Normality and congruence in fuzzy subgroups

INFORMATION

SCIENCES

59,121-129

121

(19921

Normality and Congruence in Fuzzy Subgroups B. B. MAKAMBA Department of Mathematics, University of Fort Hare, Alice, Ciskei, South Africa

V. MURAL1 Department of Mathematics, Rhodes University, Grahamstown, South Africa

Communicated by Azriel Rosenfeld

ABSTRACT It is proved that normal fuzzy subgroups and fuzzy congruence relations determine each other in a group-theoretic situation. This is analogous to the crisp-set theoretic case of groups.

1.

INTRODUCTION

Fuzzy subgroups were first defined by Rosenfeld [8]. Since then others have studied fuzzy topological groups [4], normal fuzzy subgroups [l], and fuzzy cosets [5] in various contexts. On the other hand, fuzzy relations have also been studied since a definition was proposed by Zadeh [lo]. Fuzzy ordering, fuzzy equivalence relations, and fuzzy graphs have since been considered in the literature of fuzzy mathematics [2, 3, 111. More recently, fuzzy congruence relations analogous to congruence relations in the universal algebraic setting appeared in [7]. The purpose of this paper is to unite the two notions-normality and congruence-in a fuzzy subgroup setting. In particular we prove that every fuzzy congruence relation determines a normal fuzzy subgroup. Conversely, given a normal fuzzy subgroup, we can associate a fuzzy congruence relation. The association between normal fuzzy subgroups and fuzzy congruence relations is bijective and unique. This leads to a new notion of cosets and a corresponding notion of quotient. In Sections 2 and 3 we gather results on normal fuzzy subgroups and fuzzy equivalence relations, respecOElsevier Science Publishing Co., Inc. 1992 655 Avenue of the Americas, New York, NY 10010

0020-0255/92/%03.50

B. B. MAKAMBA

122 tively, and fix notations.

Further

AND V. MURAL1

details on these two sections can be found in

151,and 161. G denotes a group (finite or infinite), and I denotes the closed unit interval that will be our valuation lattice of degree of membership. Some of our results can be extended to more general lattices. Greek letters p, V, y, etc. will denote either fuzzy subsets or fuzzy relations of G. There will not be any confusion, because one can determine from the context whether p denotes a fuzzy subset or a fuzzy relation; for a fuzzy subset, p is G -j I, and a fuzzy relation is G x G + I. If k is a fuzzy subset, by an r-cut Cl of Jo for 0 < r < 1, we mean the crisp subset (x E G: p(x) > r} of G. Similarly, if p is a relation, by an r-cut p.’ we mean the crisp relation xpry if and only if ~(x, y) > r for x,y~G.

2.

NORMAL

FUZZY

GROUPS

Let G be a nontrivial group, and let e E G denote the identity element of G. Let p: G + I be a fuzzy subgroup. For a fixed a E G, the left fuzzy coset of /J associated with a is defined as up(x) = Fl.(a-‘xl. Similarly, pa, the right fuzzy coset, is defined as pa(x) = p(ax-‘). Now we have the following. PROPOSITION 2.1. If G and t.~ are as above,

then for

all x,a E G,

the

following are equivalent.

6) pL(x) G p(a-‘xa). (ii) p(x) = p(a-‘xa). (iii) p(ax> = y(xa). (iv) up = pa. (v) C, of p is a crisp normal subgroup for every 0 < r < p(e).

We omit the proof, which is a straightforward consequence of the definitions. A fuzzy subgroup p of G is said to be a normal fuzzysubgroup if p satisfies any one of the five equivalent conditions of Proposition 2.1. REMARKS. (i) Clearly there are fuzzy subgroups that are not normal fuzzy subgroups. For if G is any non-Abelian group and H is any subgroup that is not normal, then we can get fuzzy subgroups out of H that are not normal fuzzy subgroups. (ii) In condition 6) of Proposition 2.1, it is enough to require p(a _ ‘xa) > /-L(X) for all a E G and x E supp(p.), where supp(p.) is the support of p given by {X E G: /4x) > O}.

NORMALITY

AND CONGRUENCE

IN FUZZY

SUBGROUPS

123

(iii) In condition (v) of Proposition 2.1, we can weaken the requirement that p be a fuzzy subgroup. In other words, we can state it as follows: A fuzzy subset p of G is a normal fuzzy subgroup if and only if every r-cut Cp of p for 0 < r < p(e) is a normal subgroup. 3.

FUZZY

CONGRUENCE

RELATIONS

Let 9(G) denote the set of all fuzzy relations on G. That is 9(G) s ZGxc;. For any fixed p E .9?(G), let t,, = supp(x, y), where sup stands for the supremum of ~(x, y) taken over all (x, y) E G x G. We observe that 0 < tc < 1. t,, = 0 implies that we have the empty relation, namely, ~(x, y) = 0 for all X, y E G. From now on, we assume 0 < t, < 1. We can define two operations on .R(G)x .9(G). One, called the composition and denoted by F(,v, is defined as

and the other, called the multiplication defined as

and denoted

by or..v or simply PV, is

(CLV)(X,Y)=SUP[CL(X,,Y,)A11(X2,YZ)l~ X.Y

where the supremum is taken over all representations x = x,x2 and y = y,y, for P,V E 9(G). DEFINITION

equivalence

3.1 [6].

A fuzzy relation

of x and y in G as

p on G is said to be a fuzzy

relation on G if

(i) ~(x, x) = tO for every x E G (reflexive), (ii) ~(x, y) = ~(y, xl for all x, y E G (symmetric), (iii) p 0 p < CL,where 0 denotes composition (transitive). It is readily checked that if p is a fuzzy equivalence relation, then p is idempotent for 0. That is, p 0 EL= p. Furthermore, for each t such that 0 < t < t,, the t-cut 11’ is a crisp equivalence relation. In particular, t,-cut pf” is a crisp equivalence relation and as such yields a partition of G in the crisp sense. The t(,-cut classes of G under this partition are denoted by s:, p, e etc., containing representative elements x, y, e respectively. For each to-cut class 5 for x E G, a fuzzy subset pcLz:G -+ I is defined as p=(a) = p(x, a> for all a E G. Now for each 0 < t < t,, the collection {Cr-, x E G} is a crisp partition of G. The family of fuzzy subsets {p,} for x E G, on G associated with a fuzzy

B. B. MAKAMBA

124

AND V. MURAL1

equivalence relation p, is called the fuzzy partition of G with respect to p. It is uniquely determined by k, union of p_, x E G is xo, the whole set G and the intersection of p, and pb” is pI A pk d LYwhenever ~(x, y> = (Y. For further details, we refer to 161. A fuzzy equivalence relation p on G is called a fuzzy congruence relation on G if /*p < CL. The relation pp < p can be thought of as a substitution property as is well known in crisp congruence relations on a group or a general algebra. Moreover, one can interpret in the crisp case a congruence relation as an equivalence relation E as a subset of G X G that is at the same time a subgroup of G x G. This is indeed analogously the case in the fuzzy case. A fuzzy equivalence relation that is at the same time a fuzzy subgroup of G X G is a fuzzy congruence relation. It is easily checked that for each t such that 0 < t G t,, p’ is a congruence relation if and only if p is a fuzzy congruence relation on G. Further results on congruences can be found in 171.

4.

CONGRUENCE

AND NORMAL

SUBGROUPS

We now turn our attention to the relationship between fuzzy congruence relations on G on the one hand and normal fuzzy subgroups on the other. First we have the following theorem. THEOREM 4.1.Let G be a group and p a normal fuzzy subgroup of G. If v is defined as v: G X G + I by v(x, y) = p(xy-l),

then v is a fuzzy congruence

relation on G. Proof.

v(X,y)=CL(XY-l)~~u(e)=to.

Also,

for every x E G, implying reflexivity of v. The properties of symmetry and transitivity We now show that vv Q v.

(vv)(x,y)=

v(x,X)=~(X\:-l)=~u(e)=to

of v can be verified easily.

SUP ["(XpYJA

x=x,x*

4%&)1

Y =YlYz

whereas

v(x, y) =

p( v-l) = P(x,x,Y;'Y~')

NORMALITY

AND CONGRUENCE

for every representation p(xpzY;lY;l)

IN FUZZY

SUBGROUPS

125

of x = x1x2 and y = y1y2. But =/+lY;lYlx*Y;lY;l) h+IY;l)A

P(Y,X*Y*

=P(x,Y;l)A

cL(xzy;‘)

-1

-1 Yl >

since p is normal in G

=V(x,,Yt)Av(x2rYZ).

Therefore vv(x,y)

=

sup x

[“(X,,Y,)A

4x24)1

=x,x2

Y = YlY2

Thus v is verified

to be a fuzzy congruence

relation.

The following is a sort of converse to the above proposition. congruence relation determines a normal fuzzy subgroup. THEOREM 4.2. Given a group

G and a fuzzy congruence

Every fuzzy

relation p on G,

there is a normal fuzzy subgroup v of G such that p(x, y> = v(x_V’). Proof. p is fuzzy congruent implies I_Lis reflexive. That is, ~(x, x) = to. By an earlier remark (made just after Definition 3.1), #I is a crisp congruence relation on G. Let [e]z be the class containing the identity e in the partition of G yielded by p’u. Define

V:

G+

(i) v is well-defined. (ii) 11(x, e) = ~(xSuppose

I by V(X) =p(x,e)

forall

xsG.

I, e) for all x E G.

x E [e]:. Then xp’“e and x-‘~‘~~x-‘.

/-4x-’ ,e)=to=p(x,e).

Therefore

ep’“x-

‘, implying

126

B. B. MAKAMBA

AND V. MURAL1

Suppose x P [e],ta. Then ~(1, e) < t,,, and also pL(x-‘, e) < t,,, because [e&t,, is a subgroup of G. Let t, = p(x,e) and t, = ~(x-‘, e). If t, < t,, then x E [el,t, and x E [e],t,. is a subgroup of G. Also, x-’ E [e],t,, implying x E [e],r,, because [e],t, This is a contradiction. A similar contradiction arises if we assume that t, < t,. Therefore, t, = t,. That is, &x,e)

=CL(x-‘,e).

Hence forall

V(X) = “(K’)

xtG.

(iii) v(xy> > V(X)A v(y). For

since F is a fuzzy congruence

relation,

and therefore

(iv) ~(x, e) = ~(xy, y) for all x, y E G. Let t, = ~(xy, y) and t, = p(x, e). If tI > t,, then [el,t, c[el,t2. and

XYdlY

yp’pfly-’

a

xp’le.

A similar Therefore, ~(x, e) > r, = t2 B t’, a contradiction. sults if t, < t,. Therefore, ~(x, e) = ~(xy, y). That is, e) =l*.(xY-‘Y,Y)

P(XYP9

-(xy-‘)

(v) v is a normal

subgroup

v(K’xu)

=dx,Y) =cL(x,y).

in G.

=p(K’xu,e)

=p(aa-‘xu,u) =p(xa,u) = dx,e) Thus the theorem

is complete.

forall

x,u~G.

contradiction

re-

NORMALITY

AND CONGRUENCE

IN FUZZY

SUBGROUPS

127

THEOREM 4.3. Let p be a fuzzy congruence relation on G and

to=

SUP cL(X,Y). ~,YEG

The collection (p r: x E G) is a fuzzy partition of G in the sense that

where ~Jx)

= t, for all x E G, and PL, A Pp <

to

for all

52:+p.

Furthermore, (p,: x E G} is a group under suitably defined binary operation. Then a fuzzy subset pS of G is precisely the left fuzzy coset xp of CL*associated with x E G, where e = [e]2. Proof. It is straightforward to check that V x EG pI = ITS. We now show that pU,Apy
p_(a)= t0 = PJa) -dx,a)

Therefore,

=b4Y,a),

XE~;

and

yap.

xp’“y. =-YE~=[x];

and

KEY.

But this is a contradiction to the fact that 2: n p =0. Now define a binary operation as follows.

where 2;p is that class containing xy for x E s and y E p. The multiplication is well defined. For let x, xi E a and y, y, E I. We must show that

dxy,a) =k4xlyl,a)

forall

aEG.

B. B. MAKAMBA AND V. MURAL1

128 Now

Therefore,

If aez?z;y, then &c~,a)=t,,=~(x~y,,a). If aE=p, then &CY, a1 < I, and I.L(X,y,, a) < t,. Therefore,

Similarly,

that is,

We next show that {CL_}is a group under the binary operation defined above. ,u,p, = p,, because

p,_(a) = iL(ex,a)= CL(x,a)= iu_(a>

forall aE.G.

Similarly, pL,~, = CL_. Define S- ’ = [x-~]~~o for x E G. Then p_p_- I = p,,-I = p,, because e E -1 m/G , c=m -I. Therefore, (@J- ’ = p,- 1. The associativity (p,pPXpP) = (~J(u,~~> follows from the same property in G. Finally, we recall that the left fuzzy coset of p associated with x E G is xp, defined by (W)(Y)

-;kwlY>

for all y E G.

NORMALITY

AND CONGRUENCE

IN FUZZY

SUBGROUPS

129

By Theorem 3.3, pa, which is the same as v in Theorem 4.2, is a normal fuzzy subgroup in G. We claim that xp< = CL_for all x E G, for

forall

=Fz:(Y)

This completes

LEG.

the proof.

Finally, we prove that every congruence relation arises from a normal fuzzy subgroup with the partition given by the left fuzzy cosets. THEOREM 4.4. fuzzy congruence

Let u be a normal fuzzy subgroup of G. Then there exists a relation u on G such that the fuzzy partition associated with u

is the collection (xv:

x E G} of left fuzzy cosets of v.

The proof follows from Theorem

4.1 and is omitted.

We gratefully acknowledge the contribution of those colleagues who provided stimulating discussion and suggestions (or were merely patient listeners) during the regular seminars in fuzzy mathematics at Rhodes University. REFERENCES 1. L. Biacino and G. Gerla, Closure systems and L-subalgebras, Inf. Sci. 33:181-195 (1984). 2. P. Bhattacharya and N. P. Mukherjee, Fuzzy relations and fuzzy groups, Inf. Sci. 36:267-282 (1985). 3. M. Chakraborty and S. Das, On fuzzy equivalence 1 and 2, Fuzzy Sets Syst. 11:185-193; 299-307 (1983). 4. D. H. Foster, Fuzzy topological groups, J. Mar/r. Anal. Appl. 69:549-564 (1979). 5. N. P. Mukherjee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inf. Sci. 34:225-239 (1984). 6. V. Murali, Fuzzy equivalence relations, Fuzzy Sers Cyst. 30:155-163 (1989). 7. V. Murali, On subalgebras and congruences in fuzzy algebras, to appear, Fuzzy Sets Syst. (1991). 8. A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35:512-517 (1971). 9. A. Rosenfeld, Fuzzy graphs in Fuzzy Sets and Their Applicationto Cognitiveand Decision Processes(L. A. Zadeh et al., eds.), Academic, New York, 1975, pp. 125-149. 10. L. A. Zadeh, Fuzzy sets, Inf. Control 8:338-353 (1965). 11. L. A. Zadeh, Similarity relations and fuzzy ordering, If. Sci. 3:117-200 (1971). Received 1 January 1989; revised 10 March 1989