantee pervasiveness of i Ai by
guarantying the finiteness of Σi µ(Ui ). [212]
The critical definition: Un = { x : |fn (x) − fn+1 (x)| > 1/2n } .
= 1/2n.
The critical proposition: µ(Un ) ≤
The proof of the proposition: suppose that µ(Un ) > 1/2n. There would be h ∈ C(In ) with
suppport in Un and with mean value greater than 1/2n. But then
|fn (x) − fn+1 (x)| > h(x)/2n for all x ∈ Un and, of course, for all x 6∈ Un . All of
which makes kfn − fn+1 k1 greater than 1/4n.
= 1/2i for each i =
Thus on each An we have |fi − fi+1 | ≤ ≥ n and the fi s uniformly converge
to a continuous function on each An .
p
(A little modification of the proof works for the L -norm for any finite p larger than 1. [213]
Given any two L1 -Cauchy sequences of continuous functions that pointwise converge to the
same virtual map it’s easy to check that the sequence of absolute differences converges to the
function that’s constantly 0. In particular, their L1 -norms converge to each other and we use
that to define the L1 -structure for L(In ).)
æ æ
We extend the meaning of = to arbitrary functions: g = f if they agree on a pervasive
set. We say that f : In → I is a measurable function if there is a virtual map g such that
æ
g = f.
Now for the other approach, the theory of Lebesgue measure. Theorem 24.7 (p??) may
be viewed as its foundation: any order-complete scale is the injective envelope of its subscale
generated by its extreme points. The extreme points in L(In ) are the elements of the form
χ S where S ⊆ In. Note that we’re not allowing any old subset, just those for which
χ S is measurable, or as usually said, is a measurable subset. We write S1 = æ
S2 when
æ χ
χ S1 = [214]
S2 . Easy lemmas tell us that the family of measurable subsets is a σ-algebra,
that is, a countably complete Boolean algebra. When we move to the quotient Boolean algebra
æ
obtained by identifying =-classes we obtain a complete Boolean algebra.[215]
[ 211 ] The big difference between this proof and Lax’s is that his n2 becomes my 2n (and his n4 becomes my 4n ). I don’t

know why Lax appears to prefer Bernoulli to Zeno (but, alas, looking at n^2 and 2^n I know too well how strephosymbolia
would have caused me to so appear).
[ 212 ] I don’t think I could ever have come up with this plan of attack.
[ 213 ] Note that the rapidly convergent subsequence converges pointwise to the virtual map. The Carleson-Hunt theorem says

any subsequence converges pointwise ae. If we take R as the target, instead of I the proof continues to work but, as pointed
out in footnote [??], the definition of virtual map has to be changed: replace Fσ with “countable union of compact subsets.”
[ 214 ] Eqivalently when the symmetric difference, (S \ S ) ∪ (S \ S ), is negligible.
1 2 2 1
[ 215 ] When I noticed this decades ago, I was surprised. So—amazingly—was everybody I told it to. But it’s an easy consequence

of countable completeness and the fact that an element of maximal L1 -value in a sub-family closed under countable unions is
necessarily that sub-family’s maximum element.

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The argument that the characteristic functions in L(In ) form a complete Boolean algebra
works as well for all of L(In ). In Section 23 (p??–??) we saw that such means that L(In ) is
an injective scale. Alas, it is not the injective envelope of C(In ). Lemma(24.5) (p??) says that
such would require C(In )? to be cofinal in L(In )? . But if U is a non-pervasive dense open
set [216] then there’s only one continuous function f such that χ U ≤ = f.
The problem, then, is to find a simply described scale whose injective envelope is L(In ).
One solution is the Boolean-algebra scale [217] I[B] where B is the minimal Boolean algebra
that contains a copy of the lattice of open sets in In. Start first with the sub-Boolean algebra in
the power-set of In generated by the open sets, traditionally called the family of “constructible
sets.” Such does not satisfy the minimallity condition. So reduce by the ideal of negligiable
subsets.
This reduction needs to be looked at carefully. Any construtible set is a finite union of
locally closed subsets, the latter being an intersection of an open and a closed.
[218]
MORE TO COME

[ 216 ] Take a countable dense subset and cover its j th element with an open neighborhood of measure 1/4j+.1
[ 217 ] Section 38 (p??–??)
[ 218 ] Given countable atomless Boolean algebras we can build a one-to-one isomorphism between them by creating an ascending

sequence of finite boolean algebras in each atomless algebra, each finite algebra being equipped with an isomorphism with its
corresponding finite algebra, each such isomorphism being an extension of the isomorphism between the previous pair of algebras.
Start each sequence with the two-element boolean algebra. Thereafter we follow the “back-and-forth” strategy Cantor used to
construct isomorphisms between countable “densely” ordered sets, that is, we alternatingly choose an element in one of the two
atomless algebras not yet involved in the correspondence and find an element in the other that will be “similarly situated,” that
is, will yield an extension of the isomorphism between the two subalgebras that result from the choices. First, note that a pair
of finite Boolean algebras are isomorphic iff they have the same number of atoms, moreover, any one-to-one correspondence
between their sets of atoms extends uniquely to an isomorphism between the algebras. So when we choose an element not yet in
the correspondence it generates a larger finite algebra. Some of the old atoms may be contained in the new element, some may be
disjoint from the new element, some may split (into two new atoms) because of the new element. For each such old atom chose
a splitting of its corresponding element in the other atomless algebra and obtain two sets of new atoms with a correspondence
between them. Now extended that correspondence to an isomorphism between the finite algebras thus generated.

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46. Addendum: A Few Subscorings [2015–7–27]

It is said that “subscoring” is short for “substitution underscoring,” to wit, a one-column

array wherein the underscores indicate the sub-strings to be replaced.[219]

Page ??

a/ (a|x) : .
x (u|v)
. .
(((a |⊥)|(a|x))∨ ) ∧ . . : (u|v)/ ((u|v)|(u|v) )
x / ( x | x)
. .
(((a |a)|(⊥|x))∨ ) ∧ . : . (u|v)/ ((u|v) |(u|v))
x / ( x | x)
(((⊥|>)|(⊥|x))∨ ) ∧ . (u|v)/
x/
( (⊥|(>|x))∨ ) ∧ (u|v)/ ( | )
. .
x / (x |x)
(>|x) ∧ . .
(u|v)/ (( u |u)|( v |v))
x
x . .
(u|v)/ ((u| u)|( v| v))
. .
(u|v)/ ((u|v)|(u | v))
. .
u|v

[ 219 ] The macro {\scor}[1]{\uuline{\rule[-7pt]{0pt}{0pt}#1}} (using package ulem)

is useful in their construction.

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Page ??n

. . y|x =
x = v

x y
. . (⊥|y)∨
x / (x |x)
. . . .
x / (v |v) (((⊥|y) )∧ )
. .
. .
x / (x |v) (y/ (y|((⊥|y) )∧ ))
. .
(y/ (((⊥|y)∨ |((⊥|y) )∧ ))
v
. . . .
(y/ ((((⊥|y) )∧ ) |((⊥|y) )∧ ))
.
(y/ )
.
(y/ (y|x))
.
x

x| x|y
. /( |(x|y))
(x|x)|(x|x)
. / (( |x)|( |y))
(x| x)|(x|x)
: . / (( |x)|(y| ))
(x | x)|x
/ (( |y)|(x| ))
|x
/ (( |y)|( |x))

/ ( |(y|x))

y|x

∧
x| ⊥
∧
x/ (x|(x | ⊥))
∨ ∧
x/ ((x | ⊥)|(x | ⊥))

x/ (x |⊥)

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Page ??–?? (z|x) −◦ (z|y)

(u|v) −◦ (w|x)
.
((z|x) |(z|y))∨ .
((u|v) |(w|x))∨
. .
((z | x)|(z|y))∨ . .
((u | v)|(w|x))∨
. .
((z |z)|(x |y))∨ . .
((u |w)|(v |x))∨
.
( |(x |y))∨ . . .
((u| w) |(v |x))∨
.
>|(x |y)∨ . .
(u| w) −◦ (v |x)
>|(x −◦ y)

Page ??n . .
. a/ (a/ (( |a)|( |x) ))
/ ( / ((a | )|(a|x))
. .
. a/ (a/ (( |a)|( | x)))
/ ( / (( a |a)|( |x)))
. .
/ ( / ( |( |x))) a/ (a/ ( |(a| x)))
. . .
/ ( |x) a/ (a/ ((a| a)|(a| x)))
. . .
x a/ (a/ (a|(a | x)))
. . .
a/ (a | x)

a/ (a|x)

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Page ??n x≡y

¬x ¬y

¬x ∧ > > ∧ ¬y

¬x ∧ (y ∨ ¬y) (¬x ∨ x) ∧ ¬y

¬x ∧ (x ∨ ¬y) (¬x ∨ y) ∧ ¬y

(¬x ∧ x) ∨ (¬x ∧ ¬y) (¬x ∧ ¬y) ∨ (y ∧ ¬y)

⊥ ∨ (¬x ∧ ¬y) (¬x ∧ ¬y) ∨ ⊥

¬x ∧ ¬y

x2 x∗ = x = x∗∗
xx∗ = x∗ x
x≡y
x∗ y∗

x∗2 x yy ∗2

x∗2 y xy ∗2

x∗2 yyy ∗ x∗ xxy ∗2

x∗2 xyy ∗ y ∗ x∗ x∗ xxxy ∗2

x∗2 y 3 y ∗2 ≡ x∗2 x3 y ∗2

u≡v
. .
u v
. .
/ ( | u) / (v | )
. . . .
/ ((v |v)| u) / (v |(u |u))
. . . .
/ ((v |u)| u) / ( v |(u |v))
. . . . . .
/ ((v | u)|(u| u)) / ((v | u)|(v |v))
. .
/ ((v | u)| )

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Page ??
e

e∨⊥
.
e ∨ (e ∧ e)
.
e ∨ (e ∧ e)
.
(e ∨ e) ∧ (e ∨ e)
. .
e ∧ (e ∨ e) ≥
= e ∧ (e ∨ e)

e ∧>

e
Page ??n e

e+0

e + (1−e)0

e + (1−e)0

e + (1−e)(1−e)e

e + (1−e)(1−e) e

e + (1−e)e

e + e − ee

e + e−e

Page ??n

bx − bycc bbxc − bycc

0 + bx − bycc 0 + bbxc − bycc

−byc + byc + (0 ∨ (x − byc)) −byc + byc + (0 ∨ (bxc − byc))

−byc + ((byc + 0) ∨ (byc + (x − byc))) −byc + ((byc + 0) ∨ (byc + (bxc − byc)))

−byc + (byc ∨ x) −byc + (byc ∨ bxc)

− byc + (((y ∨ 0) ∨ x) =
= −byc + (((y ∨ 0) ∨ (x ∨ 0))

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b0c

bb0cc − b − b0cc

bb0cc − b0 − b0cc

bb0cc − bb0c − b0cc

bb0cc − b0c

bbb0cc − b0cc − bb0c − bb0ccc

bb0c − b0cc − bbb0cc − bb0ccc

b0c − b0c

Page ??n

|( |x) |( |x)

( |>)|( |x) ( |>)|( |x)

( | )|(>|x) ( | )|(>|x)

((>|⊥)|(>| ))|(>|x) ((⊥|>)|(⊥| ))|(>|x)

(>|(⊥| ))|(>|x) (⊥|(>| ))|(>|x)

(>| )|(>|x) (⊥| )|(>|x)

>|( |x) (⊥|>)|( |x)

|( |x)

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(U − V ) ∪ (U 0− V 0 )

[(U − V )C ∩ (U 0− V 0 )] + [(U − V ) ∩ (U 0− V 0 )] + [(U − V ) ∩ (U 0− V 0 )C )]

[(U C ∩ (U 0− V 0 )] + [V ∩ (U 0− V 0 )] + [(U − V ) ∩ (U 0− V 0 )] + [(U − V ) ∩ U 0 C ] + [(U − V ) ∩ V 0 ]

[(X − U ) ∩ (U 0− V 0 )] + [(V − ∅) ∩ (U 0− V 0 )] + [(U − V ) ∩ (U 0− V 0 )] + [(U − V ) ∩ (X − U 0 )] + [(U − V ) ∩ (V 0− ∅)]

[U 0 \(U ∪ V 0 )] + [U 0 ∩ V )\V 0 ] + [(U ∩ U 0 )\(V ∪ V 0 )] + [U \(U 0 ∪ V )] + [(U ∪ V 0 )\V ]

[(U ∪ U 0 ) − (U ∪ V 0 )] + [(U 0 ∩ V ) − (V ∩ V 0 )] + [(U ∩ U 0 )\(V ∪ V 0 )] + [(U ∪ U 0 ) − (U 0 ∪ V )] + [(U ∪ V 0 ) − (V ∩ V 0 )]

J
J
Department of Mathematics
University of Pennsylvania
Philadelphia, PA 19104
pjf @ upenn.edu

Available at
http://www.math.upenn.edu/~pjf/analysis.pdf
And check out
http://www.math.upenn.edu/~pjf/e-pi.pdf

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