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On the definition of a fuzzy subgroup
Fuzzy Sets and Systems 51 (1992) 235-241 North-Holland
235
On the definition of a fuzzy subgroup S.K. Bhakat Siksha-Satra, Visva-Bharati University, Santiniketan, West Bengal, India
P. Das Department of Mathematics, Visva-Bharati University, Santiniketan, West Bengal, India Received March 1991 Revised August 1991
this manner is the ( • , • v q)-fuzzy subgroup which is a generalization of the concept of fuzzy subgroup defined by Rosenfeld. The rest of the paper has been devoted to the study of some fundamental properties of such fuzzy subgroups. The case when min is replaced by an arbitrary t-norm in the definition of an (~r,/3)-fuzzy subgroup will be dealt with in a subsequent paper.
Abstract: Using the idea of quasi-coincidence of a fuzzy point with a fuzzy set, some new concepts of a fuzzy subgroup are introduced and their acceptibility is investigated. Some fundamental properties of one such viable fuzzy subgroup are obtained.
2. Preliminaries
Keywords: Fuzzy algebra; fuzzy point; quasi-coincidence;
Let G be any non-empty set and let I denote the closed unit interval [0, 1].
fuzzy subgroup; fuzzy normal subgroup; fuzzy coset.
1. Introduction Using the notion of a fuzzy set introduced by Zadeh [12], Rosenfeld [10] defined a fuzzy subset A of a group G to be a fuzzy subgroup of G if for all x, y • G, (i) A(xy) >I min{A(x), A(y)} and (ii) A(x) = A(x-~). Condition (i) is equivalent to the following condition: (iii) for any two fuzzy points xt,, Yt2• A holds (Xy )min{t,,t2} • A. If we take notice of the idea of quasicoincidence of a fuzzy point with a fuzzy set, it is natural to enquire about the outcome if the two • 's in (iii) are replaced by 'quasi-coincident', q, or 'belongs to and is quasi-coincident with', i.e., • ^ q or 'belongs to or is quasi-coincident with', i.e., • v q. With this objective in view, the concept of an (tr, fl)-fuzzy subgroup is introduced where o~, /3 denote any one of • , q, • ^ q, • v q and tr 4: • A q. It has been found that the only acceptable non-trivial concept obtained in Correspondence to: S.K. Bhakat, Siksha-Satra, VisvaBharati University, Santiniketan, West Bengal, India.
Definition 2.1. A map A : G --~ I is called a fuzzy subset of G.
Definition 2.2 [7]. A fuzzy subset A of G of the form A(y) = {~(~0)
ify =x, i f y 4=x,
is said to be a fuzzy point with support x and value t and is denoted by x,. Definition 2.3 [7]. A fuzzy point x, is said to belong to (resp. be quasi-coincident with) a fuzzy subset A, written as xt • A (resp. x, qA) if A(x) >1t (resp. A(x) + t > 1). If x , • A and (resp. o r ) x , qA, then we write x , • ^ qA (resp. x , • v qA). For all tl, t2 • I, min{tl, t2} will be denoted by M(tl, t2). Definition 2.4 [10]. Let G be a group. A fuzzy subset A of G is said to be a fuzzy subgroup of G if for all x, y e G, (i) A(xy) >>-M(A(x), A(y)), (ii) A(x) = A(x-l). Lemma 2.5. For any fuzzy subset A of a group
0165-0114/92/$05.00 ~ 1992--Elsevier Science Publishers B.V. All rights reserved
S.K. Bhakat, P. Das / Definitionof a fuzzy subgroup
236
G, the following statements are equivalent: (i) For all x, y • G, A(xy) >! M(A(x), A(y)). (ii) For all x, y • G and tl, t2 • 1 - {0},
if x,,, y,: • A, then (Xy)MU,, t~) • A.
Proof. (i) :ff (ii): If x,,, yt: • A, then A(x) >i t,,
A(y) >! t2,
which implies
A(xy) >i M(A(x), A(y)), by (i). Now this is greater than M(t~, t2), and thus (Xy)M(t,, t2) • A. (ii) f f ( i ) : If XA(,), YA(y)•A, then by (ii) (Xy)M(A(x), a(y)) E A . So
A(xy) >>-M(A(x), A(y)). Remark 2.6. It follows from L e m m a 2.5 that condition (i) of Definition 2.4 may be replaced by condition (ii) of L e m m a 2.5. If we take notice of the idea of quasi-coincidence of a fuzzy point with a fuzzy set, it is natural to enquire what happens if the • ' s in the left- and right-hand side of condition (ii) of L e m m a 2.5 are replaced by q or some combination of e and q like • ^ q, • v q. With this objective in view the concept of an (o:, f)-fuzzy subgroup is introduced in the following section.
3. {a, ll)-fuzzy subgroup Unless otherwise mentioned G will denote a group with e as the identity element and a¢ f will denote any one of • , q, • v q, or • ^ q. x, ~ A will mean that xt trA does not hold.
Definition 3.1. A fuzzy subset A of G is said to be an (~, fl)-fuzzy subgroup of G (o: 4= • ^ q) if for any x, y • G and tl, t2 e I - {0}, the following hold: (i) If xt, irA, y,: erA, then (xy)Mu,,,2) flA. (ii) A ( x ) = A(x-1). Remark 3.2. The case tr = • ^ q is omitted since there exist fuzzy subsets A such that {Xt'~X t • ^ qA} is empty. In fact, if A(x) ~<0.5 for all x e G, then A is such a fuzzy subset.
Theorem 3.3. Let A be a non-zero (o:, fl)-fuzzy subgroup of G. Then (i) A(e) > 0 (ii) A o = { x • G ; A ( x ) > 0 } is a subgroup of G. Proof. (i) If possible, let A ( e ) = 0. Since A is non-zero, there exists x • G such that A ( x ) = t>0. If a e = • or • v q , then x , x7 ~c~A but (XX--1)M(t,t) ]~A, a contradiction. If a~=q, then x~, x ? 1 c~A but (xx-~)M(~,t)flA, a contradiction. Therefore A ( e ) > O. (ii) Let x , y • A o . Then A(x), A(y)>O. If possible, let A(xy) = O. If o: = • or e v q, then XA(x), YA(y)o:A but (Xy)M(Z(x),A(y)) fiA, a contradiction. Again if i x = q , then Xl, Yl o~A but (xY)M 0. So xy • Ao. Again, if x • Ao, then
A(x-~)=A(x)>O, which implies x -~ • Ao. Therefore Ao is a fuzzy subgroup of G.
Theorem 3.4. Let A be a non-zero (tr, fl)-fuzzy subgroup of G where (o:, fl)= (i) (•, q); (ii) (~, • ^ q); (iii) (iv) (v) (vi) (vii)
(q, •); (q, • ^ q); ( • v q, q); ( • v q, • ^ q ) ; ( • v q, •).
Then A = XAo, the characteristic function of Ao. Proof. If possible, let there exist an x • Ao such that A(x) < 1. (i) and (ii): Let ~ = •. Choose t • I - {0} such that t < m(1 - A(x), A(x), a(e)). Then x,, e, irA but (Xe)M(t.,)fiA where /3 = q or ^ q, a contradiction. (iii) and (iv): Let t r = q . Then x~, e~o:A but (xe)M(l,l)=XtflA where / 3 = • or • ^ q , a contradiction. (v), (vi) and (vii): Let a~= E v q . Choose t such that x , ~ A but xt (:1A. Then x, trA, e~o:A
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but (Xe)MU,1)flA where f l = q or • ^ q , a contradiction. Again in case of (vii), x~, el c~A but (Xe)Mtl.0 ~ A, a contradiction. Thus A = ~ A 0 '
Theorem 3.5. Let A be a non-zero (q, q)-fuzzy subgroup of G. Then A is constant on Ao. Proof. If possible, let there exist b • Ao such that t, = A ( b ) :/: A ( e ) = t. T h e n either tl < t or t < t~. Let t~ < t. C h o o s e t2, t3 • 1 -- { 0 } such that
1-t
and t 3 > m a x { t 2 , 1 - t l } .
Then e,~, bt~ q A but (eb)M(t2,t3) = bt2 ClA, a contradiction. Next let t
Theorem 3.6. Let H be any subgroup o f G. Let A:G---~ I be such that A ( x ) = O for all x • G - H. Then A is a (q, • v q)-fuzzy subgroup of G if any one of the following holds: (i) A is a non-zero constant on H. (ii) A ( x ) ~ > 0 . 5 and A ( x ) = A ( x -j) for all xeH.
Proof. If possible, let A ( x ) < 0 . 5 for all x • G. Since A is not constant on Ao, there is an x e A0 such that
t = A ( x ) :/: A(e) = r. T h e n either r < t or r > t. If r < t, choose 6 > 0.5 such that t+6>l
and r + 6 < l .
T h e n x6, x~ ~q A but (xx-1)Mt6.6) = e6 • v q A , a contradiction. A g a i n if r > t, we can c h o o s e 6 > 0.5 such that t+6
and r + 6 > l .
T h e n e6 q A , xl q A but
(eX)M~.j) = X~ • V q A , a contradiction. T h e r e f o r e there
exists an x • A o
such that
A ( x ) >- O.5.
Lemma 3.9. A(e) >i 0.5. Proof. If possible, let r = A ( e ) < 0.5. By L e m m a 3.8, there is an x • G such that t = A ( x ) >i 0.5. So r < t. C h o o s e tl > r such that t+tl>l
and r + t l < l .
T h e n x,,, x -t, l q A but (XX--1)M(tl,tl) = e,, • v q A , a contradiction. H e n c e A ( e ) >I 0.5.
Proof. (i) Straightforward. (ii) Let x,,, y,2qA w h e r e x, y • G and t~, t2 • 1 - {0}. If M ( h , t2) > 0 . 5 , then (xy)M(t,.t2)qA. If M(t~, t2) <<-0.5, then (Xy)MU,.,2) • A. Thus in any case (Xy)MU,,,~) • V q A . T h e r e f o r e A is a (q, • v q)-fuzzy s u b g r o u p of G.
Proof of Theorem 3.7. If possible, let t = A ( x ) < 0.5 for s o m e x • A0. Let us c h o o s e tl > 0 such that t + tt < 0.5. T h e n xl q A , eo.5+t, q A but (eX)M(I,O.5+t 0 = Xo.5+tl • V q A ,
a contradiction. T h e r e f o r e A ( x ) >I 0.5 Vx c Ao.
Theorem 3.7. Let A be a ( q , • v q)-fuzzy subgroup of G such that A is not constant on Ao. Then A ( x ) >10.5 for all x • Ao. To prove this t h e o r e m we require the following two lemmas in which the notation of T h e o r e m 3.7 is maintained.
Remark 3.10. Let H and A be as defined in T h e o r e m 3.6. T h e n A is a ( c v q, • v q)-fuzzy subgroup of G if any one of the conditions (i) and (ii) of T h e o r e m 3.6 holds. Also, if A is an ( • v q, • v q)-fuzzy s u b g r o u p of G such that A is not constant on A0, then A(x)~>0.5 for all x •Ao.
Lemma 3.8. There exists an element x • G such that A ( x ) >! 0.5.
The p r o o f can be c o n s t r u c t e d in a m a n n e r similar to those of T h e o r e m s 3.6 and 3.7.
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Example 3.11. Example of an ( • , • v q)-fuzzy subgroup. Consider Klein's 4-group G = {e, a, b, c} defined by the multiplication table
eabc
a b c
a b c
abc b c a c a b If A : G ~ I is defined by
eabc aecb bcea cbae
A(a) =0.7,
A(e)=0.6,
A(a)=0.7,
A(b)=A(c)=O.4.
Then A is an ( • , • v q)-fuzzy subgroup of G. We note that A is not an (o¢ fl)-fuzzy subgroup of G where (o~, fl) = (e, • ) , (q, • v q), ( • v q, • v q). Remark 3.12. Let A be an (• v q, • v q)-fuzzy subgroup of G. Then from Xtl , Yt: • A, it follows that x,,, Y,2 • v q A, implying
(Xy)M(t,,t2) E V q A . So A is an ( • , • v q)-fuzzy subgroup of G. Again, let A be an ( • , •)-fuzzy subgroup of G. Then xt~, y , 2 • A implies (xY)M~t,.t2)•A and thus
Therefore A is an ( • , • v q)-fuzzy subgroup of G. Remark 3.13. A is a fuzzy subgroup of G (in the sense of Rosenfeld) iff Vx, y • G,
which is equivalent to the condition Vtl, t 2 • l - { O } .
It can be readily verified that ~
Remark 3.15. The theorems and examples in this section reveal that the only non-trivial generalisation of a fuzzy subgroup defined by Rosenfeld, obtained in this way, is the concept of an ( • , • v q)-fuzzy subgroup. In the following section some fundamental properties of such fuzzy subgroups are studied.
4. ( • , • v q)-fuzzy subgroups and normal subgroups
Unless otherwise stated in this section by a fuzzy subgroup we shall mean an ( • , • v q)fuzzy subgroup of G.
Proof. Let Xn be a fuzzy subgroup of G. Then x, y • H implies xl, yl • Zn and thus (Xy)M(1,1) • v q XH which yields Xn(xy) > 0, from which we find xy • H. Also, x • H implies X n ( x - ' ) = Zn(x) = 1,
A(xy -l) >1M ( A ( x ) , A ( y ) ) ,
(xy-l)M(tl,t2)EA
A ( c ) =0.9,
Theorem 4.1. For any subset H of G, Zn is a fuzzy subgroup of G iff H is a subgroup of G.
(Xy)M(t,,t2) • V q A .
xt,,ytzEA ~
A ( b ) =0.8,
then A satisfies (1) but since A(b -1) :/:A(b), A is not an ( • , e v q)-fuzzy subgroup of G.
Let A : G---> I be defined by
xtl, Yt2 • A
defined by the multiplication table
(xy-l)M~,,.,2) • V qA
(1)
is a necessary condition for A to be an (6, • v q)-fuzzy subgroup G. However this condition is not sufficient as shown by: Example 3.14. Let G = {a, b, c} be the group
so that x -~ e H. So H is a subgroup of G. Conversely, if H is a subgroup of G, then X, is an (e, •)-fuzzy subgroup of G and so by Remark 3.12, Xrt is an (e, • v q)-fuzzy subgroup of G. Theorem 4.2. Let { A i ; i • J } be any family of fuzzy subgroups of G. Let A = N i ~ A i . Then A is a fuzzy subgroup of O. Proof. If possible, let A be not a fuzzy subgroup of G. Then there exist x, y • G and h, t• • I {0} such that xt~, Yt2 • A but (Xy)M(t~,t2) E v q A ,
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i.e.~
thus
A(xy) < M(t~, t2) and A(xy) + M(q, t2) < 1.
(f(x)f(Y))MO,.t2) • V q B
So,
(since B is a fuzzy subgroup), so that
A(xy) < 0.5.
(2)
Let
(since f is a homomorphism), and thus (Xy)M(,,.,2) • V q f - t ( B ) . Therefore f - l ( B ) is a fuzzy subgroup of G. (ii) Let xi • f ( G ) and
J, = (i • J ; A i ( x y ) >-m(tl, t2)}, J2 = {i • J: A,(xy) + M(t,, t2) > 1 and mi(xy ) < M(t~, tz)}.
(f(a))(xi) >t ti for i = 1, 2.
Then J = J~ tOJ2 and Jl N J2 = 0. If J2 = 0, then
A~(xy) >I M(t~, tz)
Since A has the 'sup property', there is a z~ • f-~(xi) such that
Vi • J,
a(zi) = sup{a(w); w • f - l ( x i ) },
which yields A(xy) >! M(q, tz), a contradiction. If J24:0, then for any i•.I2, A~(xy)+ M(fi, t2) > 1 and Ai(xy) < M(h, tl). Therefore M(tl, t2) > 0.5. Now x, • A implies A(x) >I tl and thus
Ai(x ) >!A(x) >! t~ > M(tl, t2) ~> 0.5
Vi • J.
Vi•J.
Then
Ai(xy)>~0.5
i = 1, 2.
Then zl,,, z2,, • A. Since A is a fuzzy subgroup of G, (Z1Zz)M(t,,t2) • V qA. Now
z,z2•f-~(XlX2)
an d
so
(f(A))(x,x2)>-
A(zlz2). Thus (Xy)M.,,.) • V qf(A).
Similarly,
Ai(y)>0.5
(f(xy))M(t,.t2) • V q B
Vi •J.
For, t' = A~(xy) < 0.5 implies x,., y,,, • Ai, but (xy)t,,~Q---qAi if t' < t " > 0 . 5 , which contradicts that A, is an ( • , • v q)-fuzzy subgroup of G. Thus A ( x y ) ~ 0 . 5 , which contradicts (2). So (xY)Mo,.t~) • v qA. That A(x -~) = A(x) for any x • G is obvious. Therefore A is a fuzzy subgroup of G. Theorem 4.3. Let G and G' be two groups and let f : G ~ G' be a homomorphism. Let A, B be two fuzzy subgroups of G and G', respectively. Then (i) f - l ( B ) is a fuzzy subgroup of G. (ii) If A satisfies the 'sup property', i.e., for any subset T of G there exists a to • T such that
Therefore f ( A ) is a fuzzy subgroup of f(G). Remark 4.4. If A is a fuzzy subgroup of G, then for any t • l - { 0 } , A t = { x e G ; A ( x ) ~ t } may not be a subgroup of G as can be seen from Example 3.11. Definition 4.5. A fuzzy subgroup A of G is said to be a fuzzy normal subgroup of G if
A(xax -1) >~A(a)
Vx, a • G.
(3)
Condition (3) is equivalent to the following conditions:
A(xax -1) = A(a) A(xy) = A ( y x )
Vx, a • G, Vx. y • G.
(4) (5)
then f ( A ) is a fuzzy subgroup o f f ( G ) .
Remark 4.6. For any subset H of G, ;~, is a fuzzy normal subgroup of G iff H is a normal subgroup of G. Also, the intersection of an arbitrary family of fuzzy normal subgroups of G is a fuzzy normal subgroup of G. The proof is straightforward.
Proofl Let x , y • G and t ~ , t 2 • l - { O } . Now x,,, y,. • f - ' ( B ) implies (f(x)),,, (f(Y)),2 • B, and
Remark 4.7. If A is a fuzzy normal subgroup of G in the sense of Mukherjee and Bhattacharya
A(to) = sup{A(t); t • T},
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s.K. Bhakat, P. Das / Definitionof a fuzzy subgroup
[8], then
[since A is a fuzzy normal subgroup]
A([x, y)] >~A(x)
Vx, y E G
= A ( x l l g y l 1)
where Ix, y] denotes the commutator of x, y. But this is not necessarily true if A is a fuzzy normal subgroup of G in the sense of Definition 4.5, as shown by:
Example 4.8. Consider the fuzzy subgroup A defined in Example 3.11. A is, in fact, a fuzzy normal subgroup of G. But A(a) = 0.7 > 0.6 = A([a, b]). Definition 4.9. Let A be a fuzzy subgroup of G. For any x • G, fix (resp. fl~) : G --> I defined by fl~(y) = A(yx-~), [resp..,ix(y) =A(x-ly)]
Yy e G
Theorem 4.10. Let A be a fuzzy normal subgroup of G. Let O%be the set of all fuzzy cosets of A. Then O%is a group if the composition is defined by Vx, y ~ G.
(6)
Let fl : O%--->1 be defined by
= `'iy 2.
(7)
So for any g • G,
A(gx; 1) = A(gx~ 1)
(8)
and
A(gy(')
=
A(gy~l).
Now for any g e G,
A(gyzlx2 l)
=
A(gy~lx; ')
A~,y,(g) : Ax~y~(g) vg ~ G. Thus the composition defined by (6) is well-defined. It is now easy to verify that O%is a group with respect to (6) with Ae as the identity element and Ax , as the inverse of fix for any xeG. We now show that fl is a fuzzy normal subgroup of o%. From A(Ay) I> t2
for q, t2 e I - {0}, it follows that xt,, Yt2 E A, and thus (xy)M
A(A ) = A(x)
= A(x-1)
= A(Ax-,) = A((A~)-'),
VAx ~ o%.
Thus fl is a fuzzy subgroup of O%. Moreover,
[since A is a fuzzy normal subgroup]
Proof. To verify that the composition given by (6) is well-defined, let x l, Xz, Yl, Y2 E G be such that
fly,
Hence
= A(xax -1) = A(a)
Then fl is a fuzzy normal subgroup of O%.
= fl~: and
[since A is a fuzzy normal subgroup].
A ( A ~ A ~ - , ) = A(Axox ,)
A(flx) = A(x) Vx ~ G.
fix,
=A(gy{lx7 l)
A(Ax)/> tl,
is called the fuzzy left (resp. right) coset of G determined by x and A. If A is a fuzzy normal subgroup of G, then fl~ = fl~ for all x E G.
fix" fly = flxy
[replacing g by x-(lg in (9)]
(9)
=A(Aa)
vAa, Ax~o%.
Hence A is a fuzzy normal subgroup of O%. Remark 4.11. With the same notation Theorem 4.10, let 0 : G ---> O%be defined by
O(x) = flx
of
Vx E G.
Then obviously 0 is a homomorphism. However, unlike the case of the fuzzy normal subgroup in the sense of Mukherjee and Bhattacharya [8], Ker O:/:GA = {x E G ; A ( x ) = A ( e ) } , as shown by:
[replacing g by gyfl in (8)]
=A(x71gy~ 1)
Example 4.12. Let G be the group defined in
S.K. Bhakat, P. Das / Definition of a fuzzy subgroup
Example 3.11. Let B : G --~ I be defined by B(e) = 0.6,
B(a) = 0.6,
B(b)=0.7,
B(c)=O.8.
If A~ is normal f o r i = 1 , 2 . . . . is A.
241
,n,
then so
The proof is similar to that of T h e o r e m 4.2
Then B is a fuzzy normal subgroup of G, but ker 0 = {e} +e {e, a} = GA. Theorem 4.13. Let A be a f u z z y subgroup o f ~. Then ,4 ~: G ~ I defined by Al(x) = Al(Ax) is a f u z z y subgroup o f G. I f A i is f u z z y normal, then so is A1.
The proof is straightforward. Remark 4.14. Theorems 3.3, 3.5 of Mukherjee and Bhattacharya [9] are true if A is an ( ~ , ~ v q)-fuzzy subgroup of G. However, consider Lemma 3.7 of the same which states: "Let A be a fuzzy subgroup of a finite group G. Define H = {g ~ G ; A ( g ) = A ( e ) }
and
Theorem 4.17. Let G be the internal product o f two subgroups G~ and G2. Let A , A~, A2 be f u z z y subgroups o f G, Gj and G2, respectively. Let A~, ,4i : G---~ I (i = 1, 2) be defined by
Ai(x) = A(x3 and Ai(x) = A,(x3 if x = x l x z , where xl ~ G1, x2 ~ G2. Then Ai, /[i are f u z z y subgroups o f G f o r i = 1, 2 I r A , A1, A2 are normal, then so are A~, Ai. ^
Proof. Let x, y e G. Let x = x~x:,, where xi, Yi ~ Gi for i = 1, 2. Then
v
Y =Y~Yz
xy = (x l yO(x2 yz).
Now x,, y,, • Ai implies (xi)t, (Y~)c e A, and thus (x~y~)M~,,,,) e v q A [since A is a fuzzy subgroup], which yields (Xy)M(t,t,) E V q~z[i. Therefore A~ is a fuzzy subgroup of G for i = 1, 2. If A~ is fuzzy normal, then it can be readily verified that A~ is also fuzzy normal. The case of ,4~ may be treated similarly.
I;= {x ~ G;A~= de}. Then H and K are subgroups of G. Further H=K." This is not necessarily true for an (6, ~ v q)-fuzzy subgroup of G, as shown by: Example 4.15. Let G be the group defined in Example 3.11. Let A : G ~ I be defined by A ( e ) = A ( a ) = A ( b ) = 0.6 and A ( c ) = 0.7.
Then A is an (c, 6 v q)-fuzzy subgroup of G. However, H = {x ~ G ; A ( x ) = A(e)} = {e, a, b}
is not a subgroup of G. Here ,3.~ = A. Also,
Ao(a)
= 0.6,
A,(b)
= 0.7,
A,(c)
= o.6.
Hence ii¢ +e ,~i,. Therefore H 4= K. Theorem 4.16. Let A~ be a f u z z y subgroup o f a group Gi for i = 1 , 2 . . . , n. Then A : G1× Gz x • .. x G~---->1, defined by A(x, ....
, x,) = M(A,(x,) .....
An(xn) )
where xi e Gi, for i = 1 , 2 . . . . . n, is a f u z z y subgroup o f the group Gi x (32 x • • • z G,.
References [1] M.T. Abu Osman, On the direct product of fuzzy subgroups, Fuzzy Sets and Systems 12 (1984) 87-91. [2] MT. Abu Osman, On some product of fuzzy subgroups, Fuzzy Sets and Systems 21 (1987) 79-86. [3] J.M. Anthony and M. Sherwood, Fuzzy groups redefined, J. Math. Anal. Appl. 69 (1979) 124-130. [4] P. Bhattacharya, Fuzzy subgroups, some characterizations I, J. Math. Anal. Appl. 128 (1987) 241-252. [5] P. Bhattacharya, Fuzzy subgroups, some characterisations II, Inform. Sci. 38 (1986) 293-297. [6] P.S. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl. 84 (1981) 264-269. [71 Pu Pao Ming and Liu Ying Ming, Fuzzy topology I: Neighbourhood structure of a fuzzy point and Moore-Smith convergence, J. Math. Anal. Appl. 76 (1980) 571-599. [8] N.P. Mukherjee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inform. Sci. 34 (1984) 225-239. [91 N.P. Mukherjee and P. Bhattacharya, Fuzzy groups: Some group theoretic analogs, Inform. Sci. 39 (1986) 247-268. [10] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. [11] H. Sherwood, Product of fuzzy subgroups, Fuzzy Sets and Systems U (1983) 79-89. [12] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353.
235
On the definition of a fuzzy subgroup S.K. Bhakat Siksha-Satra, Visva-Bharati University, Santiniketan, West Bengal, India
P. Das Department of Mathematics, Visva-Bharati University, Santiniketan, West Bengal, India Received March 1991 Revised August 1991
this manner is the ( • , • v q)-fuzzy subgroup which is a generalization of the concept of fuzzy subgroup defined by Rosenfeld. The rest of the paper has been devoted to the study of some fundamental properties of such fuzzy subgroups. The case when min is replaced by an arbitrary t-norm in the definition of an (~r,/3)-fuzzy subgroup will be dealt with in a subsequent paper.
Abstract: Using the idea of quasi-coincidence of a fuzzy point with a fuzzy set, some new concepts of a fuzzy subgroup are introduced and their acceptibility is investigated. Some fundamental properties of one such viable fuzzy subgroup are obtained.
2. Preliminaries
Keywords: Fuzzy algebra; fuzzy point; quasi-coincidence;
Let G be any non-empty set and let I denote the closed unit interval [0, 1].
fuzzy subgroup; fuzzy normal subgroup; fuzzy coset.
1. Introduction Using the notion of a fuzzy set introduced by Zadeh [12], Rosenfeld [10] defined a fuzzy subset A of a group G to be a fuzzy subgroup of G if for all x, y • G, (i) A(xy) >I min{A(x), A(y)} and (ii) A(x) = A(x-~). Condition (i) is equivalent to the following condition: (iii) for any two fuzzy points xt,, Yt2• A holds (Xy )min{t,,t2} • A. If we take notice of the idea of quasicoincidence of a fuzzy point with a fuzzy set, it is natural to enquire about the outcome if the two • 's in (iii) are replaced by 'quasi-coincident', q, or 'belongs to and is quasi-coincident with', i.e., • ^ q or 'belongs to or is quasi-coincident with', i.e., • v q. With this objective in view, the concept of an (tr, fl)-fuzzy subgroup is introduced where o~, /3 denote any one of • , q, • ^ q, • v q and tr 4: • A q. It has been found that the only acceptable non-trivial concept obtained in Correspondence to: S.K. Bhakat, Siksha-Satra, VisvaBharati University, Santiniketan, West Bengal, India.
Definition 2.1. A map A : G --~ I is called a fuzzy subset of G.
Definition 2.2 [7]. A fuzzy subset A of G of the form A(y) = {~(~0)
ify =x, i f y 4=x,
is said to be a fuzzy point with support x and value t and is denoted by x,. Definition 2.3 [7]. A fuzzy point x, is said to belong to (resp. be quasi-coincident with) a fuzzy subset A, written as xt • A (resp. x, qA) if A(x) >1t (resp. A(x) + t > 1). If x , • A and (resp. o r ) x , qA, then we write x , • ^ qA (resp. x , • v qA). For all tl, t2 • I, min{tl, t2} will be denoted by M(tl, t2). Definition 2.4 [10]. Let G be a group. A fuzzy subset A of G is said to be a fuzzy subgroup of G if for all x, y e G, (i) A(xy) >>-M(A(x), A(y)), (ii) A(x) = A(x-l). Lemma 2.5. For any fuzzy subset A of a group
0165-0114/92/$05.00 ~ 1992--Elsevier Science Publishers B.V. All rights reserved
S.K. Bhakat, P. Das / Definitionof a fuzzy subgroup
236
G, the following statements are equivalent: (i) For all x, y • G, A(xy) >! M(A(x), A(y)). (ii) For all x, y • G and tl, t2 • 1 - {0},
if x,,, y,: • A, then (Xy)MU,, t~) • A.
Proof. (i) :ff (ii): If x,,, yt: • A, then A(x) >i t,,
A(y) >! t2,
which implies
A(xy) >i M(A(x), A(y)), by (i). Now this is greater than M(t~, t2), and thus (Xy)M(t,, t2) • A. (ii) f f ( i ) : If XA(,), YA(y)•A, then by (ii) (Xy)M(A(x), a(y)) E A . So
A(xy) >>-M(A(x), A(y)). Remark 2.6. It follows from L e m m a 2.5 that condition (i) of Definition 2.4 may be replaced by condition (ii) of L e m m a 2.5. If we take notice of the idea of quasi-coincidence of a fuzzy point with a fuzzy set, it is natural to enquire what happens if the • ' s in the left- and right-hand side of condition (ii) of L e m m a 2.5 are replaced by q or some combination of e and q like • ^ q, • v q. With this objective in view the concept of an (o:, f)-fuzzy subgroup is introduced in the following section.
3. {a, ll)-fuzzy subgroup Unless otherwise mentioned G will denote a group with e as the identity element and a¢ f will denote any one of • , q, • v q, or • ^ q. x, ~ A will mean that xt trA does not hold.
Definition 3.1. A fuzzy subset A of G is said to be an (~, fl)-fuzzy subgroup of G (o: 4= • ^ q) if for any x, y • G and tl, t2 e I - {0}, the following hold: (i) If xt, irA, y,: erA, then (xy)Mu,,,2) flA. (ii) A ( x ) = A(x-1). Remark 3.2. The case tr = • ^ q is omitted since there exist fuzzy subsets A such that {Xt'~X t • ^ qA} is empty. In fact, if A(x) ~<0.5 for all x e G, then A is such a fuzzy subset.
Theorem 3.3. Let A be a non-zero (o:, fl)-fuzzy subgroup of G. Then (i) A(e) > 0 (ii) A o = { x • G ; A ( x ) > 0 } is a subgroup of G. Proof. (i) If possible, let A ( e ) = 0. Since A is non-zero, there exists x • G such that A ( x ) = t>0. If a e = • or • v q , then x , x7 ~c~A but (XX--1)M(t,t) ]~A, a contradiction. If a~=q, then x~, x ? 1 c~A but (xx-~)M(~,t)flA, a contradiction. Therefore A ( e ) > O. (ii) Let x , y • A o . Then A(x), A(y)>O. If possible, let A(xy) = O. If o: = • or e v q, then XA(x), YA(y)o:A but (Xy)M(Z(x),A(y)) fiA, a contradiction. Again if i x = q , then Xl, Yl o~A but (xY)M
A(x-~)=A(x)>O, which implies x -~ • Ao. Therefore Ao is a fuzzy subgroup of G.
Theorem 3.4. Let A be a non-zero (tr, fl)-fuzzy subgroup of G where (o:, fl)= (i) (•, q); (ii) (~, • ^ q); (iii) (iv) (v) (vi) (vii)
(q, •); (q, • ^ q); ( • v q, q); ( • v q, • ^ q ) ; ( • v q, •).
Then A = XAo, the characteristic function of Ao. Proof. If possible, let there exist an x • Ao such that A(x) < 1. (i) and (ii): Let ~ = •. Choose t • I - {0} such that t < m(1 - A(x), A(x), a(e)). Then x,, e, irA but (Xe)M(t.,)fiA where /3 = q or ^ q, a contradiction. (iii) and (iv): Let t r = q . Then x~, e~o:A but (xe)M(l,l)=XtflA where / 3 = • or • ^ q , a contradiction. (v), (vi) and (vii): Let a~= E v q . Choose t such that x , ~ A but xt (:1A. Then x, trA, e~o:A
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S.K. Bhakat, P. Das / Definition of a fuzzy subgroup
but (Xe)MU,1)flA where f l = q or • ^ q , a contradiction. Again in case of (vii), x~, el c~A but (Xe)Mtl.0 ~ A, a contradiction. Thus A = ~ A 0 '
Theorem 3.5. Let A be a non-zero (q, q)-fuzzy subgroup of G. Then A is constant on Ao. Proof. If possible, let there exist b • Ao such that t, = A ( b ) :/: A ( e ) = t. T h e n either tl < t or t < t~. Let t~ < t. C h o o s e t2, t3 • 1 -- { 0 } such that
1-t
and t 3 > m a x { t 2 , 1 - t l } .
Then e,~, bt~ q A but (eb)M(t2,t3) = bt2 ClA, a contradiction. Next let t
Theorem 3.6. Let H be any subgroup o f G. Let A:G---~ I be such that A ( x ) = O for all x • G - H. Then A is a (q, • v q)-fuzzy subgroup of G if any one of the following holds: (i) A is a non-zero constant on H. (ii) A ( x ) ~ > 0 . 5 and A ( x ) = A ( x -j) for all xeH.
Proof. If possible, let A ( x ) < 0 . 5 for all x • G. Since A is not constant on Ao, there is an x e A0 such that
t = A ( x ) :/: A(e) = r. T h e n either r < t or r > t. If r < t, choose 6 > 0.5 such that t+6>l
and r + 6 < l .
T h e n x6, x~ ~q A but (xx-1)Mt6.6) = e6 • v q A , a contradiction. A g a i n if r > t, we can c h o o s e 6 > 0.5 such that t+6
and r + 6 > l .
T h e n e6 q A , xl q A but
(eX)M~.j) = X~ • V q A , a contradiction. T h e r e f o r e there
exists an x • A o
such that
A ( x ) >- O.5.
Lemma 3.9. A(e) >i 0.5. Proof. If possible, let r = A ( e ) < 0.5. By L e m m a 3.8, there is an x • G such that t = A ( x ) >i 0.5. So r < t. C h o o s e tl > r such that t+tl>l
and r + t l < l .
T h e n x,,, x -t, l q A but (XX--1)M(tl,tl) = e,, • v q A , a contradiction. H e n c e A ( e ) >I 0.5.
Proof. (i) Straightforward. (ii) Let x,,, y,2qA w h e r e x, y • G and t~, t2 • 1 - {0}. If M ( h , t2) > 0 . 5 , then (xy)M(t,.t2)qA. If M(t~, t2) <<-0.5, then (Xy)MU,.,2) • A. Thus in any case (Xy)MU,,,~) • V q A . T h e r e f o r e A is a (q, • v q)-fuzzy s u b g r o u p of G.
Proof of Theorem 3.7. If possible, let t = A ( x ) < 0.5 for s o m e x • A0. Let us c h o o s e tl > 0 such that t + tt < 0.5. T h e n xl q A , eo.5+t, q A but (eX)M(I,O.5+t 0 = Xo.5+tl • V q A ,
a contradiction. T h e r e f o r e A ( x ) >I 0.5 Vx c Ao.
Theorem 3.7. Let A be a ( q , • v q)-fuzzy subgroup of G such that A is not constant on Ao. Then A ( x ) >10.5 for all x • Ao. To prove this t h e o r e m we require the following two lemmas in which the notation of T h e o r e m 3.7 is maintained.
Remark 3.10. Let H and A be as defined in T h e o r e m 3.6. T h e n A is a ( c v q, • v q)-fuzzy subgroup of G if any one of the conditions (i) and (ii) of T h e o r e m 3.6 holds. Also, if A is an ( • v q, • v q)-fuzzy s u b g r o u p of G such that A is not constant on A0, then A(x)~>0.5 for all x •Ao.
Lemma 3.8. There exists an element x • G such that A ( x ) >! 0.5.
The p r o o f can be c o n s t r u c t e d in a m a n n e r similar to those of T h e o r e m s 3.6 and 3.7.
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Example 3.11. Example of an ( • , • v q)-fuzzy subgroup. Consider Klein's 4-group G = {e, a, b, c} defined by the multiplication table
eabc
a b c
a b c
abc b c a c a b If A : G ~ I is defined by
eabc aecb bcea cbae
A(a) =0.7,
A(e)=0.6,
A(a)=0.7,
A(b)=A(c)=O.4.
Then A is an ( • , • v q)-fuzzy subgroup of G. We note that A is not an (o¢ fl)-fuzzy subgroup of G where (o~, fl) = (e, • ) , (q, • v q), ( • v q, • v q). Remark 3.12. Let A be an (• v q, • v q)-fuzzy subgroup of G. Then from Xtl , Yt: • A, it follows that x,,, Y,2 • v q A, implying
(Xy)M(t,,t2) E V q A . So A is an ( • , • v q)-fuzzy subgroup of G. Again, let A be an ( • , •)-fuzzy subgroup of G. Then xt~, y , 2 • A implies (xY)M~t,.t2)•A and thus
Therefore A is an ( • , • v q)-fuzzy subgroup of G. Remark 3.13. A is a fuzzy subgroup of G (in the sense of Rosenfeld) iff Vx, y • G,
which is equivalent to the condition Vtl, t 2 • l - { O } .
It can be readily verified that ~
Remark 3.15. The theorems and examples in this section reveal that the only non-trivial generalisation of a fuzzy subgroup defined by Rosenfeld, obtained in this way, is the concept of an ( • , • v q)-fuzzy subgroup. In the following section some fundamental properties of such fuzzy subgroups are studied.
4. ( • , • v q)-fuzzy subgroups and normal subgroups
Unless otherwise stated in this section by a fuzzy subgroup we shall mean an ( • , • v q)fuzzy subgroup of G.
Proof. Let Xn be a fuzzy subgroup of G. Then x, y • H implies xl, yl • Zn and thus (Xy)M(1,1) • v q XH which yields Xn(xy) > 0, from which we find xy • H. Also, x • H implies X n ( x - ' ) = Zn(x) = 1,
A(xy -l) >1M ( A ( x ) , A ( y ) ) ,
(xy-l)M(tl,t2)EA
A ( c ) =0.9,
Theorem 4.1. For any subset H of G, Zn is a fuzzy subgroup of G iff H is a subgroup of G.
(Xy)M(t,,t2) • V q A .
xt,,ytzEA ~
A ( b ) =0.8,
then A satisfies (1) but since A(b -1) :/:A(b), A is not an ( • , e v q)-fuzzy subgroup of G.
Let A : G---> I be defined by
xtl, Yt2 • A
defined by the multiplication table
(xy-l)M~,,.,2) • V qA
(1)
is a necessary condition for A to be an (6, • v q)-fuzzy subgroup G. However this condition is not sufficient as shown by: Example 3.14. Let G = {a, b, c} be the group
so that x -~ e H. So H is a subgroup of G. Conversely, if H is a subgroup of G, then X, is an (e, •)-fuzzy subgroup of G and so by Remark 3.12, Xrt is an (e, • v q)-fuzzy subgroup of G. Theorem 4.2. Let { A i ; i • J } be any family of fuzzy subgroups of G. Let A = N i ~ A i . Then A is a fuzzy subgroup of O. Proof. If possible, let A be not a fuzzy subgroup of G. Then there exist x, y • G and h, t• • I {0} such that xt~, Yt2 • A but (Xy)M(t~,t2) E v q A ,
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S.K. Bhakat, P. Das / Definition of a fuzzy subgroup
i.e.~
thus
A(xy) < M(t~, t2) and A(xy) + M(q, t2) < 1.
(f(x)f(Y))MO,.t2) • V q B
So,
(since B is a fuzzy subgroup), so that
A(xy) < 0.5.
(2)
Let
(since f is a homomorphism), and thus (Xy)M(,,.,2) • V q f - t ( B ) . Therefore f - l ( B ) is a fuzzy subgroup of G. (ii) Let xi • f ( G ) and
J, = (i • J ; A i ( x y ) >-m(tl, t2)}, J2 = {i • J: A,(xy) + M(t,, t2) > 1 and mi(xy ) < M(t~, tz)}.
(f(a))(xi) >t ti for i = 1, 2.
Then J = J~ tOJ2 and Jl N J2 = 0. If J2 = 0, then
A~(xy) >I M(t~, tz)
Since A has the 'sup property', there is a z~ • f-~(xi) such that
Vi • J,
a(zi) = sup{a(w); w • f - l ( x i ) },
which yields A(xy) >! M(q, tz), a contradiction. If J24:0, then for any i•.I2, A~(xy)+ M(fi, t2) > 1 and Ai(xy) < M(h, tl). Therefore M(tl, t2) > 0.5. Now x, • A implies A(x) >I tl and thus
Ai(x ) >!A(x) >! t~ > M(tl, t2) ~> 0.5
Vi • J.
Vi•J.
Then
Ai(xy)>~0.5
i = 1, 2.
Then zl,,, z2,, • A. Since A is a fuzzy subgroup of G, (Z1Zz)M(t,,t2) • V qA. Now
z,z2•f-~(XlX2)
an d
so
(f(A))(x,x2)>-
A(zlz2). Thus (Xy)M.,,.) • V qf(A).
Similarly,
Ai(y)>0.5
(f(xy))M(t,.t2) • V q B
Vi •J.
For, t' = A~(xy) < 0.5 implies x,., y,,, • Ai, but (xy)t,,~Q---qAi if t' < t " > 0 . 5 , which contradicts that A, is an ( • , • v q)-fuzzy subgroup of G. Thus A ( x y ) ~ 0 . 5 , which contradicts (2). So (xY)Mo,.t~) • v qA. That A(x -~) = A(x) for any x • G is obvious. Therefore A is a fuzzy subgroup of G. Theorem 4.3. Let G and G' be two groups and let f : G ~ G' be a homomorphism. Let A, B be two fuzzy subgroups of G and G', respectively. Then (i) f - l ( B ) is a fuzzy subgroup of G. (ii) If A satisfies the 'sup property', i.e., for any subset T of G there exists a to • T such that
Therefore f ( A ) is a fuzzy subgroup of f(G). Remark 4.4. If A is a fuzzy subgroup of G, then for any t • l - { 0 } , A t = { x e G ; A ( x ) ~ t } may not be a subgroup of G as can be seen from Example 3.11. Definition 4.5. A fuzzy subgroup A of G is said to be a fuzzy normal subgroup of G if
A(xax -1) >~A(a)
Vx, a • G.
(3)
Condition (3) is equivalent to the following conditions:
A(xax -1) = A(a) A(xy) = A ( y x )
Vx, a • G, Vx. y • G.
(4) (5)
then f ( A ) is a fuzzy subgroup o f f ( G ) .
Remark 4.6. For any subset H of G, ;~, is a fuzzy normal subgroup of G iff H is a normal subgroup of G. Also, the intersection of an arbitrary family of fuzzy normal subgroups of G is a fuzzy normal subgroup of G. The proof is straightforward.
Proofl Let x , y • G and t ~ , t 2 • l - { O } . Now x,,, y,. • f - ' ( B ) implies (f(x)),,, (f(Y)),2 • B, and
Remark 4.7. If A is a fuzzy normal subgroup of G in the sense of Mukherjee and Bhattacharya
A(to) = sup{A(t); t • T},
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s.K. Bhakat, P. Das / Definitionof a fuzzy subgroup
[8], then
[since A is a fuzzy normal subgroup]
A([x, y)] >~A(x)
Vx, y E G
= A ( x l l g y l 1)
where Ix, y] denotes the commutator of x, y. But this is not necessarily true if A is a fuzzy normal subgroup of G in the sense of Definition 4.5, as shown by:
Example 4.8. Consider the fuzzy subgroup A defined in Example 3.11. A is, in fact, a fuzzy normal subgroup of G. But A(a) = 0.7 > 0.6 = A([a, b]). Definition 4.9. Let A be a fuzzy subgroup of G. For any x • G, fix (resp. fl~) : G --> I defined by fl~(y) = A(yx-~), [resp..,ix(y) =A(x-ly)]
Yy e G
Theorem 4.10. Let A be a fuzzy normal subgroup of G. Let O%be the set of all fuzzy cosets of A. Then O%is a group if the composition is defined by Vx, y ~ G.
(6)
Let fl : O%--->1 be defined by
= `'iy 2.
(7)
So for any g • G,
A(gx; 1) = A(gx~ 1)
(8)
and
A(gy(')
=
A(gy~l).
Now for any g e G,
A(gyzlx2 l)
=
A(gy~lx; ')
A~,y,(g) : Ax~y~(g) vg ~ G. Thus the composition defined by (6) is well-defined. It is now easy to verify that O%is a group with respect to (6) with Ae as the identity element and Ax , as the inverse of fix for any xeG. We now show that fl is a fuzzy normal subgroup of o%. From A(Ay) I> t2
for q, t2 e I - {0}, it follows that xt,, Yt2 E A, and thus (xy)M
A(A ) = A(x)
= A(x-1)
= A(Ax-,) = A((A~)-'),
VAx ~ o%.
Thus fl is a fuzzy subgroup of O%. Moreover,
[since A is a fuzzy normal subgroup]
Proof. To verify that the composition given by (6) is well-defined, let x l, Xz, Yl, Y2 E G be such that
fly,
Hence
= A(xax -1) = A(a)
Then fl is a fuzzy normal subgroup of O%.
= fl~: and
[since A is a fuzzy normal subgroup].
A ( A ~ A ~ - , ) = A(Axox ,)
A(flx) = A(x) Vx ~ G.
fix,
=A(gy{lx7 l)
A(Ax)/> tl,
is called the fuzzy left (resp. right) coset of G determined by x and A. If A is a fuzzy normal subgroup of G, then fl~ = fl~ for all x E G.
fix" fly = flxy
[replacing g by x-(lg in (9)]
(9)
=A(Aa)
vAa, Ax~o%.
Hence A is a fuzzy normal subgroup of O%. Remark 4.11. With the same notation Theorem 4.10, let 0 : G ---> O%be defined by
O(x) = flx
of
Vx E G.
Then obviously 0 is a homomorphism. However, unlike the case of the fuzzy normal subgroup in the sense of Mukherjee and Bhattacharya [8], Ker O:/:GA = {x E G ; A ( x ) = A ( e ) } , as shown by:
[replacing g by gyfl in (8)]
=A(x71gy~ 1)
Example 4.12. Let G be the group defined in
S.K. Bhakat, P. Das / Definition of a fuzzy subgroup
Example 3.11. Let B : G --~ I be defined by B(e) = 0.6,
B(a) = 0.6,
B(b)=0.7,
B(c)=O.8.
If A~ is normal f o r i = 1 , 2 . . . . is A.
241
,n,
then so
The proof is similar to that of T h e o r e m 4.2
Then B is a fuzzy normal subgroup of G, but ker 0 = {e} +e {e, a} = GA. Theorem 4.13. Let A be a f u z z y subgroup o f ~. Then ,4 ~: G ~ I defined by Al(x) = Al(Ax) is a f u z z y subgroup o f G. I f A i is f u z z y normal, then so is A1.
The proof is straightforward. Remark 4.14. Theorems 3.3, 3.5 of Mukherjee and Bhattacharya [9] are true if A is an ( ~ , ~ v q)-fuzzy subgroup of G. However, consider Lemma 3.7 of the same which states: "Let A be a fuzzy subgroup of a finite group G. Define H = {g ~ G ; A ( g ) = A ( e ) }
and
Theorem 4.17. Let G be the internal product o f two subgroups G~ and G2. Let A , A~, A2 be f u z z y subgroups o f G, Gj and G2, respectively. Let A~, ,4i : G---~ I (i = 1, 2) be defined by
Ai(x) = A(x3 and Ai(x) = A,(x3 if x = x l x z , where xl ~ G1, x2 ~ G2. Then Ai, /[i are f u z z y subgroups o f G f o r i = 1, 2 I r A , A1, A2 are normal, then so are A~, Ai. ^
Proof. Let x, y e G. Let x = x~x:,, where xi, Yi ~ Gi for i = 1, 2. Then
v
Y =Y~Yz
xy = (x l yO(x2 yz).
Now x,, y,, • Ai implies (xi)t, (Y~)c e A, and thus (x~y~)M~,,,,) e v q A [since A is a fuzzy subgroup], which yields (Xy)M(t,t,) E V q~z[i. Therefore A~ is a fuzzy subgroup of G for i = 1, 2. If A~ is fuzzy normal, then it can be readily verified that A~ is also fuzzy normal. The case of ,4~ may be treated similarly.
I;= {x ~ G;A~= de}. Then H and K are subgroups of G. Further H=K." This is not necessarily true for an (6, ~ v q)-fuzzy subgroup of G, as shown by: Example 4.15. Let G be the group defined in Example 3.11. Let A : G ~ I be defined by A ( e ) = A ( a ) = A ( b ) = 0.6 and A ( c ) = 0.7.
Then A is an (c, 6 v q)-fuzzy subgroup of G. However, H = {x ~ G ; A ( x ) = A(e)} = {e, a, b}
is not a subgroup of G. Here ,3.~ = A. Also,
Ao(a)
= 0.6,
A,(b)
= 0.7,
A,(c)
= o.6.
Hence ii¢ +e ,~i,. Therefore H 4= K. Theorem 4.16. Let A~ be a f u z z y subgroup o f a group Gi for i = 1 , 2 . . . , n. Then A : G1× Gz x • .. x G~---->1, defined by A(x, ....
, x,) = M(A,(x,) .....
An(xn) )
where xi e Gi, for i = 1 , 2 . . . . . n, is a f u z z y subgroup o f the group Gi x (32 x • • • z G,.
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