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Fuzzy equivalence relation redefined
sets and systems ELSEVIER
Fuzzy Sets and Systems 79 (1996)227-233
Fuzzy equivalence relation redefined K.C. Gupta*, R.K. Gupta
Department of Mathematics, Universityof Delhi, Delhi-llO007, India Received October 1993; revisedJanuary 1994
Abstract
In this note a weak fuzzy equivalence relation is defined generally, superseding most of the established results. The concept is discussed in the setting where the min-transitivity of a fuzzy relation is replaced by the general t-norm transitivity.
Keywords: Fuzzy equivalence relation; Fuzzy partition; Complete lattice; F-indistinguishability operator.
1. Introduction
In this paper we have proposed a generalized definition of a fuzzy reflexive relation on a set. In [10] Zadeh proposed the definition of a fuzzy relation from X to Y as a fuzzy subset of X x Y. In [7] Nemitz has studied lattice-valued fuzzy relations assuming values from a Brouwerian lattice. For various types of reflexive fuzzy relations the reader is referred to [2]. The standard definition of a fuzzy reflexive relation # in X demands /~(x, x) = 1, which we have seen to be too strong. We have proposed positive values for all /~(x, x) and ~t(u, v) ~< #(x, x) for all u ~ v, and x in X. We have shown that with this definition of a fuzzy reflexive relation the redefined fuzzy equivalence relation supersedes most of the theorems proposed by Murali in [6]. It is seen that the set E(X) of all
*Corresponding author.
fuzzy redefined equivalence relations in X, instead of being complete lattice, turns out to be a superset of a countable union of an increasing chain of complete lattices. Lastly, we have discussed applicability of our redefined fuzzy equivalence relation in the setting where the min-transitivity of a fuzzy relation is replaced by the general t-norm transitivity.
2. Preliminaries
We recall some preliminary definitions and resuits.
2.1. Definition I-5]. A fuzzy relation # in a set X is a fuzzy subset of X x X. /~ is reflexive in X /f p(x,x) = 1 and ~ is symmetric in X if p ( x , y ) = /~(y, x) for all x, y in X.
0165-0114/96/$15.00 © 1996- ElsevierScienceB.V. All rights reserved SSDI 0165-01 14(95)00155-7
K.C. Gupta, R.K. Gupta / Fuzzy Sets and Systems 79 (1996) 227-233
228
The sup--rain composition 2 op of two fuzzy relations 2, p in X is the fuzzy subset of X x X defined by (2o/~)(x,y) = sup min{2(x, t),p(t,y)},
x, y ~ X.
rEX
The composition 2 o p of two fuzzy relations is associative; see [5]. A fuzzy relation p in X is transitive in X if p o p _ #. If p is a fuzzy relation in X which is reflexive, symmetric and transitive, p is called a fuzzy equivalence relation or a relation of similitude in X. 2.2. Theorem [1]. I f P is a partially ordered set with unit element and if every non-empty subset of P has a greatest lower bound in P (or dually), then P is a complete lattice. 2.3. Theorem [1]. Let y = f ( x ) be any orderpreserving mapping from a complete lattice L to itself. Then there exists a ~ L such that f(a) = a. 2.4. Definition. Let f be a mapping from the set X to the set Y. If # is a fuzzy subset of Y, the inverse imagef - 1(#) o f p is the fuzzy subset of X defined by
f - l ( p ) ( x ) = p(f(x)),
x e S.
If 2 is a fuzzy subset of X, the image f(2) of 2 is the fuzzy subset of Y defined by f(2)(y) =
sup
2(t),
iff-l(y) ¢0
t~f-~(y)
= 0
i f / - ~(y) = O.
We have in general (a) f ( f - l ( p ) ) 2 _ f - x (f(2)).
___p, and (b)
3. Fuzzy G-equivalence relation 3.1. Definition. A fuzzy relation p in X is G-reflexive iff
(1)
p(x,x) > O,
(2)
p(x, y) ~< inf p(t, t)
A fuzzy relation p in X is a G-equivalence relation in X if p is G-reflexive, symmetric and transitive in X. If a fuzzy relation p in X is G-reflexive and transitive, p is called a G-preorder in X. By pn, n = 1 , 2 , 3 , . . . , w e mean pop . . . . . p (n factors). 3.2. Theorem. If p is a fuzzy G-preorder in X, then p" = p , n = 1,2,3 . . . . . Proof. Due to transitivity, we have p o p _ p. On the other hand, (p o p)(x, y) = sup min{p(x, t), p(t, y)} t~X
>~ min {p(x, x), p(x,y)} = p(x, y),
for all x, y e X.
Therefore p o/~ = p. The rest is easy.
3.3. Definition. Let p be a G-equivalence relation in X and let 6 = infx~x p(x,x). We define the following two level relations in X: (a) If 0 ~< ~ ~< 6, define the relation F~ in X by xF~y iff p(x, y) >>.c~. (b) If 0 ~< ~ < 6, define the relation G~ in X by xG~y iff p(x, y) > ~. 3.4. Theorem. Let p be a fuzzy G-equivalence relation in X. (a) For each ~ s [0,6], F~ is an equivalence relation in X. (b) For each ~ ~ [0, 6 [, G, is an equivalence relation in X. Proof. (a) Let 0 ~< ~ ~< 6. Consider F~. For x ~ X, #(x, x) ~> 6 >~ ~ and so xF~x. Next, let xF~y. Then #(x, y) ~> ~, which implies p(y, x) >/~, and so yF~x. Finally, let xF~y, and yF~z. Then we have
p(x,z) >1(po p)(x,z) for all x ¢ y in X.
tEX
Another generalized definition of reflexivity was given by Yeh 19]. Yeh called a fuzzy relation p in X e-reflexive if p(x,x) >~ e > 0 for all x in X and weakly reflexive iff p(x, x) ~> p(x, y) for all x, y in X.
[]
= sup min{p(x,t),p(t,z)} teX
t> min {p(x, y), p(y, z) } >/~,
and
xF~z.
Similarly (b) can be proved.
[]
K.C. Gupta, R.K. Gupta / Fuzzy Sets and Systems 79 (1996) 227-233
4. Partition of a fuzzy G-equivalence relation
229
4.4. Theorem. With notations as in (4.3), p is a G-
equivalence relation in X. 4.1. Definition. Let 2 be a nonzero fuzzy subset of X. A family {2~} of fuzzy subsets of X is a partition of 2 if: (1) 2~ and 2~ are disjoint, whenever i 4:j, and (2) Ui2i = 2. If/~ is a G-equivalence relation in X, st can be decomposed by a partition as follows. Let a ~ X and let [a] = {x e X: st(a,x) > 0}. It is easy to verify that {[a]: a ~ X} is a partition of X. For a ~ X, define the fuzzy relation % in X as follows:
%(x,y) = st(x,y), =0,
Proof. Firstly, let x e X. Then st(x, x) = ~ix(x) > 0.
Next, let x 4= y ~ X. Then
st(x,y) = c~ ~< 6 = inf st(x,x). x~X
So St is G-reflexive in X. Secondly, by definition, st(x, y) = st(y, x). Finally, (st o st)(x, x) = sup min {st(x, t), st(t, x)} reX
if (x,y) e [a] x [a.]
<<.st(x,x),
if(x,y)q~[a]x[a].
We call {O'a: a e X} the G-decomposition of #. 4.2. Theorem. The G-decomposition of a G-equiva-
If x 4: y, we have (st o st)(x, y) -- sup rain {st(x, t), st(t, y)} t~X
lence relation # in X is a partition of st. Proof. Firstly, let a, 4: oh. Therefore, [a] 4: [b]. Let tTa(X,y ) > 0. Then x,y E [a]. Therefore, x,y are not in [b]. Consequently, odx, y) = 0. Similarly, if odx, y) > 0 , then o , ( x , y ) = 0 . Therefore, {%} is pairwise disjoint. Secondly, let x,y e X. If st(x,y)=O, then (x,y)¢[a]×[a] for all a ~ X . Therefore, o , ( x , y ) = 0 for all a e X , and (O,~x %)(x,y) = 0 = st(x,y). If st(x,y) > 0, then (x,y)~ Ix] x [x.] and ax(X,y) = st(x,y). If at 4: try, then o, and tr~ are disjoint and odx, y) = 0. Consequently, we get
(~?x % )(x, y) = a~(x, y) = st(x, y). Therefore, we get st = U,~x G.
[]
4.3. On the other hand, let 2 be a fuzzy subset of X such that supp(2) = X. If {2i} is a partition of 2, we construct a G-equivalence relation in X as follows. Let a ~ X. Then there exists unique 2i, such that };i.(a) > O. Let 6 = inf,~x 2io (a). Define the fuzzy subset st of X x X as follows: (a) st(x,x) = 21~(x) (b) s t ( x , y ) = a for all x 4=y in X, where ~ is a fixed number in [0, 6].
x e X.
= ~ = st(x, y).
Therefore, st °st ~ st. Consequently, st is a G-equivalence relation in X. []
5. Chain of complete lattices and fixed points It is already known that [6] the set of all fuzzy equivalence relations in X, where reflexivity is defined by st(x,x) = 1 for all x in X, forms a complete lattice under the usual order on fuzzy sets. This is not true when we consider E(X), the set of all G-equivalence relations in X. However, we show below that E(X) can be realized as a superset of the countable union of a chain of complete lattices. 5.1. Theorem. E(X), the set of all fuzzy G-equiva-
lence relations in a set X, contains a countable union of strictly increasing chain of complete lattices. Proof. Given a positive integer n, define
E1/"(X) = { 2 ~ E(X): -I ~< 2(x,x) for all x E X } . n
Clearly,
El(X) c E1/2(X) c Ells(X) ~ ...,
~j~: l E . . ( X )
~ E(X).
and
K.C Gupta, R.K. Gupta / Fuzzy Sets and Systems 79 (1996) 227-233
230
Let n be a fixed positive integer. We show that
E1/.(X) is a complete lattice. The fuzzy relation I on X defined by I(x, y) = 1 for all x, y E X, is the greatest m e m b e r of E(X). Clearly 1 ~ E1/.(X) and I is the unit element of E1/.(X). Now, let {#i} be a n o n e m p t y subset of E~/,(X). Let # = Ai#i. We show that # e E~/,(X). Firstly, let x e X. Then
1
#(x, x) = inf #i(x, x)/> - > 0. i
Then there exists a sequence {#,} in E(X) such that (1)
#. e E1/,(X), and
(2)
f ( # , ) = #,.
Proof. Let f. =fIE1/,(X). Then f . is an orderpreserving mapping from the complete lattice EI/.(X) into E1/.(X). By (2.3) there exist #, ~ E1/.(X) such that f,(#,) = #,, and this implies t h a t f ( # , ) = #,. []
n
Next, let x ¢ y ~ X, and let 6 = inft~x #(t, t). Define 6i = inft~x#i(t, t). We then have
6. Balanced mappings
#(x, y) = inf#i(x, y) ~< inf 6i
We now find the conditions under which the concept of a G-equivalence relation is preserved by the image and the preimage of a mapping connecting two sets X and Y.
i
i
=inif{inf#,(t,t)}=inf{inf#i(t,t)
}
6.1. Definition. Let X and Y be two n o n e m p t y sets. A mapping f : X × X ~ Y x Y is called a balanced mapping if
= i n f # ( t , t ) = 6. t
Thus # is G-reflexive in X. Secondly, if x, y e X, we have #(x, y) = inf #~(x, y) = inf #~(y, x) = #(y, x), i
i
(1) f ( x , y ) = ( u , u ) =~ x = y, (2) f(x,y) = (z,w) => f(y,x) = (w,z), (3) f ( x , x ) = (u,u) and f(y,y) = (v,v) =~ f(x,y) = (u,v), for all x , y ~ X and u,v,w, z e Y. The following results can easily be verified. (a) Given x ~ X, there exists u e Y such that
and # is symmetric. Finally, if x, y s X, we have (#o #)(x, y) = sup min {#(x, t), #(t, y) } t~X
=supmin{inf#i(x,t),inf#i(t,y)}
f(x, x) = (u, u). (b) If f(x,y) = (u,v), then f(x,x) = (u,u), and f(y, y) = (v, v). (c) Let f(x, y) = (u, v). Given z • X, there exists t~ ~ Y such that f(x,z) = (u,t~) and f(z,y) = (tz, v). (d) f is a one-to-one mapping from X x X into
= sup (inf min {#i(x, t ), #i(t, y) } )
Y×Y.
<<,inf (sup min {#i(x, t ), #i(t, y) }
6.2. Example. Hence is an example of a balanced mapping. Let X = {a,b,c} and Y = {1,2,3,4}. Define the m a p p i n g f f r o m X × X to Y × Y as follows:
= inf (#io #i)(x, y) ~< inf #i(x, y) i
i
= #(x,y).
Consequently, # is a G-equivalence relation in X, which lies in EI/.(X). By Definition 2.4, E1/,(X) is a complete lattice. []
5.2. Theorem. Let f be an order-preserving mapping
from E(X) to E(X) such that f(E1/.(X)) c_ E1/.(X).
f(a,a) = (1, 1),f(b,b) = (2,2),f(c,c) = (3,3), f(a, b) = (1, 2),f(b, a) = (2, 1),f(a, c) = (1, 3), f(c, a) = (3, 1),f(b, c) = (2, 3),f(c, b) = (3, 2). 6.3. Theorem. Let f be a balanced mapping from
X x X into Y x Y and let # be a fuzzy G-equivalence relation on Y. Then f - 1 ( # ) is a G-equivalence relation in X.
K.C. Gupta, R.K. Gupta / Fuzzy Sets and Systems 79 (1996) 227-233
Proof. Firstly, let x ~ X. Then f-l(~)(X,X)
= # ( f ( x , x ) ) = I~(u,u) > O,
for some u e Y. Next, take x ~ y ~ X. Let 6 = i n f x ~ x f - l ( # ) ( x , x ) , and let a = inft~y #(t, t). Then we have 6 = inf # ( f ( x , x ) ) = inf/l(tx, tx), where tx E Y, x~X
x~X
= # ( f ( x , y)) = #(u,v) <<.tr ~ 6,
where u # v e Y . reflexive. Secondly,
Consequently, f - l ( g )
is G-
= # ( f ( x , y ) ) = # ( f ( y , x)) for all x, y e X .
Finally, let x, y e X, and let f ( x , y ) = (u,v). Then we have
and so f(),) is symmetric. Finally, let (u, v) ~ Y × Y, and let f ( x , y) = (u, v). Then (f(),) of(2))(u, v) = sup min {f(),)(u, w),f(2)(w, v)}
a (#))(x,y)
weY
= sup min { f - 1 ( / ~ ) ( x , z ) , f - l ( # ) ( z , y ) } z~X
= sup min { f ( ) , ) ( f ( x , t~)),f(),)(f(tw, y))} w~Y
= sup min {#(f(x, z)), # ( f ( z , y))} z~X
= sup rain {2(x, t~), ),(t~, y) } w~Y
= sup min{#(u, tz), ~t(tz,v)},
by 6.1(c)
zEX
~< sup min{),(x, t), ),(t,y)} tEX
~< sup min {#(u, t), #(t, v)} t~Y
= (#o#)(u,v) <~ I~(u,v) = ~t(f(x,y)) = f - i (/t)(x, y), and f - 1(#) is transitive. Consequently, f - 1 (#) is a G-equivalence relation in X. [] 6.4. Theorem. Let f be a balanced mapping from X x X onto Y × Y. If). is a G-equivalence relation on X , then f(),) is a G-equivalence relation on Y. Proof. If (u, v)~ Y x Y, then there exists unique (x,y) ~ X x X such that f ( x , y ) = (u, v); 6.1(d). We therefore get f(2)(u,v) =
We then havef(2)(u,v) = ),(x,y) ~< 6 ~< a. We have proved that f(),) is G-reflexive in Y. Secondly, let f ( x , y) = (u, v). T h e n f ( y , x) = (v, u). Therefore, f(),)(u, v) = 2(x, y) = ),(y, x) = f(),)(v, u),
=f-l(ll)(y,x)
1(~) o f -
wheref(tw, tw) = (w,w)
t~X
Therefore, we get
(f-
= inf ),(t~,tw), ~> inf ),(t, t) = 6.
t~Y
f-l(#)(x,y)
Firstly, let u ~ Y, and let f ( x , x) = (u, u). T h e n f ( ) , ) ( u , u ) = 2(x,x) > 0. Next, let u :~ v ~ Y, and l e t f ( x , y ) = (u, v). Then x~y. Let 6 = inft,x ),(t, t) and let a = i n f ~ r f ( 2 ) ( w , w). Then we have We Y
>/inf #(t, t) = a.
f-l(#)(x,y)
231
sup (t,r)~ f-~(u,v)
2(t,r) = 2(x,y).
= (2 o ),)(x, y) ~< ),(x, y) = f(),)(u, v), and so f(),) is transitive. Consequently, f(),) is a Gequivalence relation in X. []
7. GF-indistinguishability operators Let us now generalize the concept of a G-equivalence relation by replacing the minimum function occurring in the definition of a transitive fuzzy relation with a continuous triangular norm (tnorm) F as follows. 7.1. Definition. Let X be a n o n e m p t y set and F be a continuous t-norm on [0,1]. A mapping 2 from X × X into [0, 1] is a GF-indistinguishability
232
K.(2 Gupta, R.K. Gupta / Fuzzy Sets and Systems 79 (1996) 227-233
operator on X iff the following three conditions hold for any x, y, z in X: (1) 2 is G-reflexive in X. (2) 2 is symmetric in X. (3) 2 is F-transitive in X; in other words, F(2(x, z), 2(z, y)) ~< 2(x, y). 7.2. Lemma. Let {2i}i~t be a family of GF-indistinguishability operators on X satisfying the condition that, given x e X, there exists ex > 0 such that ex <~hi(x, x) for all i t I. Then 2 = ('] 2i ieI
is a GF-indistinguishability operator on X. Proof. For the G-reflexivity of 4, let x e X. Then 2(x, x) = inf/,x hi(x, x)/> ex > 0. Let 61 = infx,x 2i(x,x), and let 6 = infi,t 6i. We then get inf 2 ( x , x ) = i n f ( i n f 2 , ( x , x ) ) xEX
xeX \i~l
= inf(iel \xexinf 2i(X,X))=infbi=b.iei If X ~ y in X, then 2(x,y) = inf2i(x,y) ~ inf6i = 6. iEl
iel
We have thus proved that 2 is G-reflexive in X. That 2 is symmetric and F-transitive in X can be seen from [8, proof of Proposition 3.1]. [] 7.3. Theorem. Let # be a G-reflexive and symmetric fuzzy relation in X. Then Gu, the set of all GFindistinguishability operators 2 on X such that # c_ 4, is nonempty and
#F(x,y) = inf 2(x,y),
x, y e X ,
By Lemma 7.2, #v is then a GF-indistinguishability operator on X containing #, which is clearly the least. [] In [8] Velverde proved that a fuzzy relation 2 in X is an F-indistinguishability operator on X iff there exists a family {h~}j~s of fuzzy subsets of X such that
2(x, y) = inf P(max{hj(x), hj(y)}l(min{hj(x), hi(y))). jeJ
for all x, y in X, where P is the so-called quasiinverse of the continuous t-norm F defined as
F(xFy) = sup{~ e [0, 1]: F(a,x) <~y}. This result cannot be generalized if 2 is a GFindistinguishability operator, because we may not always have 2(x, x) = 1. However, we can partition EF(X), the class of all GF-indistinguishability operators on X, in such a way that each equivalence class is represented by a unique F-indistinguishability operator on X. The following lemma is useful in that direction. 7.4. Lemma. Let 2 be a symmetric and F-transitive fuzzy relation in X. I f p(x,y) = 2(x,y)for all x ~ y in X and p(x,x) = 1 for all x in X, then p is an F-indistinguishability operator on X. Proof. Clearly, p is reflexive and symmetric. Let x, y, z e X. We show that
F(p(x,z),p(z, y)) <<.p(x, y). If x = y, the inequality follows. If either z = x or z = y, we get equality. So, let x ~ z ~ y ~ x. Then
A~G~
is the least GF-indistinguishability operator on X containing #. Proof. The fuzzy relation l(x, y) = 1 for all x, y e X is the greatest GF-indistinguishability operator 2 on X containing #. Hence Gu is nonempty. Since # is G-reflexive, given x e X, we have 0 < #(x, x) ~< 2(x, x)
for all 2 e Gu.
F(p(x, z), ptz, y)) = F(2(x, z), 2(z, y)) <~ 2(x, y) = p(x, y), and so p is F-transitive.
[]
Now, let 2, # e EF(x). We define 2 ,-~# if and only if 2(x,y) = #(x,y), whenever x ~ y in X. If p(x,x) = 1 for all x in X and p ( x , y ) = 2(x,y)
K.C. Gupta, R.K. Gupta / Fuzzy Sets and Systems 79 (1996) 227-233
whenever x # y in X, then, by Lemma 7.4, p is the unique F-indistinguishability operator on X such that p e [~.], the equivalence class containing 2. We now formulate our assertion as follows.
7.5. Theorem. Let 2 be a fuzzy relation in X. Then 2 is a GF-indistinguishability operator on X if and only if there exists a family {hj}~j offuzzy subsets of X such that inf F(max{hj(x), hi(y)}lmin{h~(x), hi(y)}), x, y e X, jeJ
is the unique F-indistinguishability operator on X representing the class of 2. []
Acknowledgements The authors take this opportunity to thank the referee who brought to their notice the ideas already proposed by R.T. Yeh and L. Velverde.
233
References [1] G. Birkhoff, Lattice Theory, 3rd edn., American Mathematical Society Colloquium Publications, Vol. XXV (AMS, Providence, R. I., 1984). I~2] M. Chakraborty and S. Das, On fuzzy equivalence 1, Fuzzy Sets and Systems 11 (1983) 185-193. [3] M. Chakraborty and S. Das, On fuzzy equivalence 2, Fuzzy Sets and Systems 12 (1983) 299-307. [4] M. Dubois and H. Prade, Fuzzy Sets and Systems (Academic Press, New York, 1980). [5] A. Kaufmann, Introduction to the Theory of Fuzzy Subsets, Vol. 1 (Academic Press, New York, 1975). [6] V. Murali, Fuzzy equivalence relations, Fuzzy Sets and Systems 30 (1989) 155-163. I-7] W.C. Nemitz, Fuzzy relations and fuzzy functions, Fuzzy Sets and Systems 19 (1986) 177-191. [8] L. Velverde, On the structure of F-indistinguishability operators, Fuzzy Sets and Systems 17 (1985) 313-328. 1-9] R.T. Yeh, Toward an algebraic theory of fuzzy relational systems, Proc. Int. Conyr. Cybern,, Namur, Belgium (1973) 205-223. 1-10] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353. [11] L.A. Zadeh, Similarity relations and fuzzy orderings, Inform. Sci. 3 (1971) 177-200.
Fuzzy Sets and Systems 79 (1996)227-233
Fuzzy equivalence relation redefined K.C. Gupta*, R.K. Gupta
Department of Mathematics, Universityof Delhi, Delhi-llO007, India Received October 1993; revisedJanuary 1994
Abstract
In this note a weak fuzzy equivalence relation is defined generally, superseding most of the established results. The concept is discussed in the setting where the min-transitivity of a fuzzy relation is replaced by the general t-norm transitivity.
Keywords: Fuzzy equivalence relation; Fuzzy partition; Complete lattice; F-indistinguishability operator.
1. Introduction
In this paper we have proposed a generalized definition of a fuzzy reflexive relation on a set. In [10] Zadeh proposed the definition of a fuzzy relation from X to Y as a fuzzy subset of X x Y. In [7] Nemitz has studied lattice-valued fuzzy relations assuming values from a Brouwerian lattice. For various types of reflexive fuzzy relations the reader is referred to [2]. The standard definition of a fuzzy reflexive relation # in X demands /~(x, x) = 1, which we have seen to be too strong. We have proposed positive values for all /~(x, x) and ~t(u, v) ~< #(x, x) for all u ~ v, and x in X. We have shown that with this definition of a fuzzy reflexive relation the redefined fuzzy equivalence relation supersedes most of the theorems proposed by Murali in [6]. It is seen that the set E(X) of all
*Corresponding author.
fuzzy redefined equivalence relations in X, instead of being complete lattice, turns out to be a superset of a countable union of an increasing chain of complete lattices. Lastly, we have discussed applicability of our redefined fuzzy equivalence relation in the setting where the min-transitivity of a fuzzy relation is replaced by the general t-norm transitivity.
2. Preliminaries
We recall some preliminary definitions and resuits.
2.1. Definition I-5]. A fuzzy relation # in a set X is a fuzzy subset of X x X. /~ is reflexive in X /f p(x,x) = 1 and ~ is symmetric in X if p ( x , y ) = /~(y, x) for all x, y in X.
0165-0114/96/$15.00 © 1996- ElsevierScienceB.V. All rights reserved SSDI 0165-01 14(95)00155-7
K.C. Gupta, R.K. Gupta / Fuzzy Sets and Systems 79 (1996) 227-233
228
The sup--rain composition 2 op of two fuzzy relations 2, p in X is the fuzzy subset of X x X defined by (2o/~)(x,y) = sup min{2(x, t),p(t,y)},
x, y ~ X.
rEX
The composition 2 o p of two fuzzy relations is associative; see [5]. A fuzzy relation p in X is transitive in X if p o p _ #. If p is a fuzzy relation in X which is reflexive, symmetric and transitive, p is called a fuzzy equivalence relation or a relation of similitude in X. 2.2. Theorem [1]. I f P is a partially ordered set with unit element and if every non-empty subset of P has a greatest lower bound in P (or dually), then P is a complete lattice. 2.3. Theorem [1]. Let y = f ( x ) be any orderpreserving mapping from a complete lattice L to itself. Then there exists a ~ L such that f(a) = a. 2.4. Definition. Let f be a mapping from the set X to the set Y. If # is a fuzzy subset of Y, the inverse imagef - 1(#) o f p is the fuzzy subset of X defined by
f - l ( p ) ( x ) = p(f(x)),
x e S.
If 2 is a fuzzy subset of X, the image f(2) of 2 is the fuzzy subset of Y defined by f(2)(y) =
sup
2(t),
iff-l(y) ¢0
t~f-~(y)
= 0
i f / - ~(y) = O.
We have in general (a) f ( f - l ( p ) ) 2 _ f - x (f(2)).
___p, and (b)
3. Fuzzy G-equivalence relation 3.1. Definition. A fuzzy relation p in X is G-reflexive iff
(1)
p(x,x) > O,
(2)
p(x, y) ~< inf p(t, t)
A fuzzy relation p in X is a G-equivalence relation in X if p is G-reflexive, symmetric and transitive in X. If a fuzzy relation p in X is G-reflexive and transitive, p is called a G-preorder in X. By pn, n = 1 , 2 , 3 , . . . , w e mean pop . . . . . p (n factors). 3.2. Theorem. If p is a fuzzy G-preorder in X, then p" = p , n = 1,2,3 . . . . . Proof. Due to transitivity, we have p o p _ p. On the other hand, (p o p)(x, y) = sup min{p(x, t), p(t, y)} t~X
>~ min {p(x, x), p(x,y)} = p(x, y),
for all x, y e X.
Therefore p o/~ = p. The rest is easy.
3.3. Definition. Let p be a G-equivalence relation in X and let 6 = infx~x p(x,x). We define the following two level relations in X: (a) If 0 ~< ~ ~< 6, define the relation F~ in X by xF~y iff p(x, y) >>.c~. (b) If 0 ~< ~ < 6, define the relation G~ in X by xG~y iff p(x, y) > ~. 3.4. Theorem. Let p be a fuzzy G-equivalence relation in X. (a) For each ~ s [0,6], F~ is an equivalence relation in X. (b) For each ~ ~ [0, 6 [, G, is an equivalence relation in X. Proof. (a) Let 0 ~< ~ ~< 6. Consider F~. For x ~ X, #(x, x) ~> 6 >~ ~ and so xF~x. Next, let xF~y. Then #(x, y) ~> ~, which implies p(y, x) >/~, and so yF~x. Finally, let xF~y, and yF~z. Then we have
p(x,z) >1(po p)(x,z) for all x ¢ y in X.
tEX
Another generalized definition of reflexivity was given by Yeh 19]. Yeh called a fuzzy relation p in X e-reflexive if p(x,x) >~ e > 0 for all x in X and weakly reflexive iff p(x, x) ~> p(x, y) for all x, y in X.
[]
= sup min{p(x,t),p(t,z)} teX
t> min {p(x, y), p(y, z) } >/~,
and
xF~z.
Similarly (b) can be proved.
[]
K.C. Gupta, R.K. Gupta / Fuzzy Sets and Systems 79 (1996) 227-233
4. Partition of a fuzzy G-equivalence relation
229
4.4. Theorem. With notations as in (4.3), p is a G-
equivalence relation in X. 4.1. Definition. Let 2 be a nonzero fuzzy subset of X. A family {2~} of fuzzy subsets of X is a partition of 2 if: (1) 2~ and 2~ are disjoint, whenever i 4:j, and (2) Ui2i = 2. If/~ is a G-equivalence relation in X, st can be decomposed by a partition as follows. Let a ~ X and let [a] = {x e X: st(a,x) > 0}. It is easy to verify that {[a]: a ~ X} is a partition of X. For a ~ X, define the fuzzy relation % in X as follows:
%(x,y) = st(x,y), =0,
Proof. Firstly, let x e X. Then st(x, x) = ~ix(x) > 0.
Next, let x 4= y ~ X. Then
st(x,y) = c~ ~< 6 = inf st(x,x). x~X
So St is G-reflexive in X. Secondly, by definition, st(x, y) = st(y, x). Finally, (st o st)(x, x) = sup min {st(x, t), st(t, x)} reX
if (x,y) e [a] x [a.]
<<.st(x,x),
if(x,y)q~[a]x[a].
We call {O'a: a e X} the G-decomposition of #. 4.2. Theorem. The G-decomposition of a G-equiva-
If x 4: y, we have (st o st)(x, y) -- sup rain {st(x, t), st(t, y)} t~X
lence relation # in X is a partition of st. Proof. Firstly, let a, 4: oh. Therefore, [a] 4: [b]. Let tTa(X,y ) > 0. Then x,y E [a]. Therefore, x,y are not in [b]. Consequently, odx, y) = 0. Similarly, if odx, y) > 0 , then o , ( x , y ) = 0 . Therefore, {%} is pairwise disjoint. Secondly, let x,y e X. If st(x,y)=O, then (x,y)¢[a]×[a] for all a ~ X . Therefore, o , ( x , y ) = 0 for all a e X , and (O,~x %)(x,y) = 0 = st(x,y). If st(x,y) > 0, then (x,y)~ Ix] x [x.] and ax(X,y) = st(x,y). If at 4: try, then o, and tr~ are disjoint and odx, y) = 0. Consequently, we get
(~?x % )(x, y) = a~(x, y) = st(x, y). Therefore, we get st = U,~x G.
[]
4.3. On the other hand, let 2 be a fuzzy subset of X such that supp(2) = X. If {2i} is a partition of 2, we construct a G-equivalence relation in X as follows. Let a ~ X. Then there exists unique 2i, such that };i.(a) > O. Let 6 = inf,~x 2io (a). Define the fuzzy subset st of X x X as follows: (a) st(x,x) = 21~(x) (b) s t ( x , y ) = a for all x 4=y in X, where ~ is a fixed number in [0, 6].
x e X.
= ~ = st(x, y).
Therefore, st °st ~ st. Consequently, st is a G-equivalence relation in X. []
5. Chain of complete lattices and fixed points It is already known that [6] the set of all fuzzy equivalence relations in X, where reflexivity is defined by st(x,x) = 1 for all x in X, forms a complete lattice under the usual order on fuzzy sets. This is not true when we consider E(X), the set of all G-equivalence relations in X. However, we show below that E(X) can be realized as a superset of the countable union of a chain of complete lattices. 5.1. Theorem. E(X), the set of all fuzzy G-equiva-
lence relations in a set X, contains a countable union of strictly increasing chain of complete lattices. Proof. Given a positive integer n, define
E1/"(X) = { 2 ~ E(X): -I ~< 2(x,x) for all x E X } . n
Clearly,
El(X) c E1/2(X) c Ells(X) ~ ...,
~j~: l E . . ( X )
~ E(X).
and
K.C Gupta, R.K. Gupta / Fuzzy Sets and Systems 79 (1996) 227-233
230
Let n be a fixed positive integer. We show that
E1/.(X) is a complete lattice. The fuzzy relation I on X defined by I(x, y) = 1 for all x, y E X, is the greatest m e m b e r of E(X). Clearly 1 ~ E1/.(X) and I is the unit element of E1/.(X). Now, let {#i} be a n o n e m p t y subset of E~/,(X). Let # = Ai#i. We show that # e E~/,(X). Firstly, let x e X. Then
1
#(x, x) = inf #i(x, x)/> - > 0. i
Then there exists a sequence {#,} in E(X) such that (1)
#. e E1/,(X), and
(2)
f ( # , ) = #,.
Proof. Let f. =fIE1/,(X). Then f . is an orderpreserving mapping from the complete lattice EI/.(X) into E1/.(X). By (2.3) there exist #, ~ E1/.(X) such that f,(#,) = #,, and this implies t h a t f ( # , ) = #,. []
n
Next, let x ¢ y ~ X, and let 6 = inft~x #(t, t). Define 6i = inft~x#i(t, t). We then have
6. Balanced mappings
#(x, y) = inf#i(x, y) ~< inf 6i
We now find the conditions under which the concept of a G-equivalence relation is preserved by the image and the preimage of a mapping connecting two sets X and Y.
i
i
=inif{inf#,(t,t)}=inf{inf#i(t,t)
}
6.1. Definition. Let X and Y be two n o n e m p t y sets. A mapping f : X × X ~ Y x Y is called a balanced mapping if
= i n f # ( t , t ) = 6. t
Thus # is G-reflexive in X. Secondly, if x, y e X, we have #(x, y) = inf #~(x, y) = inf #~(y, x) = #(y, x), i
i
(1) f ( x , y ) = ( u , u ) =~ x = y, (2) f(x,y) = (z,w) => f(y,x) = (w,z), (3) f ( x , x ) = (u,u) and f(y,y) = (v,v) =~ f(x,y) = (u,v), for all x , y ~ X and u,v,w, z e Y. The following results can easily be verified. (a) Given x ~ X, there exists u e Y such that
and # is symmetric. Finally, if x, y s X, we have (#o #)(x, y) = sup min {#(x, t), #(t, y) } t~X
=supmin{inf#i(x,t),inf#i(t,y)}
f(x, x) = (u, u). (b) If f(x,y) = (u,v), then f(x,x) = (u,u), and f(y, y) = (v, v). (c) Let f(x, y) = (u, v). Given z • X, there exists t~ ~ Y such that f(x,z) = (u,t~) and f(z,y) = (tz, v). (d) f is a one-to-one mapping from X x X into
= sup (inf min {#i(x, t ), #i(t, y) } )
Y×Y.
<<,inf (sup min {#i(x, t ), #i(t, y) }
6.2. Example. Hence is an example of a balanced mapping. Let X = {a,b,c} and Y = {1,2,3,4}. Define the m a p p i n g f f r o m X × X to Y × Y as follows:
= inf (#io #i)(x, y) ~< inf #i(x, y) i
i
= #(x,y).
Consequently, # is a G-equivalence relation in X, which lies in EI/.(X). By Definition 2.4, E1/,(X) is a complete lattice. []
5.2. Theorem. Let f be an order-preserving mapping
from E(X) to E(X) such that f(E1/.(X)) c_ E1/.(X).
f(a,a) = (1, 1),f(b,b) = (2,2),f(c,c) = (3,3), f(a, b) = (1, 2),f(b, a) = (2, 1),f(a, c) = (1, 3), f(c, a) = (3, 1),f(b, c) = (2, 3),f(c, b) = (3, 2). 6.3. Theorem. Let f be a balanced mapping from
X x X into Y x Y and let # be a fuzzy G-equivalence relation on Y. Then f - 1 ( # ) is a G-equivalence relation in X.
K.C. Gupta, R.K. Gupta / Fuzzy Sets and Systems 79 (1996) 227-233
Proof. Firstly, let x ~ X. Then f-l(~)(X,X)
= # ( f ( x , x ) ) = I~(u,u) > O,
for some u e Y. Next, take x ~ y ~ X. Let 6 = i n f x ~ x f - l ( # ) ( x , x ) , and let a = inft~y #(t, t). Then we have 6 = inf # ( f ( x , x ) ) = inf/l(tx, tx), where tx E Y, x~X
x~X
= # ( f ( x , y)) = #(u,v) <<.tr ~ 6,
where u # v e Y . reflexive. Secondly,
Consequently, f - l ( g )
is G-
= # ( f ( x , y ) ) = # ( f ( y , x)) for all x, y e X .
Finally, let x, y e X, and let f ( x , y ) = (u,v). Then we have
and so f(),) is symmetric. Finally, let (u, v) ~ Y × Y, and let f ( x , y) = (u, v). Then (f(),) of(2))(u, v) = sup min {f(),)(u, w),f(2)(w, v)}
a (#))(x,y)
weY
= sup min { f - 1 ( / ~ ) ( x , z ) , f - l ( # ) ( z , y ) } z~X
= sup min { f ( ) , ) ( f ( x , t~)),f(),)(f(tw, y))} w~Y
= sup min {#(f(x, z)), # ( f ( z , y))} z~X
= sup rain {2(x, t~), ),(t~, y) } w~Y
= sup min{#(u, tz), ~t(tz,v)},
by 6.1(c)
zEX
~< sup min{),(x, t), ),(t,y)} tEX
~< sup min {#(u, t), #(t, v)} t~Y
= (#o#)(u,v) <~ I~(u,v) = ~t(f(x,y)) = f - i (/t)(x, y), and f - 1(#) is transitive. Consequently, f - 1 (#) is a G-equivalence relation in X. [] 6.4. Theorem. Let f be a balanced mapping from X x X onto Y × Y. If). is a G-equivalence relation on X , then f(),) is a G-equivalence relation on Y. Proof. If (u, v)~ Y x Y, then there exists unique (x,y) ~ X x X such that f ( x , y ) = (u, v); 6.1(d). We therefore get f(2)(u,v) =
We then havef(2)(u,v) = ),(x,y) ~< 6 ~< a. We have proved that f(),) is G-reflexive in Y. Secondly, let f ( x , y) = (u, v). T h e n f ( y , x) = (v, u). Therefore, f(),)(u, v) = 2(x, y) = ),(y, x) = f(),)(v, u),
=f-l(ll)(y,x)
1(~) o f -
wheref(tw, tw) = (w,w)
t~X
Therefore, we get
(f-
= inf ),(t~,tw), ~> inf ),(t, t) = 6.
t~Y
f-l(#)(x,y)
Firstly, let u ~ Y, and let f ( x , x) = (u, u). T h e n f ( ) , ) ( u , u ) = 2(x,x) > 0. Next, let u :~ v ~ Y, and l e t f ( x , y ) = (u, v). Then x~y. Let 6 = inft,x ),(t, t) and let a = i n f ~ r f ( 2 ) ( w , w). Then we have We Y
>/inf #(t, t) = a.
f-l(#)(x,y)
231
sup (t,r)~ f-~(u,v)
2(t,r) = 2(x,y).
= (2 o ),)(x, y) ~< ),(x, y) = f(),)(u, v), and so f(),) is transitive. Consequently, f(),) is a Gequivalence relation in X. []
7. GF-indistinguishability operators Let us now generalize the concept of a G-equivalence relation by replacing the minimum function occurring in the definition of a transitive fuzzy relation with a continuous triangular norm (tnorm) F as follows. 7.1. Definition. Let X be a n o n e m p t y set and F be a continuous t-norm on [0,1]. A mapping 2 from X × X into [0, 1] is a GF-indistinguishability
232
K.(2 Gupta, R.K. Gupta / Fuzzy Sets and Systems 79 (1996) 227-233
operator on X iff the following three conditions hold for any x, y, z in X: (1) 2 is G-reflexive in X. (2) 2 is symmetric in X. (3) 2 is F-transitive in X; in other words, F(2(x, z), 2(z, y)) ~< 2(x, y). 7.2. Lemma. Let {2i}i~t be a family of GF-indistinguishability operators on X satisfying the condition that, given x e X, there exists ex > 0 such that ex <~hi(x, x) for all i t I. Then 2 = ('] 2i ieI
is a GF-indistinguishability operator on X. Proof. For the G-reflexivity of 4, let x e X. Then 2(x, x) = inf/,x hi(x, x)/> ex > 0. Let 61 = infx,x 2i(x,x), and let 6 = infi,t 6i. We then get inf 2 ( x , x ) = i n f ( i n f 2 , ( x , x ) ) xEX
xeX \i~l
= inf(iel \xexinf 2i(X,X))=infbi=b.iei If X ~ y in X, then 2(x,y) = inf2i(x,y) ~ inf6i = 6. iEl
iel
We have thus proved that 2 is G-reflexive in X. That 2 is symmetric and F-transitive in X can be seen from [8, proof of Proposition 3.1]. [] 7.3. Theorem. Let # be a G-reflexive and symmetric fuzzy relation in X. Then Gu, the set of all GFindistinguishability operators 2 on X such that # c_ 4, is nonempty and
#F(x,y) = inf 2(x,y),
x, y e X ,
By Lemma 7.2, #v is then a GF-indistinguishability operator on X containing #, which is clearly the least. [] In [8] Velverde proved that a fuzzy relation 2 in X is an F-indistinguishability operator on X iff there exists a family {h~}j~s of fuzzy subsets of X such that
2(x, y) = inf P(max{hj(x), hj(y)}l(min{hj(x), hi(y))). jeJ
for all x, y in X, where P is the so-called quasiinverse of the continuous t-norm F defined as
F(xFy) = sup{~ e [0, 1]: F(a,x) <~y}. This result cannot be generalized if 2 is a GFindistinguishability operator, because we may not always have 2(x, x) = 1. However, we can partition EF(X), the class of all GF-indistinguishability operators on X, in such a way that each equivalence class is represented by a unique F-indistinguishability operator on X. The following lemma is useful in that direction. 7.4. Lemma. Let 2 be a symmetric and F-transitive fuzzy relation in X. I f p(x,y) = 2(x,y)for all x ~ y in X and p(x,x) = 1 for all x in X, then p is an F-indistinguishability operator on X. Proof. Clearly, p is reflexive and symmetric. Let x, y, z e X. We show that
F(p(x,z),p(z, y)) <<.p(x, y). If x = y, the inequality follows. If either z = x or z = y, we get equality. So, let x ~ z ~ y ~ x. Then
A~G~
is the least GF-indistinguishability operator on X containing #. Proof. The fuzzy relation l(x, y) = 1 for all x, y e X is the greatest GF-indistinguishability operator 2 on X containing #. Hence Gu is nonempty. Since # is G-reflexive, given x e X, we have 0 < #(x, x) ~< 2(x, x)
for all 2 e Gu.
F(p(x, z), ptz, y)) = F(2(x, z), 2(z, y)) <~ 2(x, y) = p(x, y), and so p is F-transitive.
[]
Now, let 2, # e EF(x). We define 2 ,-~# if and only if 2(x,y) = #(x,y), whenever x ~ y in X. If p(x,x) = 1 for all x in X and p ( x , y ) = 2(x,y)
K.C. Gupta, R.K. Gupta / Fuzzy Sets and Systems 79 (1996) 227-233
whenever x # y in X, then, by Lemma 7.4, p is the unique F-indistinguishability operator on X such that p e [~.], the equivalence class containing 2. We now formulate our assertion as follows.
7.5. Theorem. Let 2 be a fuzzy relation in X. Then 2 is a GF-indistinguishability operator on X if and only if there exists a family {hj}~j offuzzy subsets of X such that inf F(max{hj(x), hi(y)}lmin{h~(x), hi(y)}), x, y e X, jeJ
is the unique F-indistinguishability operator on X representing the class of 2. []
Acknowledgements The authors take this opportunity to thank the referee who brought to their notice the ideas already proposed by R.T. Yeh and L. Velverde.
233
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