FUZZY HOMOMORPHISM AND FUZZY ISOMORPHISM

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H,PTER 5 * FUZZY HOMOMORPHISM AND FUZZY ISOMORPHISM Introduction Homomorphism and Isomorphism between two fuzzy (semi) topological semigroups are defined in this chapter. B.T.Lerner [35] studied the homomorphic and inverse images of fuzzy right topological semigroups.

1 H,PTER 5 * FUZZY HOMOMORPHISM AND FUZZY ISOMORPHISM Introduction Homomorphism and Isomorphism between two fuzzy (semi) topological semigroups are defined in this chapter. B.T.Lerner [35] studied the homomorphic and inverse images of fuzzy right topological semigroups. We prove the analogous results for induced L-fuzzy (semi) topological semigroups. In section 2 we prove that the product of a family of L-fuzzy topological semigroups is an L-fuzzy topological semigroup and the quotient space of an L-fuzzy topological semigroup is also an L-fuzzy topological semigroup. In the last section we give a brief discription about the categories of L-fuzzy topological semigroups.' * been Some of the results of this chapter has A communicated for publication in the Journal of Mathematical Analysis and Application

63 5.1 Preliminary concepts Definition 5.1.1 Let X be a semigroup. An L-fuzzy set B of X is an L-fuzzy left ideal if t± B (xy) u ^.

2 Preliminary concepts Definition Let X be a semigroup. An L-fuzzy set B of X is an L-fuzzy left ideal if t± B (xy) u ^., (y) ; an L-fuzzy right ideal if B(xy) = PB(x); and an L-fuzzy ideal if B(xy) max (x),p', (Y) ] V x,y X 11, 7 B Remark i) An L- fuzzy ideal of X is an L- fuzzy semigroup of X. ii) Every constant L-fuzzy set in a semigroup X is an L-fuzzy ideal of X. iii) When X is an abelian semigroup,an L-fuzzy left (or right) ideal becomes an L-fuzzy idea].. Definition Let X be a semigroup, a relation R on X is said to be left (or right) compatible with the operation on X if x,y== X ===>(ax,ay) E R [or (xa,ya) T R) V x,y, a r ;i, and compatible if R is both left and right compatible. Definition A compatible equivalence on a semigroup X is called a congruence.

64 If R is a congruence on a topological 5e,-j;yoLpX, then R is called a closed congruence if R is a closed subset of XxX Note The following propositions are analogous to the propositions 3.3,4.1,4.

3 64 If R is a congruence on a topological 5e,-j;yoLpX, then R is called a closed congruence if R is a closed subset of XxX Note The following propositions are analogous to the propositions 3.3,4.1,4.2 of [62], the difference being only that we have L in place of I=[0,1]. For completeness sake we indicate the proof of proposition Proposition The intersection or union of any set of L-fuzzy left ideals,right ideals or two sided ideals respectively is an L-fuzzy left,right or two sided ideal. :See proposition 3.3 [62] Proposition Let X and Y be two semigroups, f a homomorphism of X into Y.If A is an L-fuzzy semigroup (L-fuzzy ideal) of Y, then f-1(a) is an L--fuzzy semigroup (L-fuzzy ideal) of X. :See propositin 4.1 [62] Note Rosenfeld has defined, A fuzzy set p in X to have the sup property if for any subset T c X there exists to E T such that p(t0) = Sup O t) t't

65 Proposition 5.1.7 Let X and Y be :;E nigroups and a homomorphism f of X into Y.

4 65 Proposition Let X and Y be :;E nigroups and a homomorphism f of X into Y.Let G be an L-fuzzy semigr up (L-fuzzy ideal) of X that has the sup property, then the image f(g) of G is an L-fuzzy semigroup (L-fuzzy ideal) of Y. Let U,V F Y. If either f-1(u) or f-1(v) is empty then the proposition is trivially satisfied. Suppose neither f- 1 (U) nor f- 1 (V) is nonempty. Let r0= f-1(u), s0- f-1(v) be such that PG(r0)= Sup V G(t) where tef-1(u) and PG (s0) = Sup,-?G (t) where t,-=f - 1 (V) then Lf(G)(UV) = Sup 0G( w) where w = f-1(uv) Min {GO'GO = Minj c +f (G) (U), (V) f (G) l f(g) is an L-fuzzy semigroup (L-fuzzy ideal) of Y. 5.2 Fuzzy Homomorphism ( F--morphism) Definition Let (X14 1,F1),(X2,-2,F2) be two L-fuzzy ( semi ) topological semigroups. A mapping g of (X1,u1,F1) to (X2,p2,F2) is an F-morphism if

5 66 1) g is an algebraic homomorphism of X into Y ii) g is a fuzzy continuous mapping of ( X1,L1,F1) to (X2,u2,F2). And g is an F-isomorphism if i) g is algebraically an isomorphism of X into Y ii) g is fuzzy homeomorphism of (X1,P1,F1 ) to (X2,u21F2). Example Let X=(N,+) FX=the discrete L - fuzzy topology on X Let Y =(N,+) and FY =fg:y- - > [0,1] g is a constant map Define () :X ->Y O(x)=2x clearly 0 is an F-morphism of (X,1X,F) into (Y,IY,FY). Example Let X= ( "r=,+) and Y= ( =-(01, o) Let FX and FY be the associated fuzzy topologies on X and Y respectively for their usual topology. x Define 0 :X---- >Y.(l(x)=2 Clearly C.3 is an F-morphism of (X,lX,F) to (Y,1Y,FY)

67 Proposition 5.2.4 Let (X1,u1,F1) and ( X2,u2,F2) be two L-fuzzy topological semigroups and f: ( xiipi,f1)- -> (X2,?+21F2) be an F-morphism. If ^!

6 67 Proposition Let (X1,u1,F1) and ( X2,u2,F2) be two L-fuzzy topological semigroups and f: ( xiipi,f1)- -> (X2,?+21F2) be an F-morphism. If ^!2c U2 be an L-fuzzy semigroup of X2, then ' (X1, f -1 (? ^ 2 Flf-1 ( ` ' 2 ) [where F1 f-1 0. )2) is the induced L-fuzzy topology with respect to f-1 (u2) on (X1,u1,F1)] is an induced L - fuzzy topological semigroup of (X1,u1,F1). By proposition ( 5.1.4) f -1(u2) is an L-fuzzy semigroup of X1,then by proposition (2.2.3) (Xi,f(u2),Fif-1(t±2)) is an induced L-fuzzy topological semigroup of (X1,i?1,F1). Proposition Let (Xi,ui,F1) and (X2,u2,F2) be two L-fuzzy topological semigroups and f: (X1,u1, F1)- - >(X2,u2,F2 ) be an F-morphism. If ui Pi be an L-fuzzy semigroup of Xi satifies the sup property then (X2,f(U ),F2f(^,) is an induced L-fuzzy 1 topological semigroup of 1.6,u2,F2) It is clear from the propositin ( and )

7 68 Proposition Given semigroups X,Y and a homomorphism f of X onto Y and an L-fuzzy topology -'1 on Y, let X have L-fuzzy topology T where T is the inverse image under f of and let (Y,G,'"'1G) be an induced L-fuzzy topological semigroup of (Y ) then the inverse image ( X,f-1(G),Tf--1(G) ) is an induced L-fuzzy topological semigroup of (X,1XT) The proof is analogous to the proof of proposition 6.1 [20] Note. Let X, Y be two semigroups, f:x ->Y be a homomorphism of X into Y Let G be, a fuzzy semigroup of X then the membership function PG of G is f-invariant if for all xl,x2 F X such that f (x1)=f ( x2) we have p G(x1 ) G(x2 Proposition Given semigroups X,Y and a homomorphism'f of X into Y,and an L-fuzzy topology F on X, let Y have L-fuzzy topology `i where 'L1 is the image under f of F and let ( X,G, FG) be an induced L-fuzzy topological semigroup of (X,1X,F). If the membership function of G is f - invariant,the image (Y.f(G),`'_1f(G)) is an induced L-fuzzy topological semigroup of (Y, 1,,,'L

69 The proof is analogous to.the proof of proposition 6.2 (20) Theorem 5.2.8 Let {x.tu.tf.iei J be a collection of L-fuzzy 1 1 1 topological semigroups, then their product jj (Xi,ui,F.

8 69 The proof is analogous to.the proof of proposition 6.2 (20) Theorem Let {x.tu.tf.iei J be a collection of L-fuzzy topological semigroups, then their product jj (Xi,ui,F.)=( X,U,F) is also an L-fuzzy topological iei semigroup,where X =r! (Xthe usual set product and u =j pi iei iei be the product L-fuzzy set in X whose membership function is defined by u(x)=inf f- (x )I x=xiex and F is the smallest L-fuzzy topology on X for which each projection mapping p. J of X onto X.'is fuzzy continuous. J We have X is a semigroup under coordinate wise multiplication, where the associativity of multiplication follows from that of the semigroups in the collection ( X ie I l and x. y 1 EX 1. the mapping g 1 : ( x x 1. y 1.)----> x 1 y 1. of 1 1, (X 1.,u1.,F 1. )x (X 1, t) 1. F 1. ) into ( X 1.,t! 1. F 1. ) is fuzzy continuous. [each (Xi,pi, Fi) is an L--fuzzy topological semigroup let g be the mapping such that g(x,y)= xy,and

70 p g = g 1 1 Then g is fuzzy continuous by proposition 1.2.10 ji (X.,uli,F;) iei is an L-fuzzy topological semigroup. Remark From the definition of multiplication on X each p.

9 70 p g = g 1 1 Then g is fuzzy continuous by proposition ji (X.,uli,F;) iei is an L-fuzzy topological semigroup. Remark From the definition of multiplication on X each p. is J an F-morphism of X onto X. J Definition let (X,^ ±.;F) be an L-fuzzy topological semigroup,r be a closed congruence on X, then the quotient space (X\R,,, 21) of (X,p, F) is a n, L-fuzzy topological semigroup. Consider the diagram X x X f X p x p w p X\R x X\R f X\R where f: (x,y)-- -->xy of (X,u,F) x (X,p,F) into (X,u,F) is fuzzy continuous.

10 71 We have to show that f1: (X/R,; x (X,'R,,','?.L) into (X/R,r-,2L) is fuzzy continuous. let f^e L, then 0C,Jp (B),-=F (since p is fuzzy continuous) That is(p x u)rl f-1 (L± r I p1 ( B) ) T F x F That is (,u x ( f-1 (p) 1"1 f1 (p-1 (B)) F x F -1-1 That is (u x ± );-1 (p^f) (B) '- F x F ( since u x u f (p) ) That is ( p x p) = (_ x p) f--- (pof) (B) x '?L (since p is fuzzy open so is p x p) That is ( p x p) (p x u) n (px p) ( p f)(b ) LL x That is (pxp ) (pxp)il( pxp) (f15pxp )-1(B)E `LL x ''L (pof = flopxp) That is (p x p) (p x p) rl (p x p) (p x p)- 1f1-1 (B) F- 9/ x V. That is ef i'i f-1(b) _ '7L x 'U is fuzzy continuous Corollary The quotient L-fuzzy topological semigroup (X/R,,1-,9-0 of (X,u,F) is an F-morphic image of (X,,p,F). Definition Let X be a semigroup. A fuzzy relation R is said to be a fuzzy equivalence relation if:

11 72 i.r(x,x )=1,V x = X ii R (x,y) = R (y,x),w,. x,y F= X iii. R (x,z) R (x,y) r--l R(y,z) V x,y,z X A fuzzy equivalence relation of X which is compatible with the semigroup structure of X is called a fuzzy congruence of X. Example. i) If X is a semigroup, every fuzzy ideal in X is a fuzzy congruence ii)-the constant fuzzy set in a semigroup is a fuzzy congruence of X. Proposition Let R1, R2 be two fuzzy congruences of X, then R1( R2 is also a fuzzy congruence. The intersection of two fuzzy equivalence relations is a fuzzy equivalence relation and the intersection of two fuzzy semigroups is a fuzzy semigroup. R1i R2 is also a fuzzy congruence. Remark R1U R2 need ijot be a fuzzy congruence.

73 Theorem 5.2.12 Let (X,F) be a Fuzzy topological semigroup, R be a fuzzy congruence of X.

12 73 Theorem Let (X,F) be a Fuzzy topological semigroup, R be a fuzzy congruence of X. Then R with induced fuzzy topology is an induced fuzzy topological semigroup of (X,F) x (X,F) : See proposition Categories of fuzzy topological semigroups Definition Let F`1 be the collection of all L-fuzzy topological semigroups with F- morphisms and jr*- 2 be the collection of all L-fuzzy semi topological semigroups with F-morphisms, Clearly 1'- 1 and -2 consl itues sub categories of FTOP (not full sub categories). Results Monomorphisms in the above categories are precisely injective F-morphisms and the epimorphisms are F-surmorphisms.(this follows from the fact that the same is true in FTOP.(Result 1.2.2) Let 1 be the collection of all L-fuzzy topological semigroups (X,p,F) for a fixed semigroup X together with the F-morphisms and let be the collection of all L-fuzzy

13 74 topological semigroups ( X,p,F) for a fixed fuzzy semigroup p together with the F - morphisms. Clearly.r' 1 and are subcategories of 1 Remark : Using the following Functors we get an association between the category of topological semigroups a and the category of fuzzy topological semigroups,t_) Define Y1 and Y2 such tha t Y 1: ----> 7j -1(,m,T) (X,m,U)) U is a l.s.c map Similarly 2 :. ---->^' such that ^- C F2(X,m,F ) =( X,m,i(F ) where i(f)= V -1(a,11;VE F,a (0,111 2(f) = f clearly =F'2 embedds.' into D

MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA

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