Lectie de pregative
STRUCTURI ALGE BRICE (II
18.83.2023
Structuri algebrice (I) (Teste gril 2022]
1,0) Structura algolonica, adica7F.AXA -> A
1), ACC, yGA
↳ lege
xOy =
FPC,
1
compositie" saw,operate
de binara
amoscute: Semigeup
↓
alabrice all
I structuri
I multimilar
a) Eupuri(Monoizil
incapatantf
imfir
=>
aparte stabila
-
asociativilale
- -
-
-
(4,0) element neutru:
[7 e e 6a.4.x0 10x
=
=
x, Axle I
6
7x6(simetrical (ix)=
[xET.
6
a. cox=<ox= e
Motati: e=elemental neutro -
to
166
-elemental
>
multiplicatio:(6,)
M:
- simetrical invers-a
-
to f
aditio: (6,t)-elementaln:
- simetrical opus---x
=
b) Inels (I,+,) 1,+) commatio (x+y y x
I
+
grup
=
I
Ax,y
(,) monoid
-
11
distributivitate, "fatads, t
1,+) gep commutatio (4x 0)
=
apai (K,+,) (*) gap (A* A(03)
=
distributivitate ----
commtative
Exemple:a) (2,+), (R,1) gepar
(R*,);(*,); (,;) -l-
1)1,1), 0), soy By->gepcomutatio
=
Itebari: 1) Este (x*,) grup?
2) Dar
([-1,1],)?
b) G- :) ,
-1
; fmmlk ) ,
-1
:) ; 4m ,
-1 :)
121 }
*
ME IN
c) (Q :) ;
-1 +
:) ; ¢ ;) ; ftp.t:) -1
(
, ,
p - memoir psim
>" carpal dasdordessturi
modulo p
"
2 Structuraindus pe o altamultime
I copiers de structura" (
H dongmultimi a., exista:
Fie G,
f: 6 + H o
fundie bigstia;
I do(6-H1, cardinale egale).
Fie (6,0) a structura algeloicadatat/monoid saw grup):
Decause
bigective
of
->
If ": H -
6 impersa luit
-Its necesar o
sa
aflam"
Atmni I! operatic binar "pe H 9.4.
1,x) devine a structura algebizaisomora(n (6,0)
ia
f:G -> H itomodism
ste -
(bigivatmarism)
f: (6,0) -> (H,x) morism
f(xog
↓
, xx,y 6
(f) ofa) *te
Exess(daca stim (,*)):
x0y f"(f()*f(z),
=
x,y G
=
Proprietati. Fief:(6,0) -> (H,x) mostism (degreeinsande)
·
Daca (6,0) monoid, VIG)=(xEc/acstesimetrizabil ->
(VIG), oste grup,
·
Dacafste morism de monoiti->
f(2,0.00xn) f(x) 0.* f((n);
= +
Imparticulas: f(x) Af(x), =
Heh, ...., (m, ()EG.
er pie e di
·
Dacaf etoitomalism de monoizi-> f(f)=ei
f(V()) V (H)
=
Daca fets malism de grapes-> f(f)
f(xx))
· eH
=
f(x) =
im (6,) f(x) f(x) t,i(6,t) f(=x)
=
=
= = -
f(ul).
Exemple:
① Fie(R,+) gp si f:R t
R, f(x) ax
= b
+
atr*, bG R
feste bijectivecu f: i
ti, f'(n) (T.A.)
=
->
finduce per operatia,*"datade
f(f((n)of (r)) f/f() F(r))
↑
1xx
=
f(* +
f(*23)
=
-
a.Mtb b
+
u 0
=
+ - b
=
Deci:
(R,+) IR,*), MAV u
= w
+
-
b
Pb : Sedan grupwils (R,+) $i (iR,*), x*y c y 1.
+
= +
Functia of: R i, f(x) ax+b etsizomodism
- =
ob'la (R,+) la dasa,i numai daca a, b =?
(R,x)
aatis:
acasic") F maisa"
de
grape
e
R
A modism f(x y)
=>
+
f(x) f(y),Ac,y
= +
a(x y) b
+ +
(ax b) (ay
=
+
x
b)
+
=
a(x + y) b
+ ax
=
b
+
ay b
+ +
1
+
ay) b axy) 2b
+
+
= +
1
+
->
b 26
=
1
+
= 7
-
b = 1 =>
Et1
Ibijectiva deci rspemsb==1.
: Conform discutiei 0=7 -
b 1
=
0ptSE f
#
a +
T. A: Pb8
② Fie 44*1=110,4;) gsnpnl dat .
Fie f 1-1 :
,
it → 10 a)
,
i fix ) =
1¥ .
"
(G) 01=(1-14,0) ? structure algebricé opiate
,,
TA : Ara-tatica-fstebised-ioei.si afbyti f- !
l" "
este guys comutativ
Him to : •
4-1 )
,
* on /
\ a- '
Am > 0
=µ1 ,
'
•
f- :(opt →
1- i. A film :# =
Iaxes. We
feesat
A 11) Ven
-
I "( 2) 1) se -
-
- -
8+ year
At 1
y
By 1
+
it 1
1
-
x -
y +
cy
I
isthe
re
x y xy
-
- +
Deci
(40) Eso)"bretit=ixy, x+y Afre).
Pb 773 775
-
Se
dagrpus (6,0),6 (- 1,1)
=
y
siccoy-
Seo: (a) elemental neutru ly=?
(b) E070000otp=?
C) Valmik a,b =? a.4. f:(G,0) (0,0)-),
->
flud-sa tie izomalism de grai
atie:
recomandat ca in astful as problems
-
↳exatie: Este
sa
imper aflats isomoismulai!
an
(C) Dim discutia(z) d sns => a 1,b 1
=
=
T. A:
L Clasic"din I morism
ifbijective
alac a
=>
1,b
=
1)
=
(0,0),0)
-
(a) Deci f(x) #,=
teiromalism (6,0) =
(f(f) (x)
f(x) 1
=
=>
e f() 0.
=
=
b) Notam 9 5070000 to
=
f(9) f(704000.00)
=
f(z).f(t).....7(5)
=
·
I...
-
6
nee
I
2p
=........
I
i
=> f(z)
=
11
=
g
=
f()==I
=
=>
=
=
F I
i.
8in(tre I 2018
⑬3 I
Admitere
2021
-
-
Pb,908: Pe D es dalegta de composities
2, z ziz-
=
ilE, +zz) -1+
i, E1,zz E K.
Se ar: (a) elemental neutru al legi, *"ests?
(b) Multime element eld
/
simetrizabile/inversabile)
all monoidulai(C,*),
() Daca =
t atenci
ste?
Eliteti) 0.
+
*(2+i)
di
2022 de
die.Se (C,*) monoid:
specifica
Stir (04)
a grup das
(C,) movoid.
& ele neute (a= 1
Observe ca
z1 izz 1 3
2,*zz zzz
-
-
+
-
=
zzz i) j(zz i) i
+
-
=
(t,- i)(z-i)
=>
i
+
=
sugneata
z,xzz i
(1 i)(z i) - s definim
-
-
- =
f:0 -> c
f(z) z
=
-
3, bijective
f(z)
z
=
an
+
inversa
12
T. Ae I
(Severifica
f(z,*zz) f(z)).f(zz) 7
= =
monoidal) mauridal i
(G,*) H, ·
f(d 1 4 7(1) =
(a) f(x) en ->
=
= =
->
=
let
of ste isomalism de
3
because monoite ->
(b)
f(x) f(x)
=
f(V(*) U(t,) si f"(V(ki) v12,*)
=
=
=>
f((Y) v1x,y)
=
Da
((((B)
f(0) = i (([7(a))
=
V1,1) aRih
= =
(2) Fie 5+
9= i; Calculam
(dti))
f(z) 4* =
If(d+i)
=
2022
2022
E..... E
= -
E
e
2022
=> 1 =>
E -
=
1 i ws
+
=
+ iSi =
->
q2022(23)674 1
=
f(1) 1 i =
f(z)
+
g
=
1
=
Fi
-
XX
