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Abelian fuzzy group and its properties
Fuzzy Sets and Systems 64 (1994) 415-420 North-Holland
415
Abelian fuzzy group and its properties Lu Tu Department of Mathematics, Chang Chun Teachers College, Chang Chum Ji Lin. 13(X)32, China
Gu Wenxiang Department ~ Computer Science, Northeast Normal University, Chang Chun, Ji Lin, 130024, China Received February 1993 Revised September 1993
Abstract: The concepts of Abelian fuzzy group, torsion fuzzy group, torsion-free fuzzy group and mixed fuzzy group arc introduced. The properties of these fuzzy groups and their relations are discussed. Keywords: Fuzzy group; Abclian fuzzy group; torsion fuzzy group; torsion-free fuzzy group; mixed fuzzy group.
I. Introduction
The concept of fuzzy sets and fuzzy set operations was first introduced by Zadeh [6]. Fuzzy groups were introduced by Rosenfeld [5] and subsequently discussed by Anthony and Sherwood [3]. Recently, F.I. Sidky and Mishref [1] gave the definition of A-Abelian fuzzy group and some of its properties. In this paper, we gave the concepts of Abelian fuzzy group, torsion fuzzy group, torsion-free fuzzy group and mixed fuzzy group, and discussed their relations and properties.
2. Preliminaries Definition 2.1 [1, Definition 2.2]. Let G be a group and A a fuzzy set of G. Then A is called a fuzzy subgroup of G iff (i) A ( x y ) >! min(A(x), A ( y ) ) ; x, y E G: (ii) A ( x ) = A ( x ~), x e G; (iii) A l e ) = 1, where e is the identity of G. Definition 2.2 [l, Definition 2.3]. Let A be a fuzzy subset of nonempty set X, t ~ [0, 1]. Then X'A = {X ~ X I A ( x ) ~ t} is called a t-level subset of X under A. Proposition 2.3 [1, Proposition 2.5]. Let A be a f u z z y set o f group G. Then A is a f u z z y subgroup o f G iff every t-level subset G'A o f G under A is a subgroup o f G for every t ~ [0, 1]. Definition 2.4 [1, Definition 4.1 and Proposition 4.2]. Let A be a fuzzy subgroup of group G. If A ( x ) = A(yxy-~), x, y ~ G, then A is called a normal fuzzy subgroup of G. Correspondence to: Dr. T. ku, Department of Mathematics, Chang Chun Teachers Training College, Chang Chum Ji Lin 130032, People's Republic of China. 0165-0114/94/$07.00 © 1994--Elsevier Science B.V. All rights reserved SSDI 0165-0114(94)00077-K
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Proposition 2.5 [1, Proposition 4.2]. Let A be a f u z z y subgroup o f group G. Then A & a normal f u z z y subgroup o f G iff G~A is a normal subgroup o f G for every t • [0, 1]. Definition 2.6 [1, Definition 3.1]. Let A be a fuzzy subgroup of group G, x • G. Then x A is called a left fuzzy coset of A in G, where supA(z),
xz=u,
uEG;
Z~G
xA(u) =
O,
if there is no such z.
Similarly, A x is defined to be right fuzzy coset of A in G.
Proposition 2.7 [1, Proposition 3.3]. Let A be a f u z z y subgroup o f group G. Then x A = y A ( A x = A y ) , x, y E G iff A ( x - X y ) = A ( y - l x ) = 1 ( A ( x y 1) = A ( y x - ~ ) = 1). Proposition 2.8 [1, Proposition 3.5]. Let A be a f u z z y subgroup o f group G, x, y • G, and x A = yA. Then A ( x ) = A ( y ) .
Proposition 2.9 [1, Proposition 3.6]. Let A be a f u z z y subgroup o f group G, x G ~ = yG~A, x, y ~ G G~, t • [0, 1]. Then A ( x ) = A ( y ) .
3. Abelian fuzzy group and its properties Definition 3.1. Let A be a fuzzy subset of group G. If G~t is an Abelian subgroup of G for every t • [0, 1], then A is called an Abelian fuzzy subgroup of G.
Remarks 3.2. By Definition 3.1 it is easy to see that (a) A fuzzy subgroup A of group G is an Abelian fuzzy subgroup iff G is an Abelian group. (b) If A is an Abelian fuzzy subgroup of group G, then G~ is a normal subgroup of G for every t • [0, 1], thus A is a normal fuzzy subgroup of G. (c) A product of finite Abelian fuzzy subgroups of group G is a Abelian fuzzy subgroup of G.
Proof. (a) If A is an Abelian fuzzy group, then G ° = G is an Abelian group of G. Conversely, if G is an Abelian group, then for any G~t (t • [0, 1]), G~ ( c G °) is the subgroup of G °, thus G~ is an Abelian subgroup of G ° . (b) By the theory of groups, the subgroup of Abelian group G is the normal subgroup of G. (c) Let A~, A 2 , . . . , A n be fuzzy Abelian subgroups of G. Then A I A 2 • " • A n is fuzzy subgroup of G. But G is an Abelian group, so A1A2 • • • A,, is an fuzzy Abelian subgroup of G. Definition 3.3. Let A be an Abelian fuzzy subgroup of group G, for all t E [0, 1]. (a) If the elements of G~ are all elements of finite order, then A is called a torsion fuzzy subgroup of G. (b) If the elements of G~ are all elements of infinite order except the identity, then A is called a torsion-free fuzzy subgroup of G. (c) If A is neither torsion fuzzy subgroup of G nor torsion-free fuzzy subgroup of G, then A is called a mixed fuzzy subgroup of G.
Proposition 3.4. Let A be an Abelian f u z z y subgroup o f group G. Then
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(a) A is a torsion (torsion-free, mixed) fuzzy subgroup iff G is a torsion (torsion-free, mixed) Abelian group. (b) A is a mixed fuzzy subgroup of G, then there exists i ~ [0, 1], such that G~A(~G) is a mixed Abelian group, and GJA is a mixed Abelian group for every j ~ [0, i]. Specially, if GIA is a mixed Abelian group, then G~ is a mixed Abelian group for every t ~ [0, 1]. Proof. (a) Since A is a torsion (torsion-free, mixed) fuzzy subgroup of G, G ~ - - G is torsion (torsion-free, mixed) Abelian group. Conversely, if G is a torsion (torsion-free) Abelian group, for any t e [0, 1], G~ ( ~ G ) is a torsion (torsion-free) Abelian group. Thus A is a torsion (torsion-free) fuzzy Abelian group. A is a mixed fuzzy Abelian group when G is a mixed Abelian group. Definition 3.5. Let A be a fuzzy set of nonempty set X, Y ~ X and for every x c X Air(x) Then
.~ A ( x ) ,
lundefined,
A IY is called
x ~ Y,
x ~ Y,
(Aiy)O(x) : ~ A ( x ) , I
[0,
x E Y
x ~ Y.
a restriction of A in Y, and (Air)" is called a zero-extension of
A I~.
Definition 3.6. Let A be a fuzzy subgroup of group G and B a fuzzy subgroup of subgroup H of G. For every t e [0, 1], if (a) H~ is a subgroup of G~, then B is called a subgroup of A. (b) H~ is a normal subgroup of G~, then B is called a normal subgroup of A.
Proposition 3.7. l f A is a fuzzy subgroup of group G and H is a subgroup of G, then (a) A I, is a fuzzy subgroup of H and (AI.)" is a fuzzy subgroup of G; (b) Both Z l , and ( A I , ) ° are the subgroups of A; (c) When G is an Abelian group, both z l . and (AI.)" are normal subgroups of Z.
Proof. (a) For any t ~ [0, 1], H'AI,, = {x E H I A ( x ) >1t} ~ G~A, and e e H ~ l , , # & (here e is the identity of group G). For any x , y EH'AI, ,, have xy ~ e H by H is subgroup of G, and Al,,(xy 1) = A(xy i) >1min{A(x), a ( y ) } = min{AIH (x), A I , (y)} = t. Therefore xy ~ E H'AI,,, that is to say, H~I, , is a subgroup of H. Thus AIH is a fuzzy subgroup of H by Proposition 2.5. (b) For any t e [0, 1] both H~I,, and G'CAI,,),,are subgroups of G~. (c) If G is an Abelian group, then for any t E [0, 1], both Oil,, and GIAI,,),, are normal subgroups of G~.
Proposition 3.8. Let A be an Abelian fuzzy subgroup of group G and F a set of elements of finite order of G (in this case F is the maximal torsion group of G). If B = AlE, then B is a torsion fuzzy subgroup of F and B is a normal subgroup of A and for every t E [0, 1], UB is the maximal torsion subgroup of G~A. Proof. For t ~ [0, 1], F~ ( o F ) is a torsion group, hence B is a torsion fuzzy subgroup of F. Also G~a - F~ ~ G - F, thus F~ is the maximal torsion group of G'A. It is easy, by Proposition 3.8 and theory of groups, that we give the following proposition.
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Proposition 3.9. I r A is an Abelian f u z z y subgroup o f group G, F is the maximal torsion group o f G and B = A]F, then both G'A/F~ and GtA/G~,, are torsion-free groups.
Proposition 3.10. Let A be an Abefian f u z z y subgroup o f group G having the sup property, F a maximal torsion subgroup o f G and B = A[F. Then for x, y ~ G, when y - i x • F , xF~ = yF~, XG'A = yG~ for every t ~ [0, t*].
A ( y - l x ) > - t * • [O, 1] iff
Proof. If A ( y - l x ) = t*, then y - i x • Gta for every t • [0, t*]. Also y - i x e F, hence y-~x e F'B ~ G'+ Thus xF~ = yF'B,
XG'A = yG'A
for every t • [0, t*].
Conversely, for every t • [0, t*], if x F ~ = y F ' B and XG'A=yG'A, then y-~x E F ' B C F for every t* t E [0, t*]. Especially, y-~x E FBt* ~ GA. Hence B ( y ~x) = A ( y ~x) >>-t*. In the proof of Proposition 3.10, x and y are exactly symmetrical, hence if x-~y replaces y-~x, Proposition 3.10 is still valid.
Remarks 3.11. Let A be an Abelian fuzzy subgroup of group G having the sup property, F a maximal torsion subgroup of G and B = A]F. Then (a) For x, y • G, y-~x ~ F and A ( y ~x) = 1 iff XFtB =yF'B, XG'A =yG'A for t e [0, 1]. (b) For every x E F, xF'8 =x-~FtB; for every x • G, XG'A = x 1G'A.
Proof. (a) In Proposition 3.10, let t* = 1. (b) For any x • F, we always have A ( x x 1 ) = A ( e ) = l , XG'A = X
and by (a), xF~=x-~F'~.
Similarly,
I GtA.
Proposition 3.12. Let A be a f u z z y subgroup o f group G. Then for x • G, x A = A iff A ( x ) = 1. Proof. If for
x ~ G, xA = A,
then
1 = A(e) = x A ( e ) = A(x-~e) = A ( x - ~ ) = A(x). Conversely, if x • G and A ( x ) = 1, then x A = A by Equation (5) of Section 3 in [1].
Proposition 3.13. Let A be a f u z z y subgroup o f Abelian group G, F the maximal torsion subgroup o f G, B = AI~ and A ( F ) c G 1. Then for any tl, t2 • [0, 1], tx <~t2, we always have G'b = G~ and G ~ / G ~ is a
normal subgroup o f G~/G~. 12 t2 Proof. Clearly, the first conclusion is right. We prove the second conclusion: For any xG'~ ~ GA/GB,
yGtb
•
$1 fl GA/GB, we have
(yG'h)-~(xGt~)(yG'h) = (y-lxy)G'~ = xG'~.
Proposition 3.14. Let A be an Abelian f u z z y subgroup o f group G and F the maximal group o f G. I f for any ul, u2 • F, we always have A(u~lu2) = A ( u ~ l u l ) = 1, then A ( u 0 = A(u2) = 1. Proof, For any
Ul, U2
E F~ we have A(u~lu2) = A ( u ~ l u l ) = 1. Since e c F, hence
A(e l u ) = A ( u ) = l
for a n y u EF.
Thus A ( u , ) = A(u2) = A ( u ) = 1.
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Definition 3.15. Let A be a fuzzy s u b g r o u p of g r o u p G, H a s u b g r o u p of G and B a fuzzy s u b g r o u p of H. If for every t • [0, 1], G~A/H~ always m a k e s sense, then A/B=
U
t
t
GA/HR
re[0,1]
is called the factor g r o u p of A relative to B. W h e n for every t • [0, 1] we always have H~ -- H~,* then we define ~ / a s the fuzzy set of A / B such that
.~l(xH~) = A (x).
Remarks 3.16. Let ~ be a fuzzy s u b g r o u p of A / B . If A is a fuzzy s u b g r o u p of A b e l i a n g r o u p G, then is an A b e l i a n fuzzy s u b g r o u p of A / B . Proof. F o r every t • [0, 1], there exists e • H ~ # ~b (e is the identity of G), so eH~ • ~/~t/e # &, and for any xH'b, yH~ • SgtAm, here q, t2 • [t, 1], t~ ~< t2, then (xH'~)(yH~) = xyH'~ and ~l((xy)H'h) = A ( x y ) >i min{A(x), A ( y ) } ~> t, that is, (xH'h)(yH~) E ~g'Am. A g a i n , for any t • [0, 1], xH'8 • Sg'A/B (k • [t, 1]),
(eH~)(xH~) = xH~, since, for any ehl • eH~ (hi E H1), xh~ • xH~ (hk • H~), we have
(ehl)(xhk) = x (h, hk) • xH~. H e n c e eH~ is the identity of Sg'~/B since H ~ = H~. For every xH~ • ~I'A/B (k • [t, 1]), we have x-~H~ • ~'A/B, such that
(x 'H~)(xH~) = e n ~ = en~. T h e r e f o r e ~ / i s a fuzzy s u b g r o u p of A / B .
Definition 3.17. Let A be a fuzzy s u b g r o u p of g r o u p G, H a s u b g r o u p of G and B a fuzzy s u b g r o u p of H. If there is a m a p p i n g q~ f r o m G into H such that (a) q~ is a h o m o m o r p h i s m of G~ o n t o H ~ for every t • [0, 1], then A with B are h o m o m o r p h i c u n d e r q~
q~ and d e n o t e it by A - B (or A - B). (b) ~ is an i s o m o r p h i s m of G~ o n t o H ~ for every t • [0, 1], then A with B are isomorphic u n d e r q~ ~o
and d e n o t e it by A --~ B (or A ~ B).
Proposition 3.18. Let A be an Abelian torsion f u z z y subgroup o f group G having the sup property, F a maximal torsion subgroup o f G and B=AIF. Then A with a torsion-free f u z z y group C are homomorphic and G ~ / F ~ ~ H~c for any t e [0, 1]. Especially, when A ( F ) ~ G1A, ~l~/n ~-- C~, for t • [0, 1] and ~l ~- C, where ~1 is defined by Definition 3.15. * The conditions of fuzzy set ~/in Definition 3.15 have been intensified as compared with the definition of A' in Corollary 4.5 in [1]. A' in Corollary 4.5 in [1] is not a mapping in ordinary circumstances; therefore it is not a fuzzy subgroup. Example. In Example 3.2 of [1], let H = {a, b}. Then H is a normal subgroup of G, and aH= bH. But A'(aH) = A(a) - 1, A'(bH)=A(b)=tl, A'(aH)>A'(bH). Therefore A' is not a mapping. If we require that for any x,y E H~ (t E[0, 1]), B(x) = B(y), then B(x) = B(y) = I by e e H~. It explains why we require H~j = H i for every t ~ [0, 1].
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Proof. Since F is the maximal torsion group of G, G / F is a torsion-free group. By the basic h o m o m o r p h i s m t h e o r e m of groups, we can suppose that G with group H are h o m o m o r p h i c under ¢, ker ¢ = F and G / F ~- H. For any h • H, let
C(h) : sup (A(x) I q~(x) : hi. XEG
Then C is a fuzzy set of H and for any h, k • H,
C(hk) = sup {A(x) [ q~(x) = hk} x~G
/> sup {A(x,x2) [ x,x2 = x, ¢(x) = hk} X I,X2 ~ G {A(xlx2)[XlX
sup
2 =x,
~(Xl)=h,
~p(x2)=k}
xI,X2~G
~>min/su p {A(x,) i ~O(Xl)=h}, sup {A(x2) i q~(Xe)=k}} kxleG
x2EG
= min{C(h), C(k)}. H e n c e C is a fuzzy subgroup of H. Thus C is a torsion-free fuzzy subgroup of H. For every t • [0, 1], x • G~. By G / F ~-H, there is h • H such that ¢(x) = h and C(h) >~t, hence h • Hb. Thus ¢ is a mapping from G~t onto HE.. For y • H ' c C H , let C ( y ) = to (>~t). Since A has the sup property, there is u • G such that
A(u) = C(y) = sup { A ( x ) [ ~p(x) = y}. x~G
Thus u • G~ and ¢ ( u ) = y . q~(y) E H b and
Therefore q~ is a surjection from G~ onto H~. For x , y • G'A, q~(X),
C(q~(xy)) = C ( ~ ( x ) ¢ ( y ) ) >1min(C(q~(x)), C(q~(y))) >! t. g,
H e n c e ¢ is a h o m o m o r p h i c surjection from G~ onto H ~ and A - C. t t For x • G~, ~(x) = e. Then x • F, x • F fq G'A __ -- FBt and GA/FB -~ H'o By Propositions 3.13, 3.14 and 3.16 and Definition 3.17, t
S ~ A / B ~-- C t t t
and
sg ~ C.
References [1] F.I. Sidky and M.A. Mishref, Fuzzy cosets and cyclic and Abelian fuzzy subgroups, Fuzzy Sets and Systems 43 (1991) 243-250. [2] P.S. Das~ Fuzzy groups and level subgroups, J. Math. AnaL Appl. 84 (1981) 264-269. [3] J.M. Anthony and H. Sherwood, Fuzzy groups redefined, J. Math. A n a l AppL 69 (1979) 123-130. [4] N.P. Mukherjee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inf. Sci. 34 (1984) 225-239. [5] A. Rosenfeld, Fuzzy groups, J. Math. AnaL Appl. 35 (1971) 512-517. [6] L.A. Zadeh, Fuzzy sets, Inf. and Contr. 8 (1965) 338-353. [7] H. Sherwood, Products of fuzzy groups, Fuzzy sets and Systems 11 (1983) 79-89.
415
Abelian fuzzy group and its properties Lu Tu Department of Mathematics, Chang Chun Teachers College, Chang Chum Ji Lin. 13(X)32, China
Gu Wenxiang Department ~ Computer Science, Northeast Normal University, Chang Chun, Ji Lin, 130024, China Received February 1993 Revised September 1993
Abstract: The concepts of Abelian fuzzy group, torsion fuzzy group, torsion-free fuzzy group and mixed fuzzy group arc introduced. The properties of these fuzzy groups and their relations are discussed. Keywords: Fuzzy group; Abclian fuzzy group; torsion fuzzy group; torsion-free fuzzy group; mixed fuzzy group.
I. Introduction
The concept of fuzzy sets and fuzzy set operations was first introduced by Zadeh [6]. Fuzzy groups were introduced by Rosenfeld [5] and subsequently discussed by Anthony and Sherwood [3]. Recently, F.I. Sidky and Mishref [1] gave the definition of A-Abelian fuzzy group and some of its properties. In this paper, we gave the concepts of Abelian fuzzy group, torsion fuzzy group, torsion-free fuzzy group and mixed fuzzy group, and discussed their relations and properties.
2. Preliminaries Definition 2.1 [1, Definition 2.2]. Let G be a group and A a fuzzy set of G. Then A is called a fuzzy subgroup of G iff (i) A ( x y ) >! min(A(x), A ( y ) ) ; x, y E G: (ii) A ( x ) = A ( x ~), x e G; (iii) A l e ) = 1, where e is the identity of G. Definition 2.2 [l, Definition 2.3]. Let A be a fuzzy subset of nonempty set X, t ~ [0, 1]. Then X'A = {X ~ X I A ( x ) ~ t} is called a t-level subset of X under A. Proposition 2.3 [1, Proposition 2.5]. Let A be a f u z z y set o f group G. Then A is a f u z z y subgroup o f G iff every t-level subset G'A o f G under A is a subgroup o f G for every t ~ [0, 1]. Definition 2.4 [1, Definition 4.1 and Proposition 4.2]. Let A be a fuzzy subgroup of group G. If A ( x ) = A(yxy-~), x, y ~ G, then A is called a normal fuzzy subgroup of G. Correspondence to: Dr. T. ku, Department of Mathematics, Chang Chun Teachers Training College, Chang Chum Ji Lin 130032, People's Republic of China. 0165-0114/94/$07.00 © 1994--Elsevier Science B.V. All rights reserved SSDI 0165-0114(94)00077-K
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Proposition 2.5 [1, Proposition 4.2]. Let A be a f u z z y subgroup o f group G. Then A & a normal f u z z y subgroup o f G iff G~A is a normal subgroup o f G for every t • [0, 1]. Definition 2.6 [1, Definition 3.1]. Let A be a fuzzy subgroup of group G, x • G. Then x A is called a left fuzzy coset of A in G, where supA(z),
xz=u,
uEG;
Z~G
xA(u) =
O,
if there is no such z.
Similarly, A x is defined to be right fuzzy coset of A in G.
Proposition 2.7 [1, Proposition 3.3]. Let A be a f u z z y subgroup o f group G. Then x A = y A ( A x = A y ) , x, y E G iff A ( x - X y ) = A ( y - l x ) = 1 ( A ( x y 1) = A ( y x - ~ ) = 1). Proposition 2.8 [1, Proposition 3.5]. Let A be a f u z z y subgroup o f group G, x, y • G, and x A = yA. Then A ( x ) = A ( y ) .
Proposition 2.9 [1, Proposition 3.6]. Let A be a f u z z y subgroup o f group G, x G ~ = yG~A, x, y ~ G G~, t • [0, 1]. Then A ( x ) = A ( y ) .
3. Abelian fuzzy group and its properties Definition 3.1. Let A be a fuzzy subset of group G. If G~t is an Abelian subgroup of G for every t • [0, 1], then A is called an Abelian fuzzy subgroup of G.
Remarks 3.2. By Definition 3.1 it is easy to see that (a) A fuzzy subgroup A of group G is an Abelian fuzzy subgroup iff G is an Abelian group. (b) If A is an Abelian fuzzy subgroup of group G, then G~ is a normal subgroup of G for every t • [0, 1], thus A is a normal fuzzy subgroup of G. (c) A product of finite Abelian fuzzy subgroups of group G is a Abelian fuzzy subgroup of G.
Proof. (a) If A is an Abelian fuzzy group, then G ° = G is an Abelian group of G. Conversely, if G is an Abelian group, then for any G~t (t • [0, 1]), G~ ( c G °) is the subgroup of G °, thus G~ is an Abelian subgroup of G ° . (b) By the theory of groups, the subgroup of Abelian group G is the normal subgroup of G. (c) Let A~, A 2 , . . . , A n be fuzzy Abelian subgroups of G. Then A I A 2 • " • A n is fuzzy subgroup of G. But G is an Abelian group, so A1A2 • • • A,, is an fuzzy Abelian subgroup of G. Definition 3.3. Let A be an Abelian fuzzy subgroup of group G, for all t E [0, 1]. (a) If the elements of G~ are all elements of finite order, then A is called a torsion fuzzy subgroup of G. (b) If the elements of G~ are all elements of infinite order except the identity, then A is called a torsion-free fuzzy subgroup of G. (c) If A is neither torsion fuzzy subgroup of G nor torsion-free fuzzy subgroup of G, then A is called a mixed fuzzy subgroup of G.
Proposition 3.4. Let A be an Abelian f u z z y subgroup o f group G. Then
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(a) A is a torsion (torsion-free, mixed) fuzzy subgroup iff G is a torsion (torsion-free, mixed) Abelian group. (b) A is a mixed fuzzy subgroup of G, then there exists i ~ [0, 1], such that G~A(~G) is a mixed Abelian group, and GJA is a mixed Abelian group for every j ~ [0, i]. Specially, if GIA is a mixed Abelian group, then G~ is a mixed Abelian group for every t ~ [0, 1]. Proof. (a) Since A is a torsion (torsion-free, mixed) fuzzy subgroup of G, G ~ - - G is torsion (torsion-free, mixed) Abelian group. Conversely, if G is a torsion (torsion-free) Abelian group, for any t e [0, 1], G~ ( ~ G ) is a torsion (torsion-free) Abelian group. Thus A is a torsion (torsion-free) fuzzy Abelian group. A is a mixed fuzzy Abelian group when G is a mixed Abelian group. Definition 3.5. Let A be a fuzzy set of nonempty set X, Y ~ X and for every x c X Air(x) Then
.~ A ( x ) ,
lundefined,
A IY is called
x ~ Y,
x ~ Y,
(Aiy)O(x) : ~ A ( x ) , I
[0,
x E Y
x ~ Y.
a restriction of A in Y, and (Air)" is called a zero-extension of
A I~.
Definition 3.6. Let A be a fuzzy subgroup of group G and B a fuzzy subgroup of subgroup H of G. For every t e [0, 1], if (a) H~ is a subgroup of G~, then B is called a subgroup of A. (b) H~ is a normal subgroup of G~, then B is called a normal subgroup of A.
Proposition 3.7. l f A is a fuzzy subgroup of group G and H is a subgroup of G, then (a) A I, is a fuzzy subgroup of H and (AI.)" is a fuzzy subgroup of G; (b) Both Z l , and ( A I , ) ° are the subgroups of A; (c) When G is an Abelian group, both z l . and (AI.)" are normal subgroups of Z.
Proof. (a) For any t ~ [0, 1], H'AI,, = {x E H I A ( x ) >1t} ~ G~A, and e e H ~ l , , # & (here e is the identity of group G). For any x , y EH'AI, ,, have xy ~ e H by H is subgroup of G, and Al,,(xy 1) = A(xy i) >1min{A(x), a ( y ) } = min{AIH (x), A I , (y)} = t. Therefore xy ~ E H'AI,,, that is to say, H~I, , is a subgroup of H. Thus AIH is a fuzzy subgroup of H by Proposition 2.5. (b) For any t e [0, 1] both H~I,, and G'CAI,,),,are subgroups of G~. (c) If G is an Abelian group, then for any t E [0, 1], both Oil,, and GIAI,,),, are normal subgroups of G~.
Proposition 3.8. Let A be an Abelian fuzzy subgroup of group G and F a set of elements of finite order of G (in this case F is the maximal torsion group of G). If B = AlE, then B is a torsion fuzzy subgroup of F and B is a normal subgroup of A and for every t E [0, 1], UB is the maximal torsion subgroup of G~A. Proof. For t ~ [0, 1], F~ ( o F ) is a torsion group, hence B is a torsion fuzzy subgroup of F. Also G~a - F~ ~ G - F, thus F~ is the maximal torsion group of G'A. It is easy, by Proposition 3.8 and theory of groups, that we give the following proposition.
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Lu Tu, Gu Wenxiang / Abelian fuzzy group and its properties
Proposition 3.9. I r A is an Abelian f u z z y subgroup o f group G, F is the maximal torsion group o f G and B = A]F, then both G'A/F~ and GtA/G~,, are torsion-free groups.
Proposition 3.10. Let A be an Abefian f u z z y subgroup o f group G having the sup property, F a maximal torsion subgroup o f G and B = A[F. Then for x, y ~ G, when y - i x • F , xF~ = yF~, XG'A = yG~ for every t ~ [0, t*].
A ( y - l x ) > - t * • [O, 1] iff
Proof. If A ( y - l x ) = t*, then y - i x • Gta for every t • [0, t*]. Also y - i x e F, hence y-~x e F'B ~ G'+ Thus xF~ = yF'B,
XG'A = yG'A
for every t • [0, t*].
Conversely, for every t • [0, t*], if x F ~ = y F ' B and XG'A=yG'A, then y-~x E F ' B C F for every t* t E [0, t*]. Especially, y-~x E FBt* ~ GA. Hence B ( y ~x) = A ( y ~x) >>-t*. In the proof of Proposition 3.10, x and y are exactly symmetrical, hence if x-~y replaces y-~x, Proposition 3.10 is still valid.
Remarks 3.11. Let A be an Abelian fuzzy subgroup of group G having the sup property, F a maximal torsion subgroup of G and B = A]F. Then (a) For x, y • G, y-~x ~ F and A ( y ~x) = 1 iff XFtB =yF'B, XG'A =yG'A for t e [0, 1]. (b) For every x E F, xF'8 =x-~FtB; for every x • G, XG'A = x 1G'A.
Proof. (a) In Proposition 3.10, let t* = 1. (b) For any x • F, we always have A ( x x 1 ) = A ( e ) = l , XG'A = X
and by (a), xF~=x-~F'~.
Similarly,
I GtA.
Proposition 3.12. Let A be a f u z z y subgroup o f group G. Then for x • G, x A = A iff A ( x ) = 1. Proof. If for
x ~ G, xA = A,
then
1 = A(e) = x A ( e ) = A(x-~e) = A ( x - ~ ) = A(x). Conversely, if x • G and A ( x ) = 1, then x A = A by Equation (5) of Section 3 in [1].
Proposition 3.13. Let A be a f u z z y subgroup o f Abelian group G, F the maximal torsion subgroup o f G, B = AI~ and A ( F ) c G 1. Then for any tl, t2 • [0, 1], tx <~t2, we always have G'b = G~ and G ~ / G ~ is a
normal subgroup o f G~/G~. 12 t2 Proof. Clearly, the first conclusion is right. We prove the second conclusion: For any xG'~ ~ GA/GB,
yGtb
•
$1 fl GA/GB, we have
(yG'h)-~(xGt~)(yG'h) = (y-lxy)G'~ = xG'~.
Proposition 3.14. Let A be an Abelian f u z z y subgroup o f group G and F the maximal group o f G. I f for any ul, u2 • F, we always have A(u~lu2) = A ( u ~ l u l ) = 1, then A ( u 0 = A(u2) = 1. Proof, For any
Ul, U2
E F~ we have A(u~lu2) = A ( u ~ l u l ) = 1. Since e c F, hence
A(e l u ) = A ( u ) = l
for a n y u EF.
Thus A ( u , ) = A(u2) = A ( u ) = 1.
Lu Tu, Gu Wenxiang / Abelian fuzzy group and its properties
419
Definition 3.15. Let A be a fuzzy s u b g r o u p of g r o u p G, H a s u b g r o u p of G and B a fuzzy s u b g r o u p of H. If for every t • [0, 1], G~A/H~ always m a k e s sense, then A/B=
U
t
t
GA/HR
re[0,1]
is called the factor g r o u p of A relative to B. W h e n for every t • [0, 1] we always have H~ -- H~,* then we define ~ / a s the fuzzy set of A / B such that
.~l(xH~) = A (x).
Remarks 3.16. Let ~ be a fuzzy s u b g r o u p of A / B . If A is a fuzzy s u b g r o u p of A b e l i a n g r o u p G, then is an A b e l i a n fuzzy s u b g r o u p of A / B . Proof. F o r every t • [0, 1], there exists e • H ~ # ~b (e is the identity of G), so eH~ • ~/~t/e # &, and for any xH'b, yH~ • SgtAm, here q, t2 • [t, 1], t~ ~< t2, then (xH'~)(yH~) = xyH'~ and ~l((xy)H'h) = A ( x y ) >i min{A(x), A ( y ) } ~> t, that is, (xH'h)(yH~) E ~g'Am. A g a i n , for any t • [0, 1], xH'8 • Sg'A/B (k • [t, 1]),
(eH~)(xH~) = xH~, since, for any ehl • eH~ (hi E H1), xh~ • xH~ (hk • H~), we have
(ehl)(xhk) = x (h, hk) • xH~. H e n c e eH~ is the identity of Sg'~/B since H ~ = H~. For every xH~ • ~I'A/B (k • [t, 1]), we have x-~H~ • ~'A/B, such that
(x 'H~)(xH~) = e n ~ = en~. T h e r e f o r e ~ / i s a fuzzy s u b g r o u p of A / B .
Definition 3.17. Let A be a fuzzy s u b g r o u p of g r o u p G, H a s u b g r o u p of G and B a fuzzy s u b g r o u p of H. If there is a m a p p i n g q~ f r o m G into H such that (a) q~ is a h o m o m o r p h i s m of G~ o n t o H ~ for every t • [0, 1], then A with B are h o m o m o r p h i c u n d e r q~
q~ and d e n o t e it by A - B (or A - B). (b) ~ is an i s o m o r p h i s m of G~ o n t o H ~ for every t • [0, 1], then A with B are isomorphic u n d e r q~ ~o
and d e n o t e it by A --~ B (or A ~ B).
Proposition 3.18. Let A be an Abelian torsion f u z z y subgroup o f group G having the sup property, F a maximal torsion subgroup o f G and B=AIF. Then A with a torsion-free f u z z y group C are homomorphic and G ~ / F ~ ~ H~c for any t e [0, 1]. Especially, when A ( F ) ~ G1A, ~l~/n ~-- C~, for t • [0, 1] and ~l ~- C, where ~1 is defined by Definition 3.15. * The conditions of fuzzy set ~/in Definition 3.15 have been intensified as compared with the definition of A' in Corollary 4.5 in [1]. A' in Corollary 4.5 in [1] is not a mapping in ordinary circumstances; therefore it is not a fuzzy subgroup. Example. In Example 3.2 of [1], let H = {a, b}. Then H is a normal subgroup of G, and aH= bH. But A'(aH) = A(a) - 1, A'(bH)=A(b)=tl, A'(aH)>A'(bH). Therefore A' is not a mapping. If we require that for any x,y E H~ (t E[0, 1]), B(x) = B(y), then B(x) = B(y) = I by e e H~. It explains why we require H~j = H i for every t ~ [0, 1].
420
Lu Tu, Gu Wenxiang / Abelian fuzzy group and its"properties
Proof. Since F is the maximal torsion group of G, G / F is a torsion-free group. By the basic h o m o m o r p h i s m t h e o r e m of groups, we can suppose that G with group H are h o m o m o r p h i c under ¢, ker ¢ = F and G / F ~- H. For any h • H, let
C(h) : sup (A(x) I q~(x) : hi. XEG
Then C is a fuzzy set of H and for any h, k • H,
C(hk) = sup {A(x) [ q~(x) = hk} x~G
/> sup {A(x,x2) [ x,x2 = x, ¢(x) = hk} X I,X2 ~ G {A(xlx2)[XlX
sup
2 =x,
~(Xl)=h,
~p(x2)=k}
xI,X2~G
~>min/su p {A(x,) i ~O(Xl)=h}, sup {A(x2) i q~(Xe)=k}} kxleG
x2EG
= min{C(h), C(k)}. H e n c e C is a fuzzy subgroup of H. Thus C is a torsion-free fuzzy subgroup of H. For every t • [0, 1], x • G~. By G / F ~-H, there is h • H such that ¢(x) = h and C(h) >~t, hence h • Hb. Thus ¢ is a mapping from G~t onto HE.. For y • H ' c C H , let C ( y ) = to (>~t). Since A has the sup property, there is u • G such that
A(u) = C(y) = sup { A ( x ) [ ~p(x) = y}. x~G
Thus u • G~ and ¢ ( u ) = y . q~(y) E H b and
Therefore q~ is a surjection from G~ onto H~. For x , y • G'A, q~(X),
C(q~(xy)) = C ( ~ ( x ) ¢ ( y ) ) >1min(C(q~(x)), C(q~(y))) >! t. g,
H e n c e ¢ is a h o m o m o r p h i c surjection from G~ onto H ~ and A - C. t t For x • G~, ~(x) = e. Then x • F, x • F fq G'A __ -- FBt and GA/FB -~ H'o By Propositions 3.13, 3.14 and 3.16 and Definition 3.17, t
S ~ A / B ~-- C t t t
and
sg ~ C.
References [1] F.I. Sidky and M.A. Mishref, Fuzzy cosets and cyclic and Abelian fuzzy subgroups, Fuzzy Sets and Systems 43 (1991) 243-250. [2] P.S. Das~ Fuzzy groups and level subgroups, J. Math. AnaL Appl. 84 (1981) 264-269. [3] J.M. Anthony and H. Sherwood, Fuzzy groups redefined, J. Math. A n a l AppL 69 (1979) 123-130. [4] N.P. Mukherjee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inf. Sci. 34 (1984) 225-239. [5] A. Rosenfeld, Fuzzy groups, J. Math. AnaL Appl. 35 (1971) 512-517. [6] L.A. Zadeh, Fuzzy sets, Inf. and Contr. 8 (1965) 338-353. [7] H. Sherwood, Products of fuzzy groups, Fuzzy sets and Systems 11 (1983) 79-89.