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Fuzzy normal subgroups and fuzzy quotients
Fuzzy Sets and Systems 46 (1992) 121-132 North-Holland
121
Fuzzy normal subgroups and fuzzy quotients I.J. Kumar, Yadav
P.K. Saxena and Pratibha
Scientific Analysis Group, DRDO, Ministry of Defence, Metcalfe House, Delhi 110 054, India
Received May 1989 Revised October 1990 Abstract: In case of groups, a normal subgroup N of a group
G can be defined with three equivalent approaches, namely (i) as a kernel of some group homomorphism, (ii) as a subgroup commuting with every element of G, and (iii) as the subgroup N closed with respect to conjugates of elements of N. In case of generalizations of this concept to fuzzy sets, the elementwise third approach is found more suitable and hence is widely adopted by many authors. In this paper we further study this concept of fuzzy normal subgroups and introduce fuzzy quotients called 0c-fuzzyquotient groups for all ole[0, 1]. Two different definitions of these ol-fuzzy quotients have been given which are shown to lead to isomorphic structures. We also study the direct products of fuzzy (normal) subgroups (min-product as per the terminology of Sherwood (1983)) and the problem of writing a fuzzy (normal) subgroup of a direct product of some groups as a direct product of certain fuzzy (normal) subgroups, has been discussed in detail. Keywords: Fuzzy subgroup; fuzzy normal subgroup; 0l-fuzzy quotients; isomorphism; kernels; projections; f-invariance; direct-product; internal direct product; or-cuts.
1. Introduction The concept of fuzzy sets and various operations on it were first introduced by Z a d e h in [23]. Since then, fuzzy sets have been applied to diverse fields. But the study of fuzzy algebraic structures was started with the introduction of the concept of fuzzy subgroups in the pioneering p a p e r of Rosenfeld [16]. Many other concepts such as 'fuzzy bi-ideals', 'fuzzy semiprime ideals', 'fuzzy fields' and 'fuzzy vector spaces' were developed in [8, 9, 12] but these concepts did not have much scope for interesting and useful further studies because of too m a n y
constraints in their generalizations. However, the theory of fuzzy subgroups developed in both respects, theory as well as its practical applicability. The generalization of subgroups to the fuzzy subgroups of Rosenfeld was modified by Negoita and Ralescu [13] by replacing the membership set [0, 1] by appropriate lattice structures. A n t h o n y and Sherwood further redefined fuzzy groups in terms of a general ' T - n o r m ' which replaced the 'min' of its previous definitions [1]. They also applied this concept in pattern recognition problems [2]. Many other papers on fuzzy subgroups have also a p p e a r e d which generalize various concepts of group theory such as 'normal subgroups', 'quotient groups' [10, 3, 20, 21], 'direct products of groups' [3, 18], and also discuss h o m o m o r p h i s m and isomorphisms between these fuzzy structures. Even the concepts of M-subgroups, M-factorgroups, M - h o m o m o r p h i s m s and M-isomorphisms for groups with operators [6,128-137] have been generalized by G u in [4]. In the present p a p e r we further study fuzzy normal subgroups, define ol-fuzzy quotients, discuss isomorphisms between some specific structures and characterize the fuzzy subgroups of a direct product of certain groups in terms of fuzzy subgroups of these groups. The second section of this p a p e r gives preliminary definitions and results required later. More details of these concepts can be seen in the papers referred to. In Section 3, we review the definition of fuzzy normal subgroups and prove a theorem to characterize fuzzy normal subgroups of certain groups. Section 4 consists mainly of the definitions of or-fuzzy quotients of a fuzzy normal subgroup /~ in G in two different ways, namely (i) as a fuzzy subset of an ordinary set of ordinary cosets of the or-cut ol, and (ii) as a fuzzy subset of the set of o~-fuzzy cosets/~a of/~ in G. It is shown that both of these approaches lead to isomorphic fuzzy quotients. We also discuss
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Lagrange's theorem of groups in the context of fuzzy normal subgroups and fuzzy cosets. In Section 5, we discuss many observations about the direct product of fuzzy subgroups, the relationship between the fuzzy subgroups of product of groups and its projections. In case of finite groups, a subgroup A of the direct product G1 × G2 × " " × G, of groups can be written as the product A1 × A2 × • • • × A , of projections Ai of A on Gi, where orders of Gi's are pair-wise co-prime [5]. We prove a similar result for fuzzy subgroups, which is stronger than Theorem 4.1 of [18] and gives the counterpart of the 'fundamental theorem of finite Abelian groups' in case of fuzzy subgroups. We have also given another proof of Theorem 4.4 of [18] by using o~-cuts and avoiding the concepts of measures, aalgebras, Hausdorff Maximal Principle, etc., which have widely been used in the proof.
2. Preliminaries Definition 2.1. For a group G, a fuzzy subset/~ on G is called a fuzzy subgroup (shortly written as FSG) if (i) t~(xy) >- min(~(x), ~(y)) for all x, y in G, (ii) /~(e)= 1 where e is the identity element of G, and (iii) /~(x -1) I>/~(x) for all x in G. In the original definition of fuzzy subgroups in [16], the condition (ii) was absent. In [13] Negoita and Relascu replaced [0, 1] by an appropriate lattice structure but condition (ii) was not explicitly mentioned. However, some of their propositions seem to assume it. In [1] Anthony and Sherwood introduced arbitrary 'T-norms' in condition (i) in place of 'min' but they did not assume (ii). In several other papers also, such as [3, 10, 17, 20], condition (ii) has not been assumed. In [2] Anthony and Sherwood modified their own convention of [1] and included (ii) with the observation that a very natural condition '/~(e) > 0' together with (i) and (iii) lead to the satisfaction of all these three conditions if all characteristic values are normalized by /~(e) [2, Lemma 1]. We also follow this convention and use Definition 2.1 as the definition of FSG throughout this paper.
Observe that there is equality in (iii) in fact. We may also mention that even in definition of a 'T-norm', the boundary conditions (as it is called by Sessa [17]) differ from paper to paper [1, 2, 3, 17, 21], though we shall be dealing with 'min' which satisfies all the conventions of T-norm and is the strongest T-norm in the sense of [181. If /~ is a fuzzy set on a set S and if f is a function defined on S then ~t induces a fuzzy set #I on f ( S ) defined by /~I(y) =
sup
{/~(x)}
for all y ~ f ( S )
(1)
xef-l(y)
and is called the image of ~ under f. Similarly, if v is a fuzzy set on f ( S ) , then a fuzzy set on S can be defined through v I l(x) = v ( f ( x ) ) = v of(x)
for all x e S.
(2)
where o is the composition of mappings, v I ' is called the inverse image of v. The following result was shown by Rosenfeld in Proposition 5.8 of [16] under an additional hypotheses of 'supremum property' for the function f. This result was proved by Anthony and Sherwood [1, Proposition 4] in general for fuzzy subgroups with continuous T-norms. The T-norm 'min' being continuous, the following proposition holds.
Proposition 2.2. If f is a homomorphism from a group G onto a group G', then the homomorphic image #I of a fuzzy subgroup I~ on G is a fuzzy subgroup on G' and also that the homomorphic preimage v~-' of a fuzzy subgroup v on G' is a fuzzy subgroup of G. For a function f from a set S onto the set f ( S ) , a fuzzy set /~ of S is called f-invariant if /~(x) = # ( y ) whenever f ( x ) = f ( y ) for all x, y in S. The following theorem gives the correspondence between fuzzy subgroups of a group G with fuzzy subgroups of a homomorphic image of G [16].
Theorem 2.3. If f : G ---~f ( G ) is a group homomorphism, then there is a one-to-one correspondence between f-inoariant fuzzy subgroups of G and fuzzy subgroups o f f ( G ) . If f:G---~ G' is a group isomorphism (onto)
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Other preliminary definitions can be found in [7] and the papers referred to. As a general remark, we mention that by 'fuzzy sets' we would mean 'fuzzy subsets' only and we will not differentiate between these two.
groups treating them as subgroups commuting with every element of G. Meng [3] and Yu [22] also investigated FNSGs without condition (ii) of Definition 2.1. Recently, in [21] Wu studied normal T-fuzzy subgroups where 'min' of Definition 2.1 has been replaced by any general T-norm T. He also characterized normal T-fuzzy subgroups in terms of fuzzy congruences on a group. In the study of fuzzy algebraic structures, c~-cuts play an important role in the visualization of these concepts. Thus we have the following observation which has been shown by Wu [20] with a slightly different statement due to the absence of '/~(e) = 1' in his definition. We sketch the proof which is simpler than Propositions 1.1, 2.1, 3.1 of [20] particularly in the converse part.
3. Fuzzy normal subgroups
Proposition 3.2. For a group G, a f u z z y subset ~t
and /~ is a fuzzy subgroup of G, then ~ is f-invariant. Thus there is a one-to-one correspondence between FSGs of G and FSGs of G'. This correspondence ~ ~--~#Y is called an isomorphism between fuzzy subgroups/u and ~J. More precisely:
Definition 2.4. Two fuzzy subgroups (G, ~) and (G',/~') are said to be isomorphic if there exists a mapping f : G - - ~ G ' such that f is a group isomorphism (onto) and t h a t / u ' o f = ~.
We start with the natural generalization of the normalcy condition of ordinary groups [20] in the following:
Definition 3.1. For a group G, a fuzzy set on G is called a fuzzy normal subgroup (FNSG in short) of G if it is a fuzzy subgroup satisfying I~(xyx -~) ~ I~(Y)
for all x, y in G.
(3)
Some remarks concerning this definition are in order. In [20] Wu studied fuzzy normal subgroups but the condition (ii) of Definition 2.1 was absent. An FNSG /~ on G in our sense was called normal quasi-group by Wu. Liu in [10] adopted an alternate approach of defining fuzzy subgroups by treating a fuzzy subset as an ordinary set of fuzzy point-sets xx. He defined an operation 'sup-min' between two fuzzy subsets (and hence between a fuzzy set and a fuzzy point-set) of the group G. Condition (3) of the above definition was replaced by another equivalent condition I~(xy) = l~(yx)
for all x, y in G
(3')
for defining fuzzy invariant subgroups (as he called it). Again condition (ii) of Definition 2.1 was missing. Liu showed that a fuzzy invariant subgroup commutes with every fuzzy subsets on G, hence with every fuzzy point-set xx. Thus Liu's approach was in fact a sort of generalization of this concept of normal subgroups of
on G is a f u z z y normal subgroup o f G if and only if or-cuts o:~ = {x ~ G: I ~ ( x ) ~ o:} are n o n e m p t y and f o r m normal subgroups o f G, f o r all o: ~ [0, 1].
Proof. Let ~u be a fuzzy normal subgroup of G and 0~ in [0, 1] be an arbitrary number. Then the identity element e of G is always in the or-cut ol,. For any elements x, y in cr,, /~(x, y - l ) >1 min(t~(x),/~(y)) /> min(o~, or) = 0c and therefore xy -~ ~ o(,. Further, ~t(xyx -~) >1 /~(y)/> ol and hence x y x -~ ~ ol,. Thus or, is normal subgroup of G. Conversely, if all or-cuts are nonempty and normal subgroups of G, then in particular I v :~ 0 which guarantees for # ( e ) = 1. Let x, y be any two elements of G and cr =min(/~(x), /~(y)). Then xc0l~, y c t r , and therefore x y ~ o : ~ proving that /u(xy)/> min(#(x),/~(y)). Further, x -~ and y x y -1 belong to the or-cut ol, which proves that/~(x -l)/>/~(x) and kt(yxy -~) >~ kt(x) for all x, y in G. This completes the proof. We now give the following theorem.
Theorem 3.3. I f G is a group having a chain {e} =No c N 1 c N z c .
• • c NL = G
(4)
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of normal subgroups of G and L is the length of the maximal such chain, then a fuzzy normal subgroup of G is a step function from G ~ [0, 1] having at most L + 1 steps.
Abelian and it has a subgroup H of order p giving the chain {e} = N0 c H c N2 = G of length L = 2 and there is no chain with length more than 2, and hence the result follows.
Proof. Let It be a fuzzy normal subgroup of G having the chain (4) of normal subgroups of G. For trivial subgroup No = {e}, It(e) = 1 gives one step of the characteristic function It. Corresponding to the next member N1 of the chain (4), either It(x) = It(e) = 1 for all x e N~ or there exists an element xo c N 1 - {e} such that It(x0) 4: It(e). In first case It restricted to N~ has only one step. In the other case the claim is that It is constant throughout N ~ - {e} and I t ( x ) = I t ( x 0 ) < i t ( e ) = 1 for all x in N l - - { e } . This is because if It(x)~it(Xo) for x4:xo in N l - { e } then there exists a number o~ c [0, 1] in between It(x0) and It(x), a~ 4: It(xo) such that only one of the two elements x, x0 will belong to c~-cut o~, and the other will not. Thus we get the chain
Corollary 3.6. A group G of order pq, where p, q are primes, has fuzzy normal subgroups with at most three steps.
{e} = N o c c~, cN~ c . . - c N L
= G
of normal subgroups having length L + 1 which contradicts our hypotheses about the length L. Thus It is constant on N I - {e} having values different from It(e) = 1. In this case It restricted to N~ has exactly two steps in its graph. Proceeding with a similar argument we see that the consideration of each member of the chain (4) gives rise to at most one step in the graph of It. Thus we can have at most L + 1 steps, which proves the result.
Proof. There is only one chain {e} c N c G of normal subgroups where N is the cyclic subgroup of G of order max(p, q), hence the result.
The following corollary characterizes FNSGs of dihedral group D2p.
all
Corollary 3.7. For a dihedral group Dap = {xry~: X 2 = e, yP = e, yx = x y p-l, where 1 <~r <- 2, l<~s<-p} of order p (a prime) and a fuzzy normal subgroup It of G, 1 = #(e)/> It(y) . . . . .
It(xy) . . . . .
It(yp-l)
It(xyP-~).
Proof. The group D2p has only one proper normal subgroup
N = {e, y, y2 . . . . .
yp-l}
of order p. Hence It is constant on N - {e} and there are at most three steps in It which proves that # is constant with lower values (possibly) for the remaining set G - N.
Corollary 3.5. If G is a group of order p2 (p is a prime) and It is a f u z z y normal subgroup of G, then It has at most three different values in [0, 1].
Before we discuss images and preimages of fuzzy normal subgroups under a group homomorphism, we observe that Proposition 3.2 also holds for ordinary fuzzy subgroups of a group G. Further, as in T h e o r e m 3.3, the chain of maximum possible length of subgroups of the group G gives the number of possible different steps in the fuzzy subgroup It and helps in finding types of possible fuzzy subgroups of finite groups as in the corollaries above. Consider the group homomorphism f : G ~ G ' (onto). Wu showed in [20] that Proposition 2.2 holds even for fuzzy normal subgroups /~ of G. Though It(e) was not necessarily 1 there, the proof of the following theorem remains almost the same and hence is omitted. We shall need this result in the next section.
Proof. If o ( G ) = pZ, p a prime, then G is always
Proposition 3.8. For a fuzzy normal subgroup It
We now give the following specific cases as corollaries. Corollary 3.4 (Proposition 5.10 of [16]). Any fuzzy subgroup It of a cyclic group G of order p (a prime) is a constant function except possibly at the identity element of G where It is 1. Proof. Obviously, It is fuzzy normal and there is only the trivial chain {e} c G possible in place of the chain in (4) of normal subgroups of G.
l.J. Kumar et al. / Fuzzy normal subgroups and fuzzy quotients
of G and a homomorphism f : G --* G ' (onto), t~1 is also a fuzzy normal subgroup of G'. Further, if v is a fuzzy normal subgroup of G' then vf ~is also a fuzzy normal subgroup on G. The following theorem extends T h e o r e m 2.3 of Section 2 for fuzzy normal subgroups. The proof, following the case of fuzzy subgroups [16], does not use the normalcy condition and hence is omitted.
Theorem 3.9. There is one-to-one correspondence between f-invariant fuzzy normal subgroups of G and fuzzy normal subgroups of G ' = f ( G ) , where f:G---~G' = f ( G ) is a group homomorphism from group G to G'.
4. Fuzzy quotients In groups, normal subgroups are also defined as kernels of group homomorphisms. In the other words, there is a one-one correspondence between normal subgroups of a group G and group homomorphisms on G. Thus the three concepts namely 'kernels', 'homomorphisms' and 'quotients' are mutually dependent in terms of their definitions. For perceiving FNSGs through homomorphisms, we have to visualize concepts like 'fuzzy kernels' of some (fuzzy?) homomorphisms and talk of 'fuzzy-quotients' as obtained by slicing out the 'fuzzy kernels' from the domain set. If the fuzzy quotients are to be treated as ordinary groups then it becomes difficult to associate membership values to the kernel of group homomorphism to define fuzzy kernel in a natural way. If we think of the fuzzy quotients as fuzzy sets, then the question becomes how a homomorphism from a group to a fuzzy subgroup should look so that the 'kernel' of this homomorphism may become a F N S G of the domain set (group). Due to such inherent problems in visualizing these three concepts together, probably no one has defined FNSGs through homomorphisms and fuzzy kernels. However, once the FNSGs have been defined as in Definition 3.1 it becomes natural to investigate the generalization of the concept of quotient groups to the fuzzy sets even without explicitly talking of fuzzy kernels,
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homomorphisms and isomorphism theorems, etc. In this section we introduce and study u-fuzzy quotients not as just ordinary quotient groups as in [10,20] but as FNSGs of some quotient groups. Before giving the formal definition, a few more comments on this concept are in order. In [20] Wu considered S/S~ where ~ is FNSG of the group S and S, is the ~(e)-cut of S and meant it to be the fuzzy quotient group. He further visualized members of S/Su as fuzzy cosets. In [10] Liu denoted this group S by X, the FNSG by A and called it a fuzzy quotient group without any further discussion on it. Though the concept also appears in some other referred papers, a slightly different approach to this idea of fuzzy quotients appeared in [21] for T-fuzzy subgroups (T to be a regular T-norm) where an attempt was made to study it in the context of group isomorphism theorems. There Wu considered a T-fuzzy subgroup g on G (called a T-fuzzy group) as the basic structure (instead of G) and defined a normal T-fuzzy subgroup v of ~, suitably, to form a T-fuzzy subgroup ~ / v of the group s(l~)/S(v) where S(~) and S(v) are supports of /~ and v, respectively. However, getting a nontrivial T-fuzzy quotient (the author does not use this name) would put a very strong restriction on v namely that v must be zero at some points of the group where # is nonzero (otherwise S ( l O / S ( v ) = { e } ) . This idea of quotients also appears in [4] in the form of M-fuzzy quotients (M does not stand for 'min') but it is entirely in a different direction of groups with operators as in [6]. For any number c~ e [0, 1] and a fuzzy normal subgroup of a group G, the o~-cut cv, is a normal subgroup of G and hence we can talk of the quotient group G/ol,. In view of Proposition 3.8, ~o is a fuzzy normal subgroup on G/ol~ where O:G---, G/o6, is the natural homomorphism. We give the following definition. Definition 4.1. For a fuzzy normal subgroup /~ of G and o~ e [0, 1], the pair (G/o6,, go) is called an o~-fuzzy quotient of tile group G. The above definition of ol-fuzzy quotients looks a bit unnatural since elements of G/ol. are ordinary cosets of the normal subgroup having some characteristic values associated with these
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126
cosets through /~o. In what follows, we like to define the a~-fuzzy quotients as the set S of a~-fuzzy cosets and give structure to S by defining the o p e r a t i o n b e t w e e n two ol-fuzzy cosets.
Definition 4.2. L e t /~ be a fuzzy s u b g r o u p of a g r o u p G. F o r any a ~ G, a fuzzy subset #a of G is called an oc-fuzzy right coset of/~ in G if
# a ( x ) = min(~u(xa), o:)
for all x in G.
Similarly, a fuzzy subset a # of G is called an o~-fuzzy left coset o f / t in G if
al~(x) = m i n ( # ( a x ) , o~)
t h e o r e m shows that o ( G / # ~ ) = o(G/o6, ) having the s a m e n u m b e r of elements.
Theorem 4.5. I f I~ is a f u z z y normal subgroup oJ a group G and N = o¢~ is the ol-cut o f I~ for some a" ~ [0, 1], then Na = N b if and only if l~a = lib for all elements a, b in G.
Proof. L e t a, b be arbitrary e l e m e n t s of G and ~a =/~b. T h e n
t,a(b -1) = / , b ( b -1) = m i n ( / ~ ( b b - l ) , a0 = min(1, o~) = ol.
for all x in G.
where q~2 = e, ~03 = e and q),p2 = ,pq~. C o n s i d e r a fuzzy s u b g r o u p on G given by
But Ita(b -1) = min(/~(ab-1), a 0 which shows that l~(ab -1) >i ol and hence a b - l e N proving Na = Nb. Conversely, let Na = N b for any a, b in G and let x be any e l e m e n t in G. T h e n Nax = N b x . O b s e r v e that l*(ax)< ol and lt(bx)>i ol implies that Nbx = N 4: Nax which is not true. Similarly /t(ax) ~ o~ and l*(bx) < a~ implies that Nax = N ~ N b x which is false again. H e n c e either both ~(ax) and # ( b x ) are g r e a t e r than or equal to a~, or both are less than o~. In the first case,
/~(e) = ~(q~) = 1,
Ira(x) = min(/~(ax), a;) = a"
~,(v~) = t , ( ¢ )
and
T h e following e x a m p l e shows that for a fuzzy s u b g r o u p / , of G, a~-fuzzy left cosets n e e d not be equal to c o r r e s p o n d i n g ol-fuzzy right cosets.
Example 4.3. C o n s i d e r G = $3, the s y m m e t r i c group of o r d e r 3. T h e n G can be e x p r e s s e d as G = {e, 4, % ~p2, q~g,, q~p2}
= ~ , ( q , ~ ) = t , ( ~ ,2) = 0.5.
T h e n it can be seen that for cr = 1,
t*b(x) = min(/u(bx), a~) = c~,
~Pt~ = (0.5, 0.5, 0.5, 1.0, 1.0, 0.5),
where as in the o t h e r case,
/~p = (0.5, 0.5, 0.5, 1.0, 0.5, 1.0). ( H e r e the 6-tuple d e n o t e s the characteristic values of e l e m e n t s of G in order. W e shall use such notation even afterwards for r e p r e s e n t i n g fuzzy subsets on finite sets.) H e n c e ~p/~ 4:/~ap. This h a p p e n s because g is not a fuzzy n o r m a l s u b g r o u p of G = Ss. H o w e v e r :
Proposition 4.4. I f i~ is a f u z z y normal subgroup o f G and x is any element o f G, then the ol-fuzzy right coset I~x is same as the ol-fuzzy left coset xl~.
Proof. Let /~ be a fuzzy n o r m a l s u b g r o u p of G, a~G, and a ~ [ 0 , 1 ] . T h e n for any x ~ G , I~(xa) = / u ( a x ) and so t~a(x) = m i n ( ~ ( x a ) , a 0 = m i n ( / t ( a x ) , a~) = al*(x). Consider the set G/l*~ = {/~a: a ~ G} of all a~-fuzzy right cosets of /, in G. T h e following
#a(x) = I*(ax) < ol and t*b(x) = l*(bx) < ol. In this case, since Na = Nb, we have a = nb for some n e N and so
~b(x) = ~ ( b x ) = l~(n-lax) /> min(/~(n-1), t~(ax))
= I~(ax) = tta(x) = ~ ( n b x ) 1> min(/~(n), I~(bx)) = I~(bx) = ~b(x). Thus I~a(x) = I~b(x) for all x e G and the p r o o f is complete. W e give a structure to G / # ~ = {/~a: a ~ G} by defining the o p e r a t i o n * b e t w e e n two a~-fuzzy right cosets as
t*a * t*b = f~ab.
(5)
In the following e x a m p l e we see that if # is a fuzzy s u b g r o u p on G (not n o r m a l ) , then the
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Example 4.6. Consider G and g as in Example 4.3. It can be seen that for tr = 1,
Observe that the mapping O: G---)G/go` defined is a well defined map and that for x, y in G, O(xy) = gxy = Izx * gy = O(x) * O(y). Hence it is an onto group homomorphism. Thus we get
#e = (1.0, 1.0, 0.5, 0.5, 0.5, 0.5),
g * ( g a ) = sup {g(x)} =
sup
pax--.a
~J(x)= O(a)
operation * defined above in (5) is not well defined.
by 0 ( x ) = p x
gq) = (1.0, 1.0, 0.5, 0.5, 0.5, 0.5), =
gap = (0.5, 0.5, 0.5, 1.0, 0.5, 1.0),
sup
(g(x)}
{g(x)},
xe~ ~(.a)
g(PW = (0.5, 0.5, 1.0, 0.5, 1.0, 0.5),
which shows that g . = g 0 and hence by Proposition 3.8, g* is a fuzzy normal subgroup of G/go`. In the following theorem we see that (G/#o`, g*) is just another way of defining a<-fuzzy quotients.
gap2 = (0.5, 0.5, 1.0, 0,5, 1.0, 0.5), g ~ l p 2~-- (0.5, 0.5, 0.5, 1.0, 0.5, 1.0).
Here gap = gq)ap2, gap2= gq)ap. But gap * gape = ge 4: #q)ap2, #q)ap =
Theorem 4.8. The o<-fuzzy quotient ( G / o:,, #o),
g~ap2~ap = gap~.
However, the operation * is well defined in case # is a fuzzy normal subgroup. Proposition 4.7. The operation * defined in (5) on G/#o`, where g is a fuzzy normal subgroup of G and o
x~A
= mm(g(abx), tr)
where
= m m ( g ( a l b x ) , o<)
A = {x e G : x e 0-1(ga)}
= m m ( g ( x a l b ) , o<)
= {x e G: px = ga}
= mm(g(bxaO, tr)
= {x e G: Nx = Na}
= mm(g(b,xaO, o<)
= {x e G: O(x) = Na}
= m m ( g ( x a l b O, tr)
= {x e G : x e O-t(Na)}.
= m m ( g ( a , b , x ) , o<)
Thus #6of ( N a ) = g ° ( N a ) for all N a e G / c h , , and hence g ° o f = gO as desired.
= galbl(x). Thus #a * gb = gab = #a, b ~ and hence * is well defined. The set (G/#o,, *) becomes a group with ge working as the identity element and #a -~ as the inverse of the o<-fuzzy coset #a. We can define g*, a fuzzy set on G/#o` as =
sup {#(x)}.
/aa=/~x
Proof. Consider the mapping f:G/o6,--> G/#~ defined by f ( N a ) = #a where N = ct, and a e G. Then it is clear by T h e o r e m 4.5 that f is a one-one onto homomorphism. It remains to show that g 6 o f = g o . For that, consider any element ga of G/go`. Then
g ° o f ( N a ) = g°(ga) = sup {g(x)}
(#a * #b )(x ) = gab(x)
#*(#a)
where O: G---> G / ol, is the natural epimorphism, is isomorphic to the fuzzy normal subgroup (G/go`, g°), where O:G---~G/go` is the homomorphism defined by O(x ) = gx.
(6)
Another equivalent way of defining tr-fuzzy quotients is through the equivalence classes of the equivalence relation - on the group G defined as a - b iff g(ab -l) >! o<.
(7)
The equivalence classes of - coincide with the cosets of c~-cut tr,.
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In case of ordinary normal subgroups of a group G, Lagrange's T h e o r e m states that the group G is the disjoint union of cosets of N in G. In other words the set {Na: a • G } of cosets partitions G. To see a similar result in the fuzzy situation we give the following definition.
We prove this result for a more general situation of the fuzzy partition of the group G by the or-fuzzy cosets of F N S G / ~ of G. Theorem 4.10. If l~ is a fuzzy normal subgroup of a group G, then the class o%= {~a: a • G} of
ol-fuzzy cosets of t~ in G is a fuzzy partition of G. Definition 4.9. The class ~ = {#~} of fuzzy sets /~ of a set S is called a fuzzy partition of S if for all o r • [ 0 , 1], either the class o%,={a,~} of 0l-cuts o f / ~ has no non-empty m e m b e r (set) or it partitions S in ordinary set theoretic sense. The additional condition 'either ~ has no non-empty members' in the above definition is included for the following reason. If we do not assume this then we will have to assume that at least o n e / t i must be 1 somewhere in S (for finite cases) or that 1 should be a limiting point of {~i(x): i arbitrary, x • S} (for infinite cases). For otherwise one can find some /3 • [0, 1] such that all fiu's are empty. It is not important whether G is covered and partitioned by each and every o%o, o~ • [0, 1], but what matters more is that if one member of ,~o~ is nonempty it should be a partition. Naturally the number of classes of these partitions will depend on o~. In Definition 2.1 of [20] Wu defined fuzzy cosets of a fuzzy set /~. In Proposition 2.7 he characterized the Z-cuts of fuzzy cosets as ordinary cosets of )t-cuts of the fuzzy subgroup. Since/~(e) is not necessarily 1 in his definition of a FNSG, the statement in this Proposition 2.7 has some ambiguity in the sense that for any ~. between /~(e) and 1, /~a (/l, in his notation) will be empty and the product a/~a (a)~, of course) will be meaningless. However, for nonambiguous cases this proposition holds. Thus with the conventions of fuzzy cosets and FNSGs of Wu, it follows from this Proposition 2.7 [20] that for a FNSG ~ of a group G, the class {a/~: a e G} of fuzzy cosets is a fuzzy partition of G. This is what Lagrange's T h e o r e m looks like in the fuzzy context. Observe that, even for cr = 1, the Definition 2.1 [20] of 'fuzzy cosets' differs from our Definition 4.2 of 1-fuzzy cosets and the result similar to Proposition 2.7 [20] does not hold here. This is clear from Example 4.3, where l.~p = {llY2, I ~ 2} @ {I/3, (/)113} = 1j," 1/)= {e, qb}. ,p. Thus even for 1-fuzzy cosets the result of Lagrange's T h e o r e m does not follow as in [20].
Proof. Consider any number fi • [0, 1] and the set ~t~ = {/3,a: a • G} of/3-cuts o'f or-fuzzy cosets /~a of the given fuzzy normal subgroup ~ of G. If /3 is bigger than or, t h e n / 3 , , = q~ (the empty set) for all a • G, since gta(x) = min(/~(xa), ol) ~< cr for all x • G. Thus oft~ has only null set 0 as a member and we are done. So assume that/3 ~< ol. In this case, for any x • G, /~c-l(x) = min(/~(xx-l), or) = min(/~(e), or) = cr/>/3 and x • / 3 ~ , which means that G c_ U.~G/3.a. It remains to show that for any a, b in G, the sets 13.. and /3.b are either identical or disjoint. For this let x •/3.. ¢q/3ub be any element in the nonempty intersection, if possible, and let y be any arbitrary element of/3... Then
I~a(x) = min(/_t(xa), or) I> 13, lib(x) -- min(~(xb), or)/>/3 and
l~a(y) = min(~t(ay), or)/>/3. Therefore
#b(y) = min(/t(by), o~) = min(i.t(bxx-la-lay), o~) i> min(/,(bx), l~(x-la-l), g(ay), ol) --min(/t(bx), I.t(ax), l*(ay), or)
>~/3. Thus Y •flub and so /3.a~-/3.h- Similarly, choosing y in /3.b instead of /3u. above, we get fl.b~_/3.a. This shows that /3~a and /3ub are identical if they have even a single element in common. Thus ~ is a partition of G and the proof is complete. We have seen that in defining fuzzy quotients,
(G/o:,, I~') of/~ in G, It' is defined a s / t ° where O: G---~ G/o: u is the natural epimorphism. If G
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is a finite group, then for g • G / o : , , s u p / , o ,~){/~(x)} = i t o ( ~ ) is the value of It at some point x0 of A = {x e G: O(x)) = g } and hence I t ' ( g ) ~ c~ unless g = ~. T h u s tx-cut a~u, = {~}, i.e. It' is 1 only at ~ and for all of the other points of G / t r u, It' is below tr. T h o u g h in taking s u p r e m u m over infinite sets o f n u m b e r s the 'sup' m a y not actually be attained at s o m e points of the set, yet the a b o v e m e n t i o n e d observation holds for any g r o u p as is clear from:
case of o: = 1 this set S has only one m e m b e r as is clear from the following proposition which has been shown in [3]. Proposition 4.12. I f G is a group and It a f u z z y normal subgroup on G, then f o r N = 1~,, It is a constant function on Na f o r all a c G. T h e a b o v e proposition also characterizes fuzzy normal subgroups o f a g r o u p G. T h e g r o u p G can be a r r a n g e d as G=NUNaUNbU...,
Proposition 4.11. I f G is a group and /~ is a f u z z y n o r m a l s u b g r o u p o f G, then cr,, = {~} in the f u z z y quotient ( G / cr~,, it').
the disjoint union of cosets of N = 1, and tt has to be constant on each o n e of these cosets.
Proof. In view of the r e m a r k s a b o v e , it is sufficient to prove the result for nonfinite groups. T h e r e f o r e assume that G is a nonfinite g r o u p and g0 (go c G) is an e l e m e n t of G/ol~ such that It'(go) = 13/> or. D e n o t i n g ol, by N, we observe that
5. Fuzzy direct products
/3 = I t ' ( g . ) =
sup {/_t(x)}.
xeNgo
(8)
For any given arbitrary n o n z e r o positive real n u m b e r , say e, there exists s o m e n o o N such that It(nogo) > / 3 - e I> o~ - •. But then It(g,,) = It(ncT'nog,,)
In [18] S h e r w o o d defined the T ' - d i r e c t p r o d u c t of T-fuzzy subgroups It~ of s o m e groups G~, where T, T ' are s o m e T - n o r m s on [0, 1] × [0, 1]. W e consider only ' m i n ' T - n o r m and by a direct p r o d u c t of fuzzy subgroups we shall m e a n the ' m i n ' direct product. W e recall the definition. Definition 5.1. For each i = 1, 2 . . . . . n let It~ be a fuzzy s u b g r o u p o f Gi. T h e n the direct p r o d u c t It = Itl × It2 x • • • ×/~n, the function defined on G1 x G2 x • • • x Gn is given by
/> min(it(rqT'), It(nogo)) >/min(ol, It(nogo)).
It(x~, x2 . . . . . = It1 x t~2 x
If It(nogo) >i oc then min(~r, It(nogo)) = ,x > , x - e; otherwise min(ol, It(nogo)) = It(nogo) > ol - e.
x.) • • • x Itn(x~,
x2 . . . . .
- - m i n ( i t ~ ( x 0 , It2(x2) . . . . .
x~)
~(x~)).
(9)
As in Corollary 3.4 of [18] for fuzzy subgroups, the following proposition for fuzzy n o r m a l subgroups was shown by M e n g in [3] for n = 2. T h e result for any finite n also follows easily. W e just state the result.
In any case It(go) > ol - ¢. Since • is an arbitrary n u m b e r and can be c h o s e n as small as we please, we must have It(go) ~> ol which shows that go = e. Thus o l , . = {~} and h e n c e It' is 1 only at the identity e l e m e n t of the g r o u p G / o 6, and for all the other elements it is below or.
Proposition 5.2. S u p p o s e Iti subgroup o f Gi f o r each i = 1, direct p r o d u c t It = itl X l~2 X f u z z y n o r m a l subgroup o f G1
It is clear from (8) that for finding It' of an element g of G / c r , , one has to look for 'sup' out of a certain set S of values of It. F o r the specific
In case of n o n f u z z y s u b g r o u p S of a direct p r o d u c t of groups G ~ × G e × . . - x G , , the projection Si of S on each Gi is a s u b g r o u p of Gi.
is a f u z z y n o r m a l 2 . . . . . n. Then the . . . x itn is also a x G2 x • • • × G,,.
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130
To see the similar result subgroups we recall [18]:
in case
of fuzzy
Definition 5.3. L e t / , be a fuzzy subgroup on the cross product G~ × G2 x • • • × G~ of groups G~. The projection [Ai of # o n Gi, for all i = 1, 2 , . . . , n, is defined by ~i(gi) = ~(el,
e l - l , gi, e i + l , . • • , en).
e2 . . . . .
(10) The proof of the following proposition straight-forward and hence omitted.
is
f u z z y s u b g r o u p It o n G = G1 × G2 × • • • × G~ can be written as the direct p r o d u c t I~ = I~ × I~2 × • • • × I~n w h e r e each [.~i is the p r o j e c t i o n o f It o n
G,. Proof. Let k l , k z . . . . . kn be the orders of groups G 1 , G 2 . . . . , Gn respectively and /~ be a fuzzy subgroup on G = G , x ( 3 2 × . . - x G n . Then k i ' s a r e pairwise co-prime and hence ki is co-prime with any product of kj's different from k~. In view of Proposition 5.5, it suffices to show that for any element (am, a2, . . . , an) of G, t,ti(ai) >1/,t(al, a 2 , . . .
Proposition 5.4. F o r all i = 1, 2 . . . . .
n, the p r o j e c t i o n I~i o f a f u z z y n o r m a l s u b g r o u p t~ o f a direct p r o d u c t G1 × G2 × • • • × Gn o n Gi is also a f u z z y n o r m a l s u b g r o u p o f Gi.
Proposition 5.5. L e t t~ be a f u z z y s u b g r o u p o f the direct p r o d u c t G1 × Gz × • • • × Gn a n d let I~i be the p r o j e c t i o n o f It o n Gi f o r each i = 1 , 2 . . . . . n. T h e n l~ >~ tz~ × ltz × . . . × t~n. Proof. For
any
element
(al, a2,...
,an)
of
(11)
To avoid complicating the notation, we prove (11) for i = 2, say. From (10) we have ~2(a2) = ~(el,
Even in case of ordinary subgroups, a subgroup S of the direct product G1 × (32 × • - - × Go is not equal to the direct product SI×S2×...×Sn of the projections of S in general (Example 4.2 of [18]) but S is the super set of S I × S z × ' " × S ~ . This is also the case with fuzzy subgroups. The p r o o f of the following result is contained in the p r o o f of T h e o r e m 4.1 of [18], we just outline it here.
, an)
for all i = 1, 2, . . . , n.
en).
a 2 , e3 . . . . .
Since k~ and k2 are co-prime, there exist integers p, q such that p k t + qk2 = 1. Then/~2(a2) can be written as #2(az) = l~(a pk', a pk'+qk2, e3 . . . . . = / t ( a P l k', a 2pkl, e3 . . . .
en)
, en)
=/u(al, a2, e3 . . . .
, en) pk'
/> ~ ( a j , a2, e3 . . . . .
e.).
Consider now numbers k l , k2 and k 3 which are co-prime and hence there exist integers P l, q l such that p l k l k 2 + q~k3 = 1 and therefore ~ ( a l , a2, e3 . . . . .
en)
= I~( ap'k'kz+qlk~, aP'ktkz+qlk~ , a 3qlk~ , e4, . . . , en)
GI × Gz × " " × Gn,
= t~(a q'k~, a q'k~, a q~k3, e4 . . . . .
I~(a,,
=/~(al,
a 2 , a 3 , e4 . . . . .
en) q'k3
~(al,
a2, a3, e4 . . . . .
en).
a2 . . . . .
a,,)
= ~ ( ( a l , e2, e 3 , . . . , "'"
(el, ez,. ..,
e n ) ( e , , a2, e3 . . . .
Thus
en-1, an))
i> min(/~l(aO, u2(a2) . . . . = ]'~1 X /"~2 X • " " X ~ n ( a l , Hence
, e,,)
en)
, I~n(a,)) a 2 , . . . , an)
# i> ~ 1 × /A2 × " " " X ~n"
The following result shows when the equality /~ = / ~ × #2 x • • • ×/~n holds in the above proposition, and is stronger than T h e o r e m 4.1 of [18].
Theorem 5.6. I f G1, G2 . . . . . Gn are f i n i t e g r o u p s h a v i n g p a i r w i s e c o - p r i m e orders, then a
/~2(a2)/>/~(al, a2, a3, e, . . . . .
e,).
Considering the co-primeness of k~, k z , k3 and k 4 w e can show as above that /~2(a2) i>/~(al, a2, a3, a4, e5 . . . .
, e,).
Repeating the same argument we see that /~2(a2)/>/~(al, a2, • . • , an). Similarly, we can show (11). In the above case we can m a k e the following
LJ. Kumar et al. / Fuzzy normal subgroups and fuzzy quotients
more general observation. Let g ~ G be any element having non-identity arguments exactly at j places (say) m~, m2 . . . . . mj ( j < n ) and also assume that I is a positive integer ( l < n ) different from all m~'s. If g' is the element of G having all arguments as that of g except the l-th which has been replaced by any x~4:e~ of G~, then ~ ( g ) > - t ~ ( g ' ) . This follows as in the proof above by considering co-primeness of k .... km~, .... k,,, and k~ and replacing pk (i) am, by a,,, ~.k. . .""k . . i n + q k t for all i = 1, 2 . . . . . j,
(ii) ez by x qk', in g where p, q are such that pkm,k,~ 2 • • • kin, + qk~ = 1.
Corollary 5.7 (Theorem 4.1 of [18]). E v e r y f u z z y s u b g r o u p u n d e r M (rain) o f a finite n u m b e r o f cyclic g r o u p s o f distinct p r i m e p o w e r orders can be written as a direct p r o d u c t o f f u z z y subgroups o f those cyclic groups. It is shown by Example 4.2 in [18] that Theorem 5.6 does not hold if all groups Gi's are cyclic of order p (prime). But in this case /~ is isomorphic to some direct product of fuzzy subgroups /~i's (not projections of /a) on some cyclic groups of order p. This observation is proved in T h e o r e m 4.4 of [18] by using the concepts of measures and a-algebras of subsets and the Hausdorff Maximal Principle. We prove this result by using the concepts of or-cuts and Theorem 3.3. T h e o r e m 5.8. L e t p be a p r i m e a n d f o r each i = 1, 2 . . . . . k, let Gi be the cyclic g r o u p o f order p. L e t G = G~ x Gz x . . . x G~ and l~ be a f u z z y s u b g r o u p o f G. Then, f o r each i = 1, 2 . . . . . k, there is a f u z z y s u b g r o u p Iti o f s o m e cyclic g r o u p o f order p such that tt is i s o m o r p h i c to the direct p r o d u c t I~ x i~2 x • • • x t~.
Proof. Let # be a fuzzy subgroup of G. Since G is of order p~, ~ has at most k + 1 steps in its graph (Theorem 3.3) since G can have a chain of subgroups of the form (4) of length k at most. In other words the set S = {or e [0, 1]: /~(x) = cr for some x e G - { e } } , which is nothing but the 'range' of/~ restricted to G - {e}, has at most k distinct numbers from [0, 1]. Choose c~ to be the largest number from S (maybe 1). Define cri's inductively for all i = 2, 3 . . . . . k as follows. If
131
the ~-cut ~ , of /~ is a subgroup of order p', choose 0(i+1 to be the next lower number (i.e. the highest amongst the numbers not considered so far) from S. Otherwise, define c~+l = or,. This way we get k numbers c~l~>O~e~>'"~>% corresponding to the given /~. Now choose an element al in the crl-cut crw and form the subgroup H = (a~) of order p (here ( S ) denotes the subgroup of G generated by the subset S of G). Consider a2e or2, such that a 2 ¢ H ~ and define He = (a2). Since every element of a cyclic group of order p is a generator, H~ A//2 = {e} and H 1 . H 2 = H I ® H 2 is the internal direct product decomposition of the group H~H2. Consider as in 013, such that a 3 ~ H i ' H e and construct H 3 = ( a 3 ) . If H i = ( a i ) has been defined, choose a~+~o~i+~, such that ag+~¢ HI®H2®'"®Hi and form H~+~= (a,+l) (it can be seen that such an a,+~ will always exist due to the suitable choice of cry's). This gives an internal direct product decomposition (12)
G = HI ® H2 ® " " " ® I l k .
Let /~i be the restriction of /~ on /4,. for all i = 1, 2 , . . . , k, then since Hi is cyclic of order p, /ui(g) =/~(g) = cri for all g ~ Hi, g 4= e. We claim that these /~i's are the ones we are looking for. We know that the internal direct product (12) is isomorphic to the external direct product H = H1 x H2 x . . . x Hk
treating the H~'s as abstract groups. Thus the mapping O : G--e HI x H2 x . . . x Hk = H
defined by O(g) = (h~, h2 . . . . .
hk),
where g = h r . h 2 - . - h k is the unique representation of g in the internal direct product, is an isomorphism. Now /~ z/~2 x . . . x Pk is defined on H and for g 6 G, x
=
x.
. . x
~'~1 X ~'~2 X
" " " X
O(g) Irk(hi, h2 . . . . .
= min(/~(h,), ~2(h2) . . . . . = min(/~(h0,/~(h2) . . . . .
/~(h,)) /~(h~))
1 if hi = e for all i, cri if i is the largest index such that hi :~ e.
hk)
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If hi = e for all i, t h e n g = e a n d s o / ~ ( g ) = 1. If i is t h e first n u m b e r in { 1 , 2 . . . . . k} such t h a t hi 4: e, t h e n g = h l h 2 • • • hi H I H 2 " " • Hi - H , H 2 " • " H i _ , .
Since H I H 2 " • • H i is a s u b g r o u p o f o r d e r p i a n d HIH2." Hi_~ is a s u b g r o u p o f o r d e r p i - ~ a n d also b e c a u s e t h e r e can be no s u b g r o u p o f G in b e t w e e n t h e s e t w o , it follows t h a t g is c o n s t a n t on 111142" • • H i - H , H 2 . ' ' H i - , . B u t /~(ai) = or, a n d h e n c e ~ ( g ) = o6 in this case. T h u s w e get = (/~1 X ~ 2 X " • " X / ~ k ) o 0
w h e r e / ~ i is a fuzzy s u b g r o u p o f a cyclic g r o u p / 4 , . o f o r d e r p , a n d t h e p r o o f is c o m p l e t e .
Acknowledgement T h e a u t h o r s t h a n k t h e u n k n o w n r e f e r e e for h i s / h e r v a l u a b l e c o m m e n t s which h e l p e d in i m p r o v i n g this p a p e r .
References [1] J.M. Anthony and H. Sherwood, Fuzzy groups redefined, J. Math. Anal. Appl. 69 (1979) 124-130. [2] J.M. Anthony and H. Sherwood, A characterization of fuzzy subgroups, Fuzzy Sets and Systems 7 (1982) 297-305. [3] Meng Daoji, Fuzzy groups, Fuzzy Math. 2 (2) (1982) 49-60.
[4] Wenxiang Gu, Pointwise fuzzy groups with operator, Fuzzy Math. 3 (1983) 17-24. [5] M. Hail, Jr., The Theory o f Groups (Macmillan, New York, 1959). [6] N. Jacobson, Lectures in Abstract Algebra (East West Press, 1951). [7] A. Kaufmann, Introduction to the Theory o f Fuzzy Subsets, Vol. 1 (Academic Press, New York, 1975). [8] N. Kuroki, Fuzzy bi-ideals in semigroups, Comment. Math. Univ. St. Paul. XVIII (I) (1979). [9] N. Kuroki, Fuzzy semi-prime ideals in semigroups, Fuzzy Sets and Systems 8 (1982) 71-80. [10] Wang-jin Liu, Fuzzy invarient subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982) 133-139. [I1] Jiliang Ma and Chunhai Yu, Fuzzy groups I, J, Dongbei Normal Univ. Nat. Sci. Ed. 3 (1981) 21-25. [12] S. Nanda, Fuzzy fields and fuzzy linear spaces, Fuzzy Sets and Systems 19 (1986) 89-94. [13] C.V. Negoita and D.A. Ralescu, Applications o f Fuzzy Sets to Systems Analysis (Wiley, New York, 1975). [14] Zhenkai Qi, Pointwise fuzzy groups, Fuzzy Math. 1 (2) (1981) 29-36. [15] Zhang Qun, Some rudimentary opinions on several definitions of fuzzy groups, J. Henan Normal Univ. Nat. Sci. Ed. 2 (1982) 4-14. [16] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. [17] S. Sessa, Fuzzy subgroups and fuzzy ideals under triangular norm (short communication), Fuzzy Sets and Systems 13 (1984) 95-100. [18] H. Sherwood, Products of fuzzy subgroups, Fuzzy Sets and Systems 11 (1983) 79-89. [19] Congxin Wu and Zhenkai Qi, Transfer theorems for fuzzy groups, Fuzzy Math. 2 (1) (1982) 51-56. [20] Wangming Wu, Normal fuzzy subgroups, Fuzzy Math. 1 (1) (1981) 21-30. [21] Wangming Wu, Fuzzy congruences and normal fuzzy subgroups, Fuzzy Math. 3 (1988) 9-20. [22] Yandong Yu, On some properties of fuzzy groups, Fuzzy Math. 3 (4) (1983) 35-40. [23] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353.
121
Fuzzy normal subgroups and fuzzy quotients I.J. Kumar, Yadav
P.K. Saxena and Pratibha
Scientific Analysis Group, DRDO, Ministry of Defence, Metcalfe House, Delhi 110 054, India
Received May 1989 Revised October 1990 Abstract: In case of groups, a normal subgroup N of a group
G can be defined with three equivalent approaches, namely (i) as a kernel of some group homomorphism, (ii) as a subgroup commuting with every element of G, and (iii) as the subgroup N closed with respect to conjugates of elements of N. In case of generalizations of this concept to fuzzy sets, the elementwise third approach is found more suitable and hence is widely adopted by many authors. In this paper we further study this concept of fuzzy normal subgroups and introduce fuzzy quotients called 0c-fuzzyquotient groups for all ole[0, 1]. Two different definitions of these ol-fuzzy quotients have been given which are shown to lead to isomorphic structures. We also study the direct products of fuzzy (normal) subgroups (min-product as per the terminology of Sherwood (1983)) and the problem of writing a fuzzy (normal) subgroup of a direct product of some groups as a direct product of certain fuzzy (normal) subgroups, has been discussed in detail. Keywords: Fuzzy subgroup; fuzzy normal subgroup; 0l-fuzzy quotients; isomorphism; kernels; projections; f-invariance; direct-product; internal direct product; or-cuts.
1. Introduction The concept of fuzzy sets and various operations on it were first introduced by Z a d e h in [23]. Since then, fuzzy sets have been applied to diverse fields. But the study of fuzzy algebraic structures was started with the introduction of the concept of fuzzy subgroups in the pioneering p a p e r of Rosenfeld [16]. Many other concepts such as 'fuzzy bi-ideals', 'fuzzy semiprime ideals', 'fuzzy fields' and 'fuzzy vector spaces' were developed in [8, 9, 12] but these concepts did not have much scope for interesting and useful further studies because of too m a n y
constraints in their generalizations. However, the theory of fuzzy subgroups developed in both respects, theory as well as its practical applicability. The generalization of subgroups to the fuzzy subgroups of Rosenfeld was modified by Negoita and Ralescu [13] by replacing the membership set [0, 1] by appropriate lattice structures. A n t h o n y and Sherwood further redefined fuzzy groups in terms of a general ' T - n o r m ' which replaced the 'min' of its previous definitions [1]. They also applied this concept in pattern recognition problems [2]. Many other papers on fuzzy subgroups have also a p p e a r e d which generalize various concepts of group theory such as 'normal subgroups', 'quotient groups' [10, 3, 20, 21], 'direct products of groups' [3, 18], and also discuss h o m o m o r p h i s m and isomorphisms between these fuzzy structures. Even the concepts of M-subgroups, M-factorgroups, M - h o m o m o r p h i s m s and M-isomorphisms for groups with operators [6,128-137] have been generalized by G u in [4]. In the present p a p e r we further study fuzzy normal subgroups, define ol-fuzzy quotients, discuss isomorphisms between some specific structures and characterize the fuzzy subgroups of a direct product of certain groups in terms of fuzzy subgroups of these groups. The second section of this p a p e r gives preliminary definitions and results required later. More details of these concepts can be seen in the papers referred to. In Section 3, we review the definition of fuzzy normal subgroups and prove a theorem to characterize fuzzy normal subgroups of certain groups. Section 4 consists mainly of the definitions of or-fuzzy quotients of a fuzzy normal subgroup /~ in G in two different ways, namely (i) as a fuzzy subset of an ordinary set of ordinary cosets of the or-cut ol, and (ii) as a fuzzy subset of the set of o~-fuzzy cosets/~a of/~ in G. It is shown that both of these approaches lead to isomorphic fuzzy quotients. We also discuss
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Lagrange's theorem of groups in the context of fuzzy normal subgroups and fuzzy cosets. In Section 5, we discuss many observations about the direct product of fuzzy subgroups, the relationship between the fuzzy subgroups of product of groups and its projections. In case of finite groups, a subgroup A of the direct product G1 × G2 × " " × G, of groups can be written as the product A1 × A2 × • • • × A , of projections Ai of A on Gi, where orders of Gi's are pair-wise co-prime [5]. We prove a similar result for fuzzy subgroups, which is stronger than Theorem 4.1 of [18] and gives the counterpart of the 'fundamental theorem of finite Abelian groups' in case of fuzzy subgroups. We have also given another proof of Theorem 4.4 of [18] by using o~-cuts and avoiding the concepts of measures, aalgebras, Hausdorff Maximal Principle, etc., which have widely been used in the proof.
2. Preliminaries Definition 2.1. For a group G, a fuzzy subset/~ on G is called a fuzzy subgroup (shortly written as FSG) if (i) t~(xy) >- min(~(x), ~(y)) for all x, y in G, (ii) /~(e)= 1 where e is the identity element of G, and (iii) /~(x -1) I>/~(x) for all x in G. In the original definition of fuzzy subgroups in [16], the condition (ii) was absent. In [13] Negoita and Relascu replaced [0, 1] by an appropriate lattice structure but condition (ii) was not explicitly mentioned. However, some of their propositions seem to assume it. In [1] Anthony and Sherwood introduced arbitrary 'T-norms' in condition (i) in place of 'min' but they did not assume (ii). In several other papers also, such as [3, 10, 17, 20], condition (ii) has not been assumed. In [2] Anthony and Sherwood modified their own convention of [1] and included (ii) with the observation that a very natural condition '/~(e) > 0' together with (i) and (iii) lead to the satisfaction of all these three conditions if all characteristic values are normalized by /~(e) [2, Lemma 1]. We also follow this convention and use Definition 2.1 as the definition of FSG throughout this paper.
Observe that there is equality in (iii) in fact. We may also mention that even in definition of a 'T-norm', the boundary conditions (as it is called by Sessa [17]) differ from paper to paper [1, 2, 3, 17, 21], though we shall be dealing with 'min' which satisfies all the conventions of T-norm and is the strongest T-norm in the sense of [181. If /~ is a fuzzy set on a set S and if f is a function defined on S then ~t induces a fuzzy set #I on f ( S ) defined by /~I(y) =
sup
{/~(x)}
for all y ~ f ( S )
(1)
xef-l(y)
and is called the image of ~ under f. Similarly, if v is a fuzzy set on f ( S ) , then a fuzzy set on S can be defined through v I l(x) = v ( f ( x ) ) = v of(x)
for all x e S.
(2)
where o is the composition of mappings, v I ' is called the inverse image of v. The following result was shown by Rosenfeld in Proposition 5.8 of [16] under an additional hypotheses of 'supremum property' for the function f. This result was proved by Anthony and Sherwood [1, Proposition 4] in general for fuzzy subgroups with continuous T-norms. The T-norm 'min' being continuous, the following proposition holds.
Proposition 2.2. If f is a homomorphism from a group G onto a group G', then the homomorphic image #I of a fuzzy subgroup I~ on G is a fuzzy subgroup on G' and also that the homomorphic preimage v~-' of a fuzzy subgroup v on G' is a fuzzy subgroup of G. For a function f from a set S onto the set f ( S ) , a fuzzy set /~ of S is called f-invariant if /~(x) = # ( y ) whenever f ( x ) = f ( y ) for all x, y in S. The following theorem gives the correspondence between fuzzy subgroups of a group G with fuzzy subgroups of a homomorphic image of G [16].
Theorem 2.3. If f : G ---~f ( G ) is a group homomorphism, then there is a one-to-one correspondence between f-inoariant fuzzy subgroups of G and fuzzy subgroups o f f ( G ) . If f:G---~ G' is a group isomorphism (onto)
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Other preliminary definitions can be found in [7] and the papers referred to. As a general remark, we mention that by 'fuzzy sets' we would mean 'fuzzy subsets' only and we will not differentiate between these two.
groups treating them as subgroups commuting with every element of G. Meng [3] and Yu [22] also investigated FNSGs without condition (ii) of Definition 2.1. Recently, in [21] Wu studied normal T-fuzzy subgroups where 'min' of Definition 2.1 has been replaced by any general T-norm T. He also characterized normal T-fuzzy subgroups in terms of fuzzy congruences on a group. In the study of fuzzy algebraic structures, c~-cuts play an important role in the visualization of these concepts. Thus we have the following observation which has been shown by Wu [20] with a slightly different statement due to the absence of '/~(e) = 1' in his definition. We sketch the proof which is simpler than Propositions 1.1, 2.1, 3.1 of [20] particularly in the converse part.
3. Fuzzy normal subgroups
Proposition 3.2. For a group G, a f u z z y subset ~t
and /~ is a fuzzy subgroup of G, then ~ is f-invariant. Thus there is a one-to-one correspondence between FSGs of G and FSGs of G'. This correspondence ~ ~--~#Y is called an isomorphism between fuzzy subgroups/u and ~J. More precisely:
Definition 2.4. Two fuzzy subgroups (G, ~) and (G',/~') are said to be isomorphic if there exists a mapping f : G - - ~ G ' such that f is a group isomorphism (onto) and t h a t / u ' o f = ~.
We start with the natural generalization of the normalcy condition of ordinary groups [20] in the following:
Definition 3.1. For a group G, a fuzzy set on G is called a fuzzy normal subgroup (FNSG in short) of G if it is a fuzzy subgroup satisfying I~(xyx -~) ~ I~(Y)
for all x, y in G.
(3)
Some remarks concerning this definition are in order. In [20] Wu studied fuzzy normal subgroups but the condition (ii) of Definition 2.1 was absent. An FNSG /~ on G in our sense was called normal quasi-group by Wu. Liu in [10] adopted an alternate approach of defining fuzzy subgroups by treating a fuzzy subset as an ordinary set of fuzzy point-sets xx. He defined an operation 'sup-min' between two fuzzy subsets (and hence between a fuzzy set and a fuzzy point-set) of the group G. Condition (3) of the above definition was replaced by another equivalent condition I~(xy) = l~(yx)
for all x, y in G
(3')
for defining fuzzy invariant subgroups (as he called it). Again condition (ii) of Definition 2.1 was missing. Liu showed that a fuzzy invariant subgroup commutes with every fuzzy subsets on G, hence with every fuzzy point-set xx. Thus Liu's approach was in fact a sort of generalization of this concept of normal subgroups of
on G is a f u z z y normal subgroup o f G if and only if or-cuts o:~ = {x ~ G: I ~ ( x ) ~ o:} are n o n e m p t y and f o r m normal subgroups o f G, f o r all o: ~ [0, 1].
Proof. Let ~u be a fuzzy normal subgroup of G and 0~ in [0, 1] be an arbitrary number. Then the identity element e of G is always in the or-cut ol,. For any elements x, y in cr,, /~(x, y - l ) >1 min(t~(x),/~(y)) /> min(o~, or) = 0c and therefore xy -~ ~ o(,. Further, ~t(xyx -~) >1 /~(y)/> ol and hence x y x -~ ~ ol,. Thus or, is normal subgroup of G. Conversely, if all or-cuts are nonempty and normal subgroups of G, then in particular I v :~ 0 which guarantees for # ( e ) = 1. Let x, y be any two elements of G and cr =min(/~(x), /~(y)). Then xc0l~, y c t r , and therefore x y ~ o : ~ proving that /u(xy)/> min(#(x),/~(y)). Further, x -~ and y x y -1 belong to the or-cut ol, which proves that/~(x -l)/>/~(x) and kt(yxy -~) >~ kt(x) for all x, y in G. This completes the proof. We now give the following theorem.
Theorem 3.3. I f G is a group having a chain {e} =No c N 1 c N z c .
• • c NL = G
(4)
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of normal subgroups of G and L is the length of the maximal such chain, then a fuzzy normal subgroup of G is a step function from G ~ [0, 1] having at most L + 1 steps.
Abelian and it has a subgroup H of order p giving the chain {e} = N0 c H c N2 = G of length L = 2 and there is no chain with length more than 2, and hence the result follows.
Proof. Let It be a fuzzy normal subgroup of G having the chain (4) of normal subgroups of G. For trivial subgroup No = {e}, It(e) = 1 gives one step of the characteristic function It. Corresponding to the next member N1 of the chain (4), either It(x) = It(e) = 1 for all x e N~ or there exists an element xo c N 1 - {e} such that It(x0) 4: It(e). In first case It restricted to N~ has only one step. In the other case the claim is that It is constant throughout N ~ - {e} and I t ( x ) = I t ( x 0 ) < i t ( e ) = 1 for all x in N l - - { e } . This is because if It(x)~it(Xo) for x4:xo in N l - { e } then there exists a number o~ c [0, 1] in between It(x0) and It(x), a~ 4: It(xo) such that only one of the two elements x, x0 will belong to c~-cut o~, and the other will not. Thus we get the chain
Corollary 3.6. A group G of order pq, where p, q are primes, has fuzzy normal subgroups with at most three steps.
{e} = N o c c~, cN~ c . . - c N L
= G
of normal subgroups having length L + 1 which contradicts our hypotheses about the length L. Thus It is constant on N I - {e} having values different from It(e) = 1. In this case It restricted to N~ has exactly two steps in its graph. Proceeding with a similar argument we see that the consideration of each member of the chain (4) gives rise to at most one step in the graph of It. Thus we can have at most L + 1 steps, which proves the result.
Proof. There is only one chain {e} c N c G of normal subgroups where N is the cyclic subgroup of G of order max(p, q), hence the result.
The following corollary characterizes FNSGs of dihedral group D2p.
all
Corollary 3.7. For a dihedral group Dap = {xry~: X 2 = e, yP = e, yx = x y p-l, where 1 <~r <- 2, l<~s<-p} of order p (a prime) and a fuzzy normal subgroup It of G, 1 = #(e)/> It(y) . . . . .
It(xy) . . . . .
It(yp-l)
It(xyP-~).
Proof. The group D2p has only one proper normal subgroup
N = {e, y, y2 . . . . .
yp-l}
of order p. Hence It is constant on N - {e} and there are at most three steps in It which proves that # is constant with lower values (possibly) for the remaining set G - N.
Corollary 3.5. If G is a group of order p2 (p is a prime) and It is a f u z z y normal subgroup of G, then It has at most three different values in [0, 1].
Before we discuss images and preimages of fuzzy normal subgroups under a group homomorphism, we observe that Proposition 3.2 also holds for ordinary fuzzy subgroups of a group G. Further, as in T h e o r e m 3.3, the chain of maximum possible length of subgroups of the group G gives the number of possible different steps in the fuzzy subgroup It and helps in finding types of possible fuzzy subgroups of finite groups as in the corollaries above. Consider the group homomorphism f : G ~ G ' (onto). Wu showed in [20] that Proposition 2.2 holds even for fuzzy normal subgroups /~ of G. Though It(e) was not necessarily 1 there, the proof of the following theorem remains almost the same and hence is omitted. We shall need this result in the next section.
Proof. If o ( G ) = pZ, p a prime, then G is always
Proposition 3.8. For a fuzzy normal subgroup It
We now give the following specific cases as corollaries. Corollary 3.4 (Proposition 5.10 of [16]). Any fuzzy subgroup It of a cyclic group G of order p (a prime) is a constant function except possibly at the identity element of G where It is 1. Proof. Obviously, It is fuzzy normal and there is only the trivial chain {e} c G possible in place of the chain in (4) of normal subgroups of G.
l.J. Kumar et al. / Fuzzy normal subgroups and fuzzy quotients
of G and a homomorphism f : G --* G ' (onto), t~1 is also a fuzzy normal subgroup of G'. Further, if v is a fuzzy normal subgroup of G' then vf ~is also a fuzzy normal subgroup on G. The following theorem extends T h e o r e m 2.3 of Section 2 for fuzzy normal subgroups. The proof, following the case of fuzzy subgroups [16], does not use the normalcy condition and hence is omitted.
Theorem 3.9. There is one-to-one correspondence between f-invariant fuzzy normal subgroups of G and fuzzy normal subgroups of G ' = f ( G ) , where f:G---~G' = f ( G ) is a group homomorphism from group G to G'.
4. Fuzzy quotients In groups, normal subgroups are also defined as kernels of group homomorphisms. In the other words, there is a one-one correspondence between normal subgroups of a group G and group homomorphisms on G. Thus the three concepts namely 'kernels', 'homomorphisms' and 'quotients' are mutually dependent in terms of their definitions. For perceiving FNSGs through homomorphisms, we have to visualize concepts like 'fuzzy kernels' of some (fuzzy?) homomorphisms and talk of 'fuzzy-quotients' as obtained by slicing out the 'fuzzy kernels' from the domain set. If the fuzzy quotients are to be treated as ordinary groups then it becomes difficult to associate membership values to the kernel of group homomorphism to define fuzzy kernel in a natural way. If we think of the fuzzy quotients as fuzzy sets, then the question becomes how a homomorphism from a group to a fuzzy subgroup should look so that the 'kernel' of this homomorphism may become a F N S G of the domain set (group). Due to such inherent problems in visualizing these three concepts together, probably no one has defined FNSGs through homomorphisms and fuzzy kernels. However, once the FNSGs have been defined as in Definition 3.1 it becomes natural to investigate the generalization of the concept of quotient groups to the fuzzy sets even without explicitly talking of fuzzy kernels,
125
homomorphisms and isomorphism theorems, etc. In this section we introduce and study u-fuzzy quotients not as just ordinary quotient groups as in [10,20] but as FNSGs of some quotient groups. Before giving the formal definition, a few more comments on this concept are in order. In [20] Wu considered S/S~ where ~ is FNSG of the group S and S, is the ~(e)-cut of S and meant it to be the fuzzy quotient group. He further visualized members of S/Su as fuzzy cosets. In [10] Liu denoted this group S by X, the FNSG by A and called it a fuzzy quotient group without any further discussion on it. Though the concept also appears in some other referred papers, a slightly different approach to this idea of fuzzy quotients appeared in [21] for T-fuzzy subgroups (T to be a regular T-norm) where an attempt was made to study it in the context of group isomorphism theorems. There Wu considered a T-fuzzy subgroup g on G (called a T-fuzzy group) as the basic structure (instead of G) and defined a normal T-fuzzy subgroup v of ~, suitably, to form a T-fuzzy subgroup ~ / v of the group s(l~)/S(v) where S(~) and S(v) are supports of /~ and v, respectively. However, getting a nontrivial T-fuzzy quotient (the author does not use this name) would put a very strong restriction on v namely that v must be zero at some points of the group where # is nonzero (otherwise S ( l O / S ( v ) = { e } ) . This idea of quotients also appears in [4] in the form of M-fuzzy quotients (M does not stand for 'min') but it is entirely in a different direction of groups with operators as in [6]. For any number c~ e [0, 1] and a fuzzy normal subgroup of a group G, the o~-cut cv, is a normal subgroup of G and hence we can talk of the quotient group G/ol,. In view of Proposition 3.8, ~o is a fuzzy normal subgroup on G/ol~ where O:G---, G/o6, is the natural homomorphism. We give the following definition. Definition 4.1. For a fuzzy normal subgroup /~ of G and o~ e [0, 1], the pair (G/o6,, go) is called an o~-fuzzy quotient of tile group G. The above definition of ol-fuzzy quotients looks a bit unnatural since elements of G/ol. are ordinary cosets of the normal subgroup having some characteristic values associated with these
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126
cosets through /~o. In what follows, we like to define the a~-fuzzy quotients as the set S of a~-fuzzy cosets and give structure to S by defining the o p e r a t i o n b e t w e e n two ol-fuzzy cosets.
Definition 4.2. L e t /~ be a fuzzy s u b g r o u p of a g r o u p G. F o r any a ~ G, a fuzzy subset #a of G is called an oc-fuzzy right coset of/~ in G if
# a ( x ) = min(~u(xa), o:)
for all x in G.
Similarly, a fuzzy subset a # of G is called an o~-fuzzy left coset o f / t in G if
al~(x) = m i n ( # ( a x ) , o~)
t h e o r e m shows that o ( G / # ~ ) = o(G/o6, ) having the s a m e n u m b e r of elements.
Theorem 4.5. I f I~ is a f u z z y normal subgroup oJ a group G and N = o¢~ is the ol-cut o f I~ for some a" ~ [0, 1], then Na = N b if and only if l~a = lib for all elements a, b in G.
Proof. L e t a, b be arbitrary e l e m e n t s of G and ~a =/~b. T h e n
t,a(b -1) = / , b ( b -1) = m i n ( / ~ ( b b - l ) , a0 = min(1, o~) = ol.
for all x in G.
where q~2 = e, ~03 = e and q),p2 = ,pq~. C o n s i d e r a fuzzy s u b g r o u p on G given by
But Ita(b -1) = min(/~(ab-1), a 0 which shows that l~(ab -1) >i ol and hence a b - l e N proving Na = Nb. Conversely, let Na = N b for any a, b in G and let x be any e l e m e n t in G. T h e n Nax = N b x . O b s e r v e that l*(ax)< ol and lt(bx)>i ol implies that Nbx = N 4: Nax which is not true. Similarly /t(ax) ~ o~ and l*(bx) < a~ implies that Nax = N ~ N b x which is false again. H e n c e either both ~(ax) and # ( b x ) are g r e a t e r than or equal to a~, or both are less than o~. In the first case,
/~(e) = ~(q~) = 1,
Ira(x) = min(/~(ax), a;) = a"
~,(v~) = t , ( ¢ )
and
T h e following e x a m p l e shows that for a fuzzy s u b g r o u p / , of G, a~-fuzzy left cosets n e e d not be equal to c o r r e s p o n d i n g ol-fuzzy right cosets.
Example 4.3. C o n s i d e r G = $3, the s y m m e t r i c group of o r d e r 3. T h e n G can be e x p r e s s e d as G = {e, 4, % ~p2, q~g,, q~p2}
= ~ , ( q , ~ ) = t , ( ~ ,2) = 0.5.
T h e n it can be seen that for cr = 1,
t*b(x) = min(/u(bx), a~) = c~,
~Pt~ = (0.5, 0.5, 0.5, 1.0, 1.0, 0.5),
where as in the o t h e r case,
/~p = (0.5, 0.5, 0.5, 1.0, 0.5, 1.0). ( H e r e the 6-tuple d e n o t e s the characteristic values of e l e m e n t s of G in order. W e shall use such notation even afterwards for r e p r e s e n t i n g fuzzy subsets on finite sets.) H e n c e ~p/~ 4:/~ap. This h a p p e n s because g is not a fuzzy n o r m a l s u b g r o u p of G = Ss. H o w e v e r :
Proposition 4.4. I f i~ is a f u z z y normal subgroup o f G and x is any element o f G, then the ol-fuzzy right coset I~x is same as the ol-fuzzy left coset xl~.
Proof. Let /~ be a fuzzy n o r m a l s u b g r o u p of G, a~G, and a ~ [ 0 , 1 ] . T h e n for any x ~ G , I~(xa) = / u ( a x ) and so t~a(x) = m i n ( ~ ( x a ) , a 0 = m i n ( / t ( a x ) , a~) = al*(x). Consider the set G/l*~ = {/~a: a ~ G} of all a~-fuzzy right cosets of /, in G. T h e following
#a(x) = I*(ax) < ol and t*b(x) = l*(bx) < ol. In this case, since Na = Nb, we have a = nb for some n e N and so
~b(x) = ~ ( b x ) = l~(n-lax) /> min(/~(n-1), t~(ax))
= I~(ax) = tta(x) = ~ ( n b x ) 1> min(/~(n), I~(bx)) = I~(bx) = ~b(x). Thus I~a(x) = I~b(x) for all x e G and the p r o o f is complete. W e give a structure to G / # ~ = {/~a: a ~ G} by defining the o p e r a t i o n * b e t w e e n two a~-fuzzy right cosets as
t*a * t*b = f~ab.
(5)
In the following e x a m p l e we see that if # is a fuzzy s u b g r o u p on G (not n o r m a l ) , then the
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Example 4.6. Consider G and g as in Example 4.3. It can be seen that for tr = 1,
Observe that the mapping O: G---)G/go` defined is a well defined map and that for x, y in G, O(xy) = gxy = Izx * gy = O(x) * O(y). Hence it is an onto group homomorphism. Thus we get
#e = (1.0, 1.0, 0.5, 0.5, 0.5, 0.5),
g * ( g a ) = sup {g(x)} =
sup
pax--.a
~J(x)= O(a)
operation * defined above in (5) is not well defined.
by 0 ( x ) = p x
gq) = (1.0, 1.0, 0.5, 0.5, 0.5, 0.5), =
gap = (0.5, 0.5, 0.5, 1.0, 0.5, 1.0),
sup
(g(x)}
{g(x)},
xe~ ~(.a)
g(PW = (0.5, 0.5, 1.0, 0.5, 1.0, 0.5),
which shows that g . = g 0 and hence by Proposition 3.8, g* is a fuzzy normal subgroup of G/go`. In the following theorem we see that (G/#o`, g*) is just another way of defining a<-fuzzy quotients.
gap2 = (0.5, 0.5, 1.0, 0,5, 1.0, 0.5), g ~ l p 2~-- (0.5, 0.5, 0.5, 1.0, 0.5, 1.0).
Here gap = gq)ap2, gap2= gq)ap. But gap * gape = ge 4: #q)ap2, #q)ap =
Theorem 4.8. The o<-fuzzy quotient ( G / o:,, #o),
g~ap2~ap = gap~.
However, the operation * is well defined in case # is a fuzzy normal subgroup. Proposition 4.7. The operation * defined in (5) on G/#o`, where g is a fuzzy normal subgroup of G and o
x~A
= mm(g(abx), tr)
where
= m m ( g ( a l b x ) , o<)
A = {x e G : x e 0-1(ga)}
= m m ( g ( x a l b ) , o<)
= {x e G: px = ga}
= mm(g(bxaO, tr)
= {x e G: Nx = Na}
= mm(g(b,xaO, o<)
= {x e G: O(x) = Na}
= m m ( g ( x a l b O, tr)
= {x e G : x e O-t(Na)}.
= m m ( g ( a , b , x ) , o<)
Thus #6of ( N a ) = g ° ( N a ) for all N a e G / c h , , and hence g ° o f = gO as desired.
= galbl(x). Thus #a * gb = gab = #a, b ~ and hence * is well defined. The set (G/#o,, *) becomes a group with ge working as the identity element and #a -~ as the inverse of the o<-fuzzy coset #a. We can define g*, a fuzzy set on G/#o` as =
sup {#(x)}.
/aa=/~x
Proof. Consider the mapping f:G/o6,--> G/#~ defined by f ( N a ) = #a where N = ct, and a e G. Then it is clear by T h e o r e m 4.5 that f is a one-one onto homomorphism. It remains to show that g 6 o f = g o . For that, consider any element ga of G/go`. Then
g ° o f ( N a ) = g°(ga) = sup {g(x)}
(#a * #b )(x ) = gab(x)
#*(#a)
where O: G---> G / ol, is the natural epimorphism, is isomorphic to the fuzzy normal subgroup (G/go`, g°), where O:G---~G/go` is the homomorphism defined by O(x ) = gx.
(6)
Another equivalent way of defining tr-fuzzy quotients is through the equivalence classes of the equivalence relation - on the group G defined as a - b iff g(ab -l) >! o<.
(7)
The equivalence classes of - coincide with the cosets of c~-cut tr,.
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In case of ordinary normal subgroups of a group G, Lagrange's T h e o r e m states that the group G is the disjoint union of cosets of N in G. In other words the set {Na: a • G } of cosets partitions G. To see a similar result in the fuzzy situation we give the following definition.
We prove this result for a more general situation of the fuzzy partition of the group G by the or-fuzzy cosets of F N S G / ~ of G. Theorem 4.10. If l~ is a fuzzy normal subgroup of a group G, then the class o%= {~a: a • G} of
ol-fuzzy cosets of t~ in G is a fuzzy partition of G. Definition 4.9. The class ~ = {#~} of fuzzy sets /~ of a set S is called a fuzzy partition of S if for all o r • [ 0 , 1], either the class o%,={a,~} of 0l-cuts o f / ~ has no non-empty m e m b e r (set) or it partitions S in ordinary set theoretic sense. The additional condition 'either ~ has no non-empty members' in the above definition is included for the following reason. If we do not assume this then we will have to assume that at least o n e / t i must be 1 somewhere in S (for finite cases) or that 1 should be a limiting point of {~i(x): i arbitrary, x • S} (for infinite cases). For otherwise one can find some /3 • [0, 1] such that all fiu's are empty. It is not important whether G is covered and partitioned by each and every o%o, o~ • [0, 1], but what matters more is that if one member of ,~o~ is nonempty it should be a partition. Naturally the number of classes of these partitions will depend on o~. In Definition 2.1 of [20] Wu defined fuzzy cosets of a fuzzy set /~. In Proposition 2.7 he characterized the Z-cuts of fuzzy cosets as ordinary cosets of )t-cuts of the fuzzy subgroup. Since/~(e) is not necessarily 1 in his definition of a FNSG, the statement in this Proposition 2.7 has some ambiguity in the sense that for any ~. between /~(e) and 1, /~a (/l, in his notation) will be empty and the product a/~a (a)~, of course) will be meaningless. However, for nonambiguous cases this proposition holds. Thus with the conventions of fuzzy cosets and FNSGs of Wu, it follows from this Proposition 2.7 [20] that for a FNSG ~ of a group G, the class {a/~: a e G} of fuzzy cosets is a fuzzy partition of G. This is what Lagrange's T h e o r e m looks like in the fuzzy context. Observe that, even for cr = 1, the Definition 2.1 [20] of 'fuzzy cosets' differs from our Definition 4.2 of 1-fuzzy cosets and the result similar to Proposition 2.7 [20] does not hold here. This is clear from Example 4.3, where l.~p = {llY2, I ~ 2} @ {I/3, (/)113} = 1j," 1/)= {e, qb}. ,p. Thus even for 1-fuzzy cosets the result of Lagrange's T h e o r e m does not follow as in [20].
Proof. Consider any number fi • [0, 1] and the set ~t~ = {/3,a: a • G} of/3-cuts o'f or-fuzzy cosets /~a of the given fuzzy normal subgroup ~ of G. If /3 is bigger than or, t h e n / 3 , , = q~ (the empty set) for all a • G, since gta(x) = min(/~(xa), ol) ~< cr for all x • G. Thus oft~ has only null set 0 as a member and we are done. So assume that/3 ~< ol. In this case, for any x • G, /~c-l(x) = min(/~(xx-l), or) = min(/~(e), or) = cr/>/3 and x • / 3 ~ , which means that G c_ U.~G/3.a. It remains to show that for any a, b in G, the sets 13.. and /3.b are either identical or disjoint. For this let x •/3.. ¢q/3ub be any element in the nonempty intersection, if possible, and let y be any arbitrary element of/3... Then
I~a(x) = min(/_t(xa), or) I> 13, lib(x) -- min(~(xb), or)/>/3 and
l~a(y) = min(~t(ay), or)/>/3. Therefore
#b(y) = min(/t(by), o~) = min(i.t(bxx-la-lay), o~) i> min(/,(bx), l~(x-la-l), g(ay), ol) --min(/t(bx), I.t(ax), l*(ay), or)
>~/3. Thus Y •flub and so /3.a~-/3.h- Similarly, choosing y in /3.b instead of /3u. above, we get fl.b~_/3.a. This shows that /3~a and /3ub are identical if they have even a single element in common. Thus ~ is a partition of G and the proof is complete. We have seen that in defining fuzzy quotients,
(G/o:,, I~') of/~ in G, It' is defined a s / t ° where O: G---~ G/o: u is the natural epimorphism. If G
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l.J. Kumar et al. / Fuzzy normal subgroups and fuzzy quotients
is a finite group, then for g • G / o : , , s u p / , o ,~){/~(x)} = i t o ( ~ ) is the value of It at some point x0 of A = {x e G: O(x)) = g } and hence I t ' ( g ) ~ c~ unless g = ~. T h u s tx-cut a~u, = {~}, i.e. It' is 1 only at ~ and for all of the other points of G / t r u, It' is below tr. T h o u g h in taking s u p r e m u m over infinite sets o f n u m b e r s the 'sup' m a y not actually be attained at s o m e points of the set, yet the a b o v e m e n t i o n e d observation holds for any g r o u p as is clear from:
case of o: = 1 this set S has only one m e m b e r as is clear from the following proposition which has been shown in [3]. Proposition 4.12. I f G is a group and It a f u z z y normal subgroup on G, then f o r N = 1~,, It is a constant function on Na f o r all a c G. T h e a b o v e proposition also characterizes fuzzy normal subgroups o f a g r o u p G. T h e g r o u p G can be a r r a n g e d as G=NUNaUNbU...,
Proposition 4.11. I f G is a group and /~ is a f u z z y n o r m a l s u b g r o u p o f G, then cr,, = {~} in the f u z z y quotient ( G / cr~,, it').
the disjoint union of cosets of N = 1, and tt has to be constant on each o n e of these cosets.
Proof. In view of the r e m a r k s a b o v e , it is sufficient to prove the result for nonfinite groups. T h e r e f o r e assume that G is a nonfinite g r o u p and g0 (go c G) is an e l e m e n t of G/ol~ such that It'(go) = 13/> or. D e n o t i n g ol, by N, we observe that
5. Fuzzy direct products
/3 = I t ' ( g . ) =
sup {/_t(x)}.
xeNgo
(8)
For any given arbitrary n o n z e r o positive real n u m b e r , say e, there exists s o m e n o o N such that It(nogo) > / 3 - e I> o~ - •. But then It(g,,) = It(ncT'nog,,)
In [18] S h e r w o o d defined the T ' - d i r e c t p r o d u c t of T-fuzzy subgroups It~ of s o m e groups G~, where T, T ' are s o m e T - n o r m s on [0, 1] × [0, 1]. W e consider only ' m i n ' T - n o r m and by a direct p r o d u c t of fuzzy subgroups we shall m e a n the ' m i n ' direct product. W e recall the definition. Definition 5.1. For each i = 1, 2 . . . . . n let It~ be a fuzzy s u b g r o u p o f Gi. T h e n the direct p r o d u c t It = Itl × It2 x • • • ×/~n, the function defined on G1 x G2 x • • • x Gn is given by
/> min(it(rqT'), It(nogo)) >/min(ol, It(nogo)).
It(x~, x2 . . . . . = It1 x t~2 x
If It(nogo) >i oc then min(~r, It(nogo)) = ,x > , x - e; otherwise min(ol, It(nogo)) = It(nogo) > ol - e.
x.) • • • x Itn(x~,
x2 . . . . .
- - m i n ( i t ~ ( x 0 , It2(x2) . . . . .
x~)
~(x~)).
(9)
As in Corollary 3.4 of [18] for fuzzy subgroups, the following proposition for fuzzy n o r m a l subgroups was shown by M e n g in [3] for n = 2. T h e result for any finite n also follows easily. W e just state the result.
In any case It(go) > ol - ¢. Since • is an arbitrary n u m b e r and can be c h o s e n as small as we please, we must have It(go) ~> ol which shows that go = e. Thus o l , . = {~} and h e n c e It' is 1 only at the identity e l e m e n t of the g r o u p G / o 6, and for all the other elements it is below or.
Proposition 5.2. S u p p o s e Iti subgroup o f Gi f o r each i = 1, direct p r o d u c t It = itl X l~2 X f u z z y n o r m a l subgroup o f G1
It is clear from (8) that for finding It' of an element g of G / c r , , one has to look for 'sup' out of a certain set S of values of It. F o r the specific
In case of n o n f u z z y s u b g r o u p S of a direct p r o d u c t of groups G ~ × G e × . . - x G , , the projection Si of S on each Gi is a s u b g r o u p of Gi.
is a f u z z y n o r m a l 2 . . . . . n. Then the . . . x itn is also a x G2 x • • • × G,,.
l.J. Kumar et al. / Fuzzy normal subgroups and f u z z y quotients
130
To see the similar result subgroups we recall [18]:
in case
of fuzzy
Definition 5.3. L e t / , be a fuzzy subgroup on the cross product G~ × G2 x • • • × G~ of groups G~. The projection [Ai of # o n Gi, for all i = 1, 2 , . . . , n, is defined by ~i(gi) = ~(el,
e l - l , gi, e i + l , . • • , en).
e2 . . . . .
(10) The proof of the following proposition straight-forward and hence omitted.
is
f u z z y s u b g r o u p It o n G = G1 × G2 × • • • × G~ can be written as the direct p r o d u c t I~ = I~ × I~2 × • • • × I~n w h e r e each [.~i is the p r o j e c t i o n o f It o n
G,. Proof. Let k l , k z . . . . . kn be the orders of groups G 1 , G 2 . . . . , Gn respectively and /~ be a fuzzy subgroup on G = G , x ( 3 2 × . . - x G n . Then k i ' s a r e pairwise co-prime and hence ki is co-prime with any product of kj's different from k~. In view of Proposition 5.5, it suffices to show that for any element (am, a2, . . . , an) of G, t,ti(ai) >1/,t(al, a 2 , . . .
Proposition 5.4. F o r all i = 1, 2 . . . . .
n, the p r o j e c t i o n I~i o f a f u z z y n o r m a l s u b g r o u p t~ o f a direct p r o d u c t G1 × G2 × • • • × Gn o n Gi is also a f u z z y n o r m a l s u b g r o u p o f Gi.
Proposition 5.5. L e t t~ be a f u z z y s u b g r o u p o f the direct p r o d u c t G1 × Gz × • • • × Gn a n d let I~i be the p r o j e c t i o n o f It o n Gi f o r each i = 1 , 2 . . . . . n. T h e n l~ >~ tz~ × ltz × . . . × t~n. Proof. For
any
element
(al, a2,...
,an)
of
(11)
To avoid complicating the notation, we prove (11) for i = 2, say. From (10) we have ~2(a2) = ~(el,
Even in case of ordinary subgroups, a subgroup S of the direct product G1 × (32 × • - - × Go is not equal to the direct product SI×S2×...×Sn of the projections of S in general (Example 4.2 of [18]) but S is the super set of S I × S z × ' " × S ~ . This is also the case with fuzzy subgroups. The p r o o f of the following result is contained in the p r o o f of T h e o r e m 4.1 of [18], we just outline it here.
, an)
for all i = 1, 2, . . . , n.
en).
a 2 , e3 . . . . .
Since k~ and k2 are co-prime, there exist integers p, q such that p k t + qk2 = 1. Then/~2(a2) can be written as #2(az) = l~(a pk', a pk'+qk2, e3 . . . . . = / t ( a P l k', a 2pkl, e3 . . . .
en)
, en)
=/u(al, a2, e3 . . . .
, en) pk'
/> ~ ( a j , a2, e3 . . . . .
e.).
Consider now numbers k l , k2 and k 3 which are co-prime and hence there exist integers P l, q l such that p l k l k 2 + q~k3 = 1 and therefore ~ ( a l , a2, e3 . . . . .
en)
= I~( ap'k'kz+qlk~, aP'ktkz+qlk~ , a 3qlk~ , e4, . . . , en)
GI × Gz × " " × Gn,
= t~(a q'k~, a q'k~, a q~k3, e4 . . . . .
I~(a,,
=/~(al,
a 2 , a 3 , e4 . . . . .
en) q'k3
~(al,
a2, a3, e4 . . . . .
en).
a2 . . . . .
a,,)
= ~ ( ( a l , e2, e 3 , . . . , "'"
(el, ez,. ..,
e n ) ( e , , a2, e3 . . . .
Thus
en-1, an))
i> min(/~l(aO, u2(a2) . . . . = ]'~1 X /"~2 X • " " X ~ n ( a l , Hence
, e,,)
en)
, I~n(a,)) a 2 , . . . , an)
# i> ~ 1 × /A2 × " " " X ~n"
The following result shows when the equality /~ = / ~ × #2 x • • • ×/~n holds in the above proposition, and is stronger than T h e o r e m 4.1 of [18].
Theorem 5.6. I f G1, G2 . . . . . Gn are f i n i t e g r o u p s h a v i n g p a i r w i s e c o - p r i m e orders, then a
/~2(a2)/>/~(al, a2, a3, e, . . . . .
e,).
Considering the co-primeness of k~, k z , k3 and k 4 w e can show as above that /~2(a2) i>/~(al, a2, a3, a4, e5 . . . .
, e,).
Repeating the same argument we see that /~2(a2)/>/~(al, a2, • . • , an). Similarly, we can show (11). In the above case we can m a k e the following
LJ. Kumar et al. / Fuzzy normal subgroups and fuzzy quotients
more general observation. Let g ~ G be any element having non-identity arguments exactly at j places (say) m~, m2 . . . . . mj ( j < n ) and also assume that I is a positive integer ( l < n ) different from all m~'s. If g' is the element of G having all arguments as that of g except the l-th which has been replaced by any x~4:e~ of G~, then ~ ( g ) > - t ~ ( g ' ) . This follows as in the proof above by considering co-primeness of k .... km~, .... k,,, and k~ and replacing pk (i) am, by a,,, ~.k. . .""k . . i n + q k t for all i = 1, 2 . . . . . j,
(ii) ez by x qk', in g where p, q are such that pkm,k,~ 2 • • • kin, + qk~ = 1.
Corollary 5.7 (Theorem 4.1 of [18]). E v e r y f u z z y s u b g r o u p u n d e r M (rain) o f a finite n u m b e r o f cyclic g r o u p s o f distinct p r i m e p o w e r orders can be written as a direct p r o d u c t o f f u z z y subgroups o f those cyclic groups. It is shown by Example 4.2 in [18] that Theorem 5.6 does not hold if all groups Gi's are cyclic of order p (prime). But in this case /~ is isomorphic to some direct product of fuzzy subgroups /~i's (not projections of /a) on some cyclic groups of order p. This observation is proved in T h e o r e m 4.4 of [18] by using the concepts of measures and a-algebras of subsets and the Hausdorff Maximal Principle. We prove this result by using the concepts of or-cuts and Theorem 3.3. T h e o r e m 5.8. L e t p be a p r i m e a n d f o r each i = 1, 2 . . . . . k, let Gi be the cyclic g r o u p o f order p. L e t G = G~ x Gz x . . . x G~ and l~ be a f u z z y s u b g r o u p o f G. Then, f o r each i = 1, 2 . . . . . k, there is a f u z z y s u b g r o u p Iti o f s o m e cyclic g r o u p o f order p such that tt is i s o m o r p h i c to the direct p r o d u c t I~ x i~2 x • • • x t~.
Proof. Let # be a fuzzy subgroup of G. Since G is of order p~, ~ has at most k + 1 steps in its graph (Theorem 3.3) since G can have a chain of subgroups of the form (4) of length k at most. In other words the set S = {or e [0, 1]: /~(x) = cr for some x e G - { e } } , which is nothing but the 'range' of/~ restricted to G - {e}, has at most k distinct numbers from [0, 1]. Choose c~ to be the largest number from S (maybe 1). Define cri's inductively for all i = 2, 3 . . . . . k as follows. If
131
the ~-cut ~ , of /~ is a subgroup of order p', choose 0(i+1 to be the next lower number (i.e. the highest amongst the numbers not considered so far) from S. Otherwise, define c~+l = or,. This way we get k numbers c~l~>O~e~>'"~>% corresponding to the given /~. Now choose an element al in the crl-cut crw and form the subgroup H = (a~) of order p (here ( S ) denotes the subgroup of G generated by the subset S of G). Consider a2e or2, such that a 2 ¢ H ~ and define He = (a2). Since every element of a cyclic group of order p is a generator, H~ A//2 = {e} and H 1 . H 2 = H I ® H 2 is the internal direct product decomposition of the group H~H2. Consider as in 013, such that a 3 ~ H i ' H e and construct H 3 = ( a 3 ) . If H i = ( a i ) has been defined, choose a~+~o~i+~, such that ag+~¢ HI®H2®'"®Hi and form H~+~= (a,+l) (it can be seen that such an a,+~ will always exist due to the suitable choice of cry's). This gives an internal direct product decomposition (12)
G = HI ® H2 ® " " " ® I l k .
Let /~i be the restriction of /~ on /4,. for all i = 1, 2 , . . . , k, then since Hi is cyclic of order p, /ui(g) =/~(g) = cri for all g ~ Hi, g 4= e. We claim that these /~i's are the ones we are looking for. We know that the internal direct product (12) is isomorphic to the external direct product H = H1 x H2 x . . . x Hk
treating the H~'s as abstract groups. Thus the mapping O : G--e HI x H2 x . . . x Hk = H
defined by O(g) = (h~, h2 . . . . .
hk),
where g = h r . h 2 - . - h k is the unique representation of g in the internal direct product, is an isomorphism. Now /~ z/~2 x . . . x Pk is defined on H and for g 6 G, x
=
x.
. . x
~'~1 X ~'~2 X
" " " X
O(g) Irk(hi, h2 . . . . .
= min(/~(h,), ~2(h2) . . . . . = min(/~(h0,/~(h2) . . . . .
/~(h,)) /~(h~))
1 if hi = e for all i, cri if i is the largest index such that hi :~ e.
hk)
132
l.J. Kumar et al. / Fuzzy normal subgroups and f u z z y quotients
If hi = e for all i, t h e n g = e a n d s o / ~ ( g ) = 1. If i is t h e first n u m b e r in { 1 , 2 . . . . . k} such t h a t hi 4: e, t h e n g = h l h 2 • • • hi H I H 2 " " • Hi - H , H 2 " • " H i _ , .
Since H I H 2 " • • H i is a s u b g r o u p o f o r d e r p i a n d HIH2." Hi_~ is a s u b g r o u p o f o r d e r p i - ~ a n d also b e c a u s e t h e r e can be no s u b g r o u p o f G in b e t w e e n t h e s e t w o , it follows t h a t g is c o n s t a n t on 111142" • • H i - H , H 2 . ' ' H i - , . B u t /~(ai) = or, a n d h e n c e ~ ( g ) = o6 in this case. T h u s w e get = (/~1 X ~ 2 X " • " X / ~ k ) o 0
w h e r e / ~ i is a fuzzy s u b g r o u p o f a cyclic g r o u p / 4 , . o f o r d e r p , a n d t h e p r o o f is c o m p l e t e .
Acknowledgement T h e a u t h o r s t h a n k t h e u n k n o w n r e f e r e e for h i s / h e r v a l u a b l e c o m m e n t s which h e l p e d in i m p r o v i n g this p a p e r .
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