Advances in Algebra

ISSN 0973-6964 Volume 9, Number 1 (2016), pp. 11-15

© Research India Publications
http://www.ripublication.com

A study on the Classes of Semirings and Ordered

Semirings

N. Sulochana*1, M. Amala2 and T.Vasanthi3

1
Dept of Mathematics, K.S.R.M College of Engineering, Kadapa, Andhra Pradesh, India
2,3
Dept. of Applied Mathematics, Yogi Vemana University, Kadapa, Andhra Pradesh, India
E‐mail: sulochananagam@gmail.com, amalamaduri@gmail.com and vasanthitm@gmail.com

Abstract

In this paper we have proved that if S is an E–inverse semiring and (S,) is a

rectangular band, then we get two different outcomes under two different conditions.

AMS Mathematics Subject Classification (2010): 20M10, 16Y60.

Keywords: Minimum, Rectangular band, Semilattice, Weak commutative.

1. INTRODUCTION
The concept of semirings was first introduced by Vandiver in 1934. Semirings arise
naturally in such diverse areas of mathematics as combinatorics, functional analysis,
topology, graph theory, Euclidean geometry, probability theory, commutative, non-
commutative ring theory and the mathematical modeling of quantum physics and
parallel computation systems. . The developments of semirings and ordered semirings
in this direction require semigroup techniques. It is well known that if the
multiplicative structure of an ordered semiring is a rectangular band, then its additive
structure is a band. In the recent papers on ordered semirings, the works of
M.Satyanarayana [8,9,10] has studied how far the properties of multiplicative
structure are reflected in the additive structure and vice-versa.
In this paper we have two sections. Section one contains classes of E-inverse
semirings and section two deals with classes of Ordered left regular semirings.

2. PRELIMINARIES:
Definition 2.1:
A semiring is called a E –inverse semiring if ab + acb = ab for every a, b, c in S.

Definition 2.2:
A semigroup (S, •) is weak commutative if abc = bac for all a, b, c in S.
12 N. Sulochana, M.Amala and T.Vasanthi

Definition 2.3:
A semigroup (S, •) is rectangular band if a = aba for all a, b in S.

Definition 2.4:
In a semiring S, an element a is Multiplicatively Subidempotent if a + a2 = a. A semiring S is
Multiplicatively Subidempotent if and only if each of its elements is Multiplicatively
Subidempotent.

Definition 2.5:
A semigroup (S, •) is left (right) singular if a + b = a (a + b = b) for all a, b in S.

Definition 2.6:
A semigroup (S, •) is a band if a2 = a for all a in S.

Definition 2.7:
An element x in a t.o.s.r is minimum (maximum) if x  a (x  a) for every a in S.

Definition 2.8:
A semigroup (S, +) is semilattice if (S, +) is a band and commutative.
i.e a + a = a and a + b = b + a for all a, b in S.

Definition 2.9:
In a totally ordered semiring (S, +, •, )
(i) (S, +, ) is positively totally ordered (p.t.o), if a + b  a, b for all a, b in S.
(ii) (S, •, ) is positively totally ordered (p.t.o), if ab  a, b for all a, b in S.
(iii) (S, +, ) is negatively totally ordered (n.t.o), if a + b ≤ a, b for all a, b in S.
(iv) (S, •, ) is positively totally ordered (n.t.o), if ab ≤ a, b for all a, b in S.

3. CLASSES OF E – INVERSE SEMIRINGS:

The concept of an E-inversive semigroup was constructed by G.Thierrin [13] and developed
by Lallement and Petrich. But Petrich [6, 7] was studied in some what different form of
Lallement and Petrich. This type of semigroups recently reapeared in papers Hall and Munn,
F.Catino and M.M.Miccoli, Margolis and Pin and H.Mitsch.

Theorem 3.1: Let S be an E – inverse semiring and (S, ·) be a rectangular band.

(i) If (S, ·) is right cancellative, then (S, ·) is multiplicatively subidempotent.
(ii) If (S, ·) is weak commutative, then a2 + a = a2 for all a in S.

Proof: (i) Given that (S, ·) is rectangular band then aba = a for all a, b in S
From the definition of E-inverse semiring ab + acb = ab------ (1)
 abc + acbc = abc  abc + ac = abc  ab + a = ab  aba + a2 = aba
 a + a2 = a
Hence (S, ·) is multiplicatively subidempotent

(ii) By hypothesis S is an E – inverse semiring then ab + acb = ab -------(i)

Since (S, ·) is weak commutative then ab + abc = ab  bab +babc = bab
By using the definition of rectangular band in above it takes the form b + bc = b
 b2 + bcb = b2  b2 + b = b2 -------(ii)
A study on the Classes of Semirings and Ordered Semirings 13

Now let us check that a2 + a = a2

For this we take the equation (i) as ab + cab = ab  aba + caba = aba  a + ca = a
 a2 + aca = a2  a2 + a = a2 --------(iii)
Now let us take, a + ca = a  ac + cac = ac  ac + c = ac  cac + c2 = cac
 c + c2 = c
Since c3 = c we have c2 + c3 = c2  c2 + c = c2 --------------(iv)
Thus we obtained a2 + a = a2 for all a, b, c in S

Theorem 3.2 : Suppose S is an E –inverse semiring. If ‘1’ is multiplicative and also additive
identity. Then
(i) ab + ab = ab.
(ii) ac = a and cb = b.

Proof: (i) By the definition of E –inverse semiring ab + acb = ab for all a, b in S ---- (I)
It implies a [b + cb] = ab  a [1 + c] b = ab
Since 1 is multiplicative identity and also additive identity then 1 + c = c
Thus we obtain acb = ab Now equation (1) can take the form as ab + ab = ab

(ii) Let us check whether ac = a

For this let us consider b =1 in equation (I) we get a.1 + ac.1 = a.1
 a.1 + ac = a  a (1 + c) = a  ac = a
Now we proceed on to show that cb = b
For this let us take a = 1in first condition it leads to 1.b + 1.cb = 1.b
 (1 + 1.c) b = b  cb = b
Hence ac = a and cb = b

Theorem 3.3 : If S is an E –inverse semiring and (S, ·) is left singular semigroup, then (S, +)
is a band.

Proof : Consider (S, ·) is left singular which implies that cb = c and ab = a

Suppose if S is an E –inverse semiring then ab + acb = ab
 a (b + cb) = ab  a (b + c) = ab  ab + ac = ab  a + a = a for all a in S
Hence (S, +) is band

CLASSES OF TOTALLY ORDERED LEFT REGULAR SEMIRINGS:

This section contains some results on totally ordered left regular semirings using the
properties like p.t.o, n.t.o, etc. A left regular semiring is a semiring in which a + ab + b = a
for all a, b in S.

Theorem 4.1: Let S be a commutative idempotent semiring. Define a relation ≤ on S such

that a ≤ b if and only if S is a left regular semiring. If ‘e’ is multiplicative identity which is
also an additive identity then S is a totally ordered semiring and e is a maximum element of S.

Proof: Let us define a relation ≤ on S such that a ≤ b if and only if S is a left regular semiring
and also ‘e’ is multiplicative identity which is also an additive identity
Since S is an idempotent semiring we have a + a = a and a2 = a for all ‘a’ in S
Assume S is a left regular semiring then a + ab + b = a for all a in S
14 N. Sulochana, M.Amala and T.Vasanthi

Now we have to check that whether (S, +, ·, ≤) is a totally ordered semiring

We have a + a.a + a = a + a + a = a + a = a
Thus a ≤ a implies ‘≤’ is reflexive
Let a ≤ b and b ≤ a  a + ab + b = a and b + ba + a = b
Again consider a = a + ab + b = a [e + b] + b = ab = a [b + e] = ab + a
Since (S, ·) is commutative
a = ba + a = b [e + a] + a = b + ba + a = b
 ‘≤’ is anti symmetric

Let a ≤ b and b ≤ c  a + ab + b = a and b + bc + c = b

To prove a ≤ c for this we have to prove that a + ac + c = a
Consider a = a + ab + b = a + a [b + bc + c] + b + bc + c
= a + ab + abc + ac + b [e + c] + c = a + ab [e + c] + ac + bc + c
= a + abc + ac + bc + c = a + a [b + e]c + bc + c = a + a[e + b]c + bc + c
= a + ac + abc + bc + c = a + [a + ab + b] c + c = a + ac + c
i.e a ≤ c which implies ‘≤’ is transitive

Next we prove that compatibility with respect to multiplication

For this we have to prove that a ≤ b implies a + ab + b = a then for any c in S ac ≤ bc
Now a ≤ b implies a + ab + b = a  (a + ab + b)c = ac  ac + abc + bc = ac
Since S is idempotent semiring implies ac + abcc + bc = ac  ac + acbc + bc = ac
Thus ac ≤ bc
Similarly we can prove that ca ≤ cb
Next we prove that compatibility with respect to addition
For this we have to prove that a ≤ b implies a + ab + b = a then for any c in S
a+c≤b+c
Now a ≤ b implies a + ab + b = a  (a + ab + b) + c = a + c
Also we know that if S is a left regular semiring with multiplicative and additive identity ‘e’
then (S, +) is left singular thus a + b = a for all a, b in S
(a + c) + (a + c) (b + c) + b + c = a + c
Thus a + c ≤ b + c
Similarly we can prove that c + a ≤ c + b
Therefore (S, +, ·, ≤)) is a totally ordered semiring
Now a + a.e + e = a + a + e = a + a = a  a ≤ e for all a in S
Hence e be the maximum element

Theorem 4.2: Let S be a totally ordered left regular and (S, +) is semilattice. If (S, +) is p.t.o
(n.t.o), then (S, ) is n.t.o (p.t.o).

Proof: Given that S is left regular then a + ab + b = a, for all a, b in S

Since (S, +) is semilattice then (S, +) is additive idempotent and commutative this implies a +
a (b + b) + b = a  a + ab + ab + b = a  ab + a + ab + b = a
 ab + a = a - - - (A)
 a = ab + a ≥ ab  a ≥ ab
Suppose ab > b  ab + a ≥ b + a  a ≥ b + a  a ≥ a + b  a + b ≤ a
Which contradicts the hypothesis that (S, +) is p.t.o  ab ≤ b
Therefore ab ≤ a and ab ≤ b
Hence (S, ) is n.t.o
A study on the Classes of Semirings and Ordered Semirings 15

Similarly we can prove that (S, ) is p.t.o if (S, +) is n.t.o.

Theorem 4.3: Let S be a totally ordered left regular semiring. If (S, +) is p.t.o (n.t.o.) which
contains multiplicative identity 1, then 1 is minimum (maximum) element.
Proof: Given that S is a totally ordered left regular semiring then
a + a.1 + 1 = a, for all a, 1 in S implies a + a + 1 = a
Suppose (S, +) is p.t.o then we have a + 1 ≥ 1  a + a + 1 ≥ a + 1
a≥a+1≥1a≥1
Thus 1 is the minimum element.
Also if (S, +) is n.t.o i.e., a + 1 ≤ 1 which implies a + a + 1 ≤ a + 1
a≤a+1≤1 a≤1
Therefore 1 is the maximum element

REFERENCES

[1] V. R. Daddi and Y.S. Pawar, “On Completely Regular Ternary Semiring”, Novi Sad
J. Math. ol. 42, No. 2, 2012, 1-7.
[2] Jonathan S.Golan, “Semirings and Affine Equations over Them: Theory and
Applications”, Kluwer Academic publishers.
[3] Jonathan S.Golan, “ Semirings and their Applications” , Kluwer Academic
Publishers.
[4] M.P.Grillet :“Semirings with a completely semisimple additive semigroups”. Jour.
Aust. Math. Soc., Vol.20 (1975), 257-267.
[5] N.Kehayopulu, “On a characterization of regular duo le Semi groups”, Maths.
Balkonica 7 (1977), 181- 186.
[6] M. Petrich, “Introduction to semigroups” Merill”, Columbus, Ohio (1973).
[7] M. Petrich, “Archimedean classes in an ordered Semigroup IV”. Czechoslovak
Mathematical Jour. 37(112) 1987, praha, 86 -119.
[8] M.Satyanarayana, “Naturally totally ordered semigroups” Pacific Journal of
Mathematics, Vol.77, No. 1, 1978.
[9] M.Satyanarayana , “Positively ordered semigroups”. Lecture notes in Pure and
Applied Mathematics, Marcel Dekker, Inc., Vol.42 (1979).
[10] M.Satyanarayana, “On the additive semigroup of ordered semirings”, Semigroup
forum vol.31 (1985), 193- 199.
[11] M. K.Sen, S. K. Maity and K. P. Shum, “Clifford Semirings And Generalized
Clifford Semirings” Taiwanese Journal of Mathematics Vol. 9, No. 3, pp. 433-444,
September 2005.
[12] M.K. Sen, S.K.Maity, “Semirings Embedded in a Completely Regular Semiring”,
Acta Univ. Palacki. Olomuc., Fac.rer. nat., Mathematica 43 (2004) 141–146.
[13] G. Theirrin, “The Sytantic monoid of a hypercode”, Semigroup form 6 (1973), 227 -
231.
16 N. Sulochana, M.Amala and T.Vasanthi