INFORMATION SCIENCES 65,33-43 (1992) 33

Direct Products and Isomo~hism of Fuzzy Subgroups

B. B. MAKAMBA*

Department of Mathematics, University of Fort Hare, Alice. Ciskei

Department of Mathematics, Rhodes University, Gruhamstown. 6140, South Africa

ABSTRACT

The concepts of internal direct product and external direct product of fuzzy subgroups of a group 9 are established as well as that of fuzzy isomorphism of fuzzy subgroups (of possibly different groups). It is shown that the internal direct product of two fuzzy subgroups of a group Y is isomorphic to their external direct product. The fuzzy counterparts of the first and third isomorphism theorems are also proved.

Communists by Azriel Bosenfeld.

1. INTRODUCTION

Fuzzy subgroups were first defined by A. Rosenfeld [ 11. Since then various people have studied analogues of results from classical group theory. In this paper the notion of a fuzzy internal direct product of two fuzzy subgroups is introduced. H. Sherwood [6] introduced the notion of an external direct product of two or more fuzzy subgroups, although he did not use the words external direct.

We also define what we mean by the statement that two fuzzy subgroups of possibly different groups are fuzzy isomorphic.

Let p and v be fuzzy subgroups of a group 3. Suppose j&dv is an internal direct product of p and Y and that p x v is an external direct product of p and v. We shall prove that #3v and p x v are fuzzy isomorphic.

Finally, analogues of the First and Third Isomorphism Theorems from group theory are stated and proved.

*The author is a member of Professor W. Kotze’s research group on Fuzzy Sets with Applications in Topology and Algebra at Rhodes University, Grahamstown.

OElsevier Science Publishing Co., Inc. 1992 655 Avenue of the Americas, New York, NY 10010 0020-0255/92/$5.00

34 B. B. MAKAMBA

2. PRELIMINARIES

DEFINITION 2.1 [ 11. Let 9 be a group. A map p: 9 -+ [0, l] is called a fuzzy subgroup of 9 if

(i) cl(xY) 2 min(p(x), p(Y)) for all x, YE 9, (ii) p(x) = ~(x-‘) for all xE 9’.

In this paper, 3 will always denote a group, and e will always denote the identity element of 9.

DEFINITION 2.2 [2]. Let p be a fuzzy subgroup of a group 9. Then p is called fuzzy normal if p(a- ‘xa) 2 p( x)Va, XE 9. We also say that p is a normal fuzzy subgroup of 9. Clearly, p is fuzzy normal if and only if b(a-‘xa) = p(x) for all a, xE 9.

Let p be a fuzzy subset of ‘s”. By the support of p we mean the set supp( CL) = { XE 9 : p(x) > 0). If p is a fuzzy subgroup of 9, then supp( cl) is a subgroup of 9. If p is fuzzy normal, then supp(p) is a normal subgroup of 9.

DEFINITION 2.3 [5]. Let p be a fuzzy subgroup of 9, and 0 < (Y < k(e). Let pa = { XE %’ : p(x) 2 a}. Then $ is a subgroup of 9, called a level subgroup of ~1 corresponding to IX.

PROPOSITION 2.4. Let p be a fuzzy subgroup of 97. Then

suPP(p) = u cc*. 0 < a Q P(e)

PROPOSITION 2.5 [2, Theorems 3.6 and 3.91. A fuzzy subgroup p of ‘5” is normal if and only if $’ is a normal subgroup of 9 for all cr~[O, p(e)].

DEFINITION 2.6 [ 11. Let g and H be groups and f: 9 + H a homomor- phism. Let p be a fuzzy subgroup of 9. The image of p under f, f (p), is a fuzzy subset of f ( 9 ) defined by

fk)(f(x)) = sup CL(Y). f(H=f(N

DEFINITION 2.7 [2]. Let v be a fuzzy subgroup of g. Let XE 9. A left fuzzy co-set of v associated with x, denoted by xv, is a fuzzy subset of ?? defined by (xv)(y) = v( x- ‘y) for all YE 9. A right fuzzy co-set of v is similarly defined.

DIRECT PRODUCTS 35

If v is fuzzy normal, then the set < = (xv: XE 9 > is a group under the binary operation defined by (xv)(yv) = xyv, x, ye 9. Also, xv = vx for all XE 9.

DEFINITION 2.8 f4]. Let ~1 be a fuzzy subgroup of 9 and v a normal fuzzy subgroup of B, such that p(e) = v(e) and v < p. The quotient p/v is a fuzzy subset of 5 = { xv: XE 9 ) defined by cc/ v( xv) = p(x) for all XE 9.

IL/ Y is a fuzzy subgroup of 9 whenever v is fuzzy normal.

If p and v are subgroups of 9 such that v < ~1, (i.e., v(x) < p(x) for all XE 9) we say that p contains v or v is contained in p.

3. FUZZY PRODUCTS

DEFINITION 3.1. Let p and u be fuzzy subgroups of g. The product ,LW: 9 +[O, 11 is defined by pv(x) = supX,,,,,[min(~(x,), Y(x~))], XE 8. (See, e.g., Zadeh [3]).

PROFQSITION 3.2. Let p,v,s be fuzzy subgroups of 8. Then

(PH(X) = sup [min(~(xOyv(x2)~E(x3))] x= X,X~X,

If p and Y’ are fuzzy subgroups of Y, PV is not necessarily a fuzzy subgroup of 9.

EXAMPLE 3.3. Let 22 = S,, the symmetric-group on 3 symbols.

S, = (e, a, a2, b, ab, a2b),

where

b2 = e and a3 = e.

LetH={e,b),K={e,ab).NandKaresubgroupsof g.Letpandube characteristic functions determined by the subsets H and K, respectively. Then p and v are fuzzy subgroups of 9. Let XE 9.

v(x) = min(k4xl)9v(x2)) f or some x, , x2 E 3 satisfying x = x, x2.

36 B. B. MAKAMBA

Now

while

/.w( a2b) = 0,

and

~~(~‘)=~~(b(ab))=min(~(b),v(ab))

= 1

pv(b) = pv(be) = min(p(b),v(e))

= 1.

So p(a*b) c pi y( b) and therefore FY is not a fuzzy subgroup of S,.

PRowsrrro~ 3.4. Let p and v be fuzzy subgroups of g. If p is fuzzy normal, then JLV = VP.

PROPOSITION 3.5. Let II. be a fuzzy subset of 9’. Then P is a fuzzy subgroup of g if and only if p2 = p and P(X) = P( x- ‘) for all XE 9.

PROPOSITION 3.6. Let p and v be fuzzy subgroups of 9”. Then pv = VP if and on/y if ~.tv is a fuzzy subgroup of 3.

It now follows from Propositions 3.4 and 3.6 that:

THEOREM 3.7. Let p and v be fuzzy subgroups of normal. Then pv is a fuzzy subgroup of 9.

THEOREM 3.8. Let ~1 and v be fuzzy subgroups of

3 where P is fuzzy

$9 where j.t is fuzzy normal. Assume also that p(e) = v(e). Then pv is the smallest fuzzy subgroup of i$ containing both 111 and v, i.e., if 4 is a fuzzy subgroup of 22 such that p<t and v=gE, then .$ >p-

PROPOSITION 3.9. Let ~1, P,, p2 be fuzzy subgroups of 9 such that P= P111*- Then suppb.4 = SUPP(F&UPP(L~~).

PROPOSITION 3.10. Let 1, P,, p2 be fuzzy subgroups of 9 with ~1~ and pZ fuzzy normal and p,(e) = p*(e). Suppose also that P~ACL~ = (cL,)~, where

and

Then supp(,~) = supp(~,)@&upp(yl,), where @ denotes the internal direct product.

DIRECT PRODUCTS 37

4. FUZZY INTERNAL DIRECT PRODUCTS

DEFINITION 4.1. Let CL, p,, p2 be fuzzy subgroups of ‘5”) with CL(~) > 0. Then p is the fuzzy internal direct product of p, and p2, and we write

B = c~@k, if

(i) CL, and pz are fuzzy normal,

(ii) k(e) = k(e) (iii) p = k ~2

(iv) CL, A cc2 = (de.

THEOREM 4.2. Let 9 be a finite group. Let p1 and ,L+ be fuzzy subgroups of $9 such that g,(e) = &e) > 0 and pi p2 is afuzw subwow of $9’. Then p,p2 is a fuzzy interval direct product if and only if for each t~(0, pl p2(e)], (pl p2)’ is an internal direct product of $, and p\.

Proof. Suppose ~1, p2 is a fuzzy internal direct product. Then pl and cl2 are fuzzy normal and

It is easy to show that (p,p2)’ = pip\. Since p1 and fi2 are fuzzy normal, all their level subgroups are normal subgroups of Y. Hence c(: and c(: are normal subgroups of 9. Let XG& n hi. Then p,(x)A pz(x) 2 t > 0. Hence _y = e and (p, p$ = &6X&, where @ denotes the internal direct product.

Conversely, suppose (chicly)’ = jki@& for all t@, rifiz(e)l. We must prove that CL, and p, are fuzzy normal and

PlAP2 = 1

deL x= e_

0, x#e

,& and & are normal subgroups of Y for all tE[O, h,(e)]. Hence Y, and cr2 are fuzzy normal.

Let xesupp(p,)-{e). Thus xecl: for some W0,&319 =d x4,4, since $, n & = { e}. Hence 0 6 p2( x) < t.

We show that: p2(x) = 0. If 0 c p2(x) < t, then let p2(x) = t,. So clz(e) 2 t, > 0, t, c t, and & G ~7. Also XE& and XE& and so X&J n ~7 = (e}. Thus x= e. This is a contradiction since x~supp( p,) - { e) . Thus

p2(~) = 0. Hence min(p,(x), p2(x)) = 0 for all xEsuPP(&)- (e). If

38 B. B. MAKAMBA

x$supp(&, then pi(x) = 0 and min(Cc,(xh 1.4~9 = 0.

So min(p,(x),p2(x))= {i”‘)’ “,;z. 9

5. FUZZY ISOMORPHISM

DEFINITION 5.1, Let 9, and gz be any two groups. Let p be a fuzzy subgroup of C!?i and v a fuzzy subgroup of 9Q. A fuzzy homomo~hism f: p -+ v is a homomo~hism f: supp(p) + supp(v) such that p(x) = v( f (x)) for all x~supp(p). If f is a bijection, then f is a fuzzy isomorphism from 1 onto v. p and v are then said to be (fuzzy) isomorphic and we write p = v.

DEFINITION 5.2. Let p and v be fuzzy subgroups of the groups iB, and 9’*, respectively. Let f: p + v be a fuzzy homomo~hism. Define p,: Y, --,

(0, 11 by

xczkerf

x$kerf,

where ker f is the kernel of f: supp( p) + supp( v). By, as defined here, is a fuzzy subgroup of 9,. pE is called the fuzzy kernel off.

PROPOSITION 5.3. Let pE be as given in Definition 5.2; then pE is fuzzy normal.

DEFINITION 5.4. Let p be a fuzzy subgroup of 9. p is fuzzy Abelian if fit is Abelian for all t E (0, p(e)] .

NOTE. A weaker definition of fuzzy Abelian was introduced in [4]. Defini- tion 5.4 is not too strong since, in many cases, a property possessed by a fuzzy set is also possessed by its support or weak o-cuts,

PROPWTION 5.5. Let p be a fuzzy subgroup of 27. Then p is fuzzy Abelian if and only is supp( p) is Abelian.

Proof. Suppose supp( k) is Abelian. & E supp,z for all tE(0, p(e)]. Then p* is Abelian for all tE(O, p(e)], i.e., p is fuzzy Abelian.

DIRECT PRODUCTS 39

Conversely, suppose p’ is Abelian for all t~(0, p(e)]:

SUPP(P) = u 4. O<tGp(e)

Let a, besupp(~). Then @E# and k#, for some t,, t,f(O, de)]. Suppose t, <t,. Then @C ptl, and a, bEpt I* ab = ba. Hence supp( CL) is Abelian.

PROPOSITION 5.6. Let ~1 and v be fuzzy subgroups of the groups $7, and g2, respectively, such that t.~ = v. Then p is fuzzy Abeiian if and only if B is fuzzy Abelian.

PROPOSITION 5.7. Let p and v be fuzzy subgroups of the groups Yg and g2, respectively, such that p = v. Let

Ep= {x~ 3,: p(x) = pL(.e))

Ev={xE CY2: v(x)=v(e))

Then Ep is isomorphic to Ev.

PROPOSITION 5.8. Let p, and p, be fuzzy subgroups of the groups p!i and $Y2, respeciively, such that pI = p2. Then ,u{ = & for 0 < t < p,(e)A p2(e).

Proof. Let f: pl --) p, be an isomorphism. So f: supp( h,) -+ supp( & is an isomorphism such that r,(x) = pZ(f( x)), x~supp(p,). Define g: &, + ki by

g=fl rf,*

Let xep\ then ~,(x)a t, and cL2(f(x))Z t *f(x)Ect’,, i.e., g(x)E&. Thus g is a well-defined isomorphism.

DEFINITION 5.9 [6]. Let p and Y be fuzzy subgroups of the groups 9, and gz2, respectively. The fuzzy external direct product of p and Y is a map

~XV: %‘X g-+[O, l] defined by pXv(x,, x,) = min(~(x,),~(x*)), x+ gi, i= 1,2.

~XY is a fiuzy subgroup of g1 X g2. (See [6]).

PROPOSITION 5.10. Let p and v be fuzzy subgroups of the groups 3, and g2, respectively. Then supp(.u Xv) = supp( P) X supp( Y), where X denotes the classical external direct product.

THEOREM 5. Il. Let fi and v be fuzzy subgroups of 9. If pv is a fuzzy internal direct product, then @v = p Xv.

40 B. B. MAKAMBA

Proof. By Proposition 3.10, supp( &v) = supp~@suppv. By Proposition 5.10, supp(pXv) = supp(~)Xsupp(v). Define f:supp(~Xv)+supp(@8v) by f(a, b) = ab, a~supp(p) and b~supp(v). From group theory, f is an isomor- phism. We only have to show that

CLXv(a,b)=Cl~‘v(f(a,b)),(a,b)EsuPP(~LXV).

/&@v( f( a, b)) = &0v( ab) = clv( ab)

= .b”YTY [min(44, V(Y))]

> 0 since aEsupp( cl) ; bEsupp( v) .

So we can assume that x~supp( p) and ~~supp( v). Then

ab = xy~supp( ~)@supp( v) =) a = x and b = Y.

so

~~v(f(u,b))=min(~(u),v(b))

= pxv( a, b)

and

/.4xv=p63bJ.

THEOREM 5.12 (The First Isomorphism Theorem). Let 9? and H be groups. Let ~1 and v be fuzzy subgroups of 3 and H, respectively. Let f: p --) v be a surjective fuzzy homomorphism with fuzzy kernel pE. Then hE=fh4 andfW= v.

Proof. First we show that f(p) = v. f(p) is a fuzzy subset of f ( 3) defined by

so

f(c()(f(x)) = sup cc(y). f(Y)=f(x)

f(cr)(fW = sup V(f(Y)) =v(f(x))- f(y)=f(x)

pE < p and pE is fuzzy normal. Clearly pE(e) = p(e) and xpE = pEx, for all

DIRECT PRODUCTS 41

XE 9. Define g:suppr/pE -‘suppv by g(xcL&= f(x). If x/.+=uPP/J//JE, then

Thus f( x)~supp V.

Suppose xk = yk Then y- ‘x,u, = ep, and

Hence x-‘yEkerf and f(x-‘y)= f(e), or f(x)=f(y). Thus g is well defined. It is easy to show that g is injective. Let yczsupp(v), then y = f(x) for some xtzsupp(& and y = g(xp&. Furthermore x~~~supp~/~~ since p/p& xpE) = p(x) > 0. Thus g is onto. Clearly g is a homomo~hism. So g is a classical isomorphism. Now

v(g(wE)) = v(fW

= IL(x)

Hence p/pE = v = f(p).

Remark. Let p, v and f be as given in Theorem 5.12. Define ?r: suppp-+

suppru/‘~r, by r(x) = xcls- Then T is a fuzzy homomo~hism from ~1 onto

~/~~.

CL/&( +>> = dPE(W.E) = Ax)

for all xfsupp CL. Also, g,r = f, as fuzzy homomorphisms.

THEOREM 5.13 (The Third Isomorphism Theorem). Let v < p < E be

42 B. B. MAKAMBA

normal fuzzy subgroups of 59 such that v(e) = p(e) = f(e). Then

(9 p/v(ev) = 4 lv(ev), (ii) p/v is fuzzy normal and p/v < 4 /v, and

(iii) ~Ic~=(t;Iv)I(~/v).

Proof.

I::) p/v(-) =A4 = ,563 = tIv(ev).

P/V(XV) =/4x) G t(x) = tIv(xv). p/v((av)-‘xvav) =p/v(a-‘vxvav)

=p/v(a-‘xav) =p(a-‘xa) = p( x) since p is fuzzy normal =p/v(xv)for all a v, XVEE:= { yv: YE 9e).

(iii) Let y: [ -+ t /CL be the natural fuzzy homomorphism, i.e. y: supp [ + supp 4 /cc is a classical homomorphism given by y(x) = xp, x~supp t.

Clearly, t(x)= [/p(y(x)) for all x~supp4:. Define \k:supp([/v)/(j~/v)

+SUPP~/P by WXV(P/V))= Y(X). If XV(C~/V)ESUPP(~/~)/(~/~), then

* El&(x)) > 0

i.e., y(x)~supp[/p. Suppose XV(II./V) = yv(~/v). Then xv(p/v)(gv) = yv(p/v)(gv) for all ge 9. Hence p/v(x-‘gv) = p/v(y-‘gv) or p(x-‘g) = p( y- ‘g) for all gE 9. Therefore xc1 = yp and y(x) = y(y). So \k is well defined.

Clearly * is a classical homomorphism.

W.++v(h+))) = W-4r(a)>

= E IcL(w)

= t(a)

= t/v(av)

= (tlv)l(Plv)(av(Plv))

for all av(p/v)Esupp( [ /v>/<~/ v). Hence \k is a fuzzy homomorphism. It is easy to show that !P is surjective. Suppose +( av( p / v)) = +( bv( p / v)). Then

DIRECT PRODUCTS 43

r(a)=y(b) or y(ab-‘)=ep. Hence ab-‘p=ep=)ap=bp and therefore p(a-‘x) = p(b-‘x) for all XE 9. Then p/v(a-‘xv) = p/v(b-‘xv) and av(p/v)(xv) = bv(p,/v)(xv) for all XE 9. Thus av(p/v) = bu(p/v) and 3 is an injection.

REFERENCES

1. A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35512-517 (1971). 2. N. P. Mukberjee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets,

Inform. Sci. 34:225-239 (1984). 3. L. A. Zadeh, Fuzzy sets, Inform. Control 8:338-353 (1965). 4. P. Bhattacharya and N. P. Mukherjee, Fuzzy groups: some group theoretic analogs II,

Inform. Sci. 41:77-91 (1987). 5. P. S. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl. 84:264-269

(1981). 6. H. Sherwood, Product of fuzzy subgroups, Fuzzy Sets and Syst. 11:79-89 (1983).

Received 1 May 1990; revised 25 October 1990