Unattainable solutions of a fuzzy relation equation

FUZZY

sets and systems ELSEVIER

Fuzzy Sets and Systems 99 (1998) 193-196

Unattainable solutions of a fuzzy relation equation Hideyuki Imai*, Ken Kikuchi, Masaaki Miyakoshi Divisionof Systemsand InformationEngineering,Hokkaido University,Sapporo060. Japan Received August 1996; revised December 1996

Abstract In a fuzzy relation equation, the properties of the solution set have been investigated especially when the basic spaces are all finite sets. However, when the basic spaces are infinite, a few properties are known. In this paper, we use the concepts of attainability and unattainability of a solution to clarify some properties of the solution set and show the condition for existence of a partially attainable solution and an unattainable one. © 1998 Elsevier Science B.V. All rights reserved

Keywords: Relation; Attainable solution; Unattainable solution

1. Introduction

in the case that U and V are both finite sets, it is shown that the solution set is completely determined by the greatest solution and the set of minimal solutions. However, when the cardinality of either U or V is infinite, a few properties about the solution set are investigated [2-4, 9]. In this paper, we use the concept of attainability to clarify some properties of the solution set of Eq. (1).

Let U and V be nonempty sets, and let L/~(U), Lf(V), and L~'(U x V) be the collections of fuzzy sets o f U, V, and U x V, respectively. Then, an equation

XoA=B,

(1)

is called a fuzzy relation equation, where A E Aa(U x V) and BE Ae(V) are given and X E L P ( U ) is unknown, and o denotes the V-A composition. A fuzzy setX satisfying the equation above is called a solution o f the equation. If/~x : U ~ / , #A : U x V ~ I, and/zB : V ~ I are their membership functions where I denotes the closed interval [0, 1], Eq. (1) is as follows:

(VvEV)(V(#x(U)AI~A(U'V))=I~B(V)) " u ~ v The solution set of a fuzzy relation equation has been investigated by many researchers [1-6, 8, 9], and several important properties are shown. Especially,

2. Preliminaries We now recapitulate some underlying definitions and important results for Eq. (I). Definition 1. Let #x and # r be membership functions of fuzzy set X, Y E ~ ( U ) , respectively. Then, the partial order ~<, the join V, and the meet A, are defined as follows:

*Corresponding author. 0165-0114/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved PH S01 65-0114(97)00020-1

~x ~<~r ¢~ (rue U)(l~x(u)<<. ~r(u)), l~xVl~r: U g u ~ ~x(u)Vllr(u)EL Bx A#y: UDu~--~#x(u)A#r(u)EL

H. Imai et al./Fuzzy Sets and Systems 99 (1998) 193-196

194

Note that lax <<.Pr is equivalent to X C Y for X,Y~Zf(U).

I f 3£usc¢ 0 holds, then 3£u°sc¢ O where 3£~c° denotes the set of minimal element of 3£usc. Moreover,

Definition 2. Let 3£c ~ ( U ) be the solution set of Eq. (1). The greatest solution of Eq. (1) is an element GE3£ such that /~x~
X E 3£ ~ (~Xusc E 3£us~)(~xu,c <<.~x )

Definition 3. For a, bE[0, 1]

(Vv c v ) ( V u ~ U)(~x(U) A re(u, v) <~~B(v))

aotb& ~ 1 ifa~
3£#0 ~ 2E3£,

holds.

3. Attainability of a solution

When X E £P(U) is a solution of Eq. (1),

holds. Moreover, when U and V are both finite sets,

(Vvc v )( 3uv ~ u)(~x(uv ) A m(u~, v) = ~e(v) ) holds. Thus, we introduce the concepts of attainability and unattainability of a solution. Definition 4. Let X E L,e(U) be a solution of Eq. (1), and let V/ be a nonvoid subset of V, then, X is attainable for V1 ~z~ (VVl ~ V1 )( ~uv I ~ U)(J~x(uv I ) A ]AA(Uvl , V 1 )

where

= #e(vl)),

A

vEV

and then, f( is the greatest solution of Eq. (1). Theorem 2 (Higashi and Klir [1]). When U and V

are both finite sets, 3£~ 0 implies 3£0 ¢ O, and then,

X63£ ¢, (S~'E3£°)(~2~/~x~
X is unattainable for V1 ¢:~ (VVl E//"i )(VuE U)(12x(u) A lO(u, Vl )< #B(Vl )). Moreover, the set of solutions which is attainable for V1 C V is denoted by ae(+), and the set of so~V I lutions which is unattainable for V1 c V is denoted by 3£(v~-). Note that when the set U and V are both finite, all solutions are attainable for V, that is, 3£ = 3£(v+). Definition 5. Let X E £~°(U) be a solution of Eq. (1), and let V1 and V2 be a nonvoid subsets of V satisfying V, A Vz = 0 and V1 U I"2 = V, then X is an attainable solution ,~ X E 3£(v+),

the index set U be a metric compact space, and

X is a partially attainable solution ae(+) n ~:(-)

3£~c & {XE3~ [ #x is upper semi-continuous on U}.

X is an unattainable solution ~

Theorem 3 (Wagenknecht and Hartmann [9]). Let

¢:~ X E~VL ''~v2 ,

XE3£~v-).

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195

Note that for a, b E I,

In [7], an attainability is used for consideration about the extension principle for fuzzy sets. In a fuzzy relation equation, the following properties about the set of attainable solutions are known.

actb = sup{x E [0, 1] [ a A x ~
Theorem 4 (Luo [3] and Miyakoshi et al. [4]). Let

aflb --- sup{x E [0, 1]la A x < b}.

(5)

and (6)

be the 9reatest solution 9iven in Theorem 1, then, Lemma 2. Let V1 and V2 be subsets of V satisfying V1 N V2 = 0 and I/1 to V2 = V, then Theorem 5 (Luo [3]).

XE~(+)N~(vT) =¢, pX~
v )(#x, <~#x <. p~ ). X E "(+) x v ¢* (~xg E 3~(+)

where

A fuzzy set Xg in Theorem 5 is called the reachable quasi-minimum solution of Eq. (1).

~G,(u)= ( / k (#A(U,V)~B(V))) vl E V~

Theorem 6 (Imai et al. [2]). I f V is a finite set, then,

+) , , x ° # 0 .

/k(v2AVz(12A(U'/))fl#B(/))) ) "

These properties are about the set of attainable solutions. However, a few properties about the set of partially attainable solutions and one of unattainable solutions are known [2, 4]. The following lemma is fundamental. Lemma 1 (Miyakoshi et al. [4]). L e t X and Y be solutions of Eq. (1), and let V1 and V2 be subsets of V, then, y(+) (2) V1 C V2 ::~ ~~(+) v i ~ v2 ,

i~x<~py

~(+) ~ yE~(+) and XEztv, v,,

(3)

#x>~py

and X E ~ ( 7 ) =~ rE3~(v71.

(4)

Proof. IfXE~(v +) n ~(v7), then (k/I) 1 E V 1 ) ( k / u E

U)(].lx(U )/k ~lA(U, 1)1) ~ #B(UI ))

and (VU2 E Vz)(k/u E

U)(~lx(U )/k ]2A(U, 1)2) <

]2B(/)2))

hold. Hence, ~ x ( u ) ~ / k (~A(u, v l ) ~ 8 ( v l ) ) v, E V~

and

A

v2l uB(v2))

t,2 E V2

4. Properties of the set of partially attainable and unattainable solutions

In this section, we show some properties about a partially attainable solution and an unattainable one. The following definition is useful for characterizing such kinds of solutions. Definition 6. For a, b E/, we define t-operator as

follows:

aflb&{1 b

if a < b , otherwise.

hold by (5) and (6).

[]

Lemma 3. Let V1 and V2 be subsets of V satisfyin 9

V1 N V2 = 0 and V1 U V2 = V, and let G1 be the solution of Eq. ( 1) defined in Lemma 2, then X(+) N ~(-)

~ "(+)

Proof. Let X be an element of 3E(+)nv,3E(v2), then

#x ~<#G, ~< #2, holds, where A" is the greatest solution defined in Theorem 1. By applying (3) in Lemma 1, we can complete the proof. []

H. Imai et al./Fuzzy Sets and Systems 99 (1998) 193-196

196

For the set o f partially attainable solutions and one o f unattainable solutions, we obtain the following. T h e o r e m 7.

Let V1 and V2 be subsets of V satisfyin9 5. C o n c l u s i o n

V1 ~ V2 = 0 and Vl U V2 = V, then

-,-(+)

(V

(u) -

<0

where GI E ~q~(U ) is the fuzzy set defined in Lemma 2. Proof. It suffices to prove the only if part, so let XE3E(v+) M 3~(v~-). From Lemma 3, there exist V3 and V4 satisfying V1 C V3, V3 A V4 = 0, and V3 U V4 = V, such that, G1 E ~v+)A ~(vT), holds. The subset Ul c U is defined as follows: ul EU1 ¢:~ ( 3 v 3 E ( V 3 n V~))(#c,(ul)ApA(ul,v3)

=a,(v3)), where V~ denotes the complement o f a set V1. For fixed e > 0 , let a fuzzy set X1 E Z a ( U ) be defined as follows:

px,(u)

f ~uG,(u), t (1 - ~)#a,(u) + ~#x(u),

u ~ U~, ue Ul.

Then, it immediately follows that r *~,, ~e(+) Yl tz: V~ N 3E(v~) and

V (~o,(u) - ~x,(u)) <~, uEU

and the theorem can be proved.

[]

In the case o f V1 = V in Theorem 7, then, GI = and

^

...(+)

¢~ X E zt v

holds. Therefore, Theorem 7 is a generalization o f Theorem 4.

In this paper, we get the necessary and sufficient condition for the existence o f a partially attainable and an unattainable solution. Theorem 7 is valid without any assumption o f the basic spaces. In future works, it is necessary to investigate more precise structure o f the solution set o f a fuzzy relation equation.

References

[1] M. Higashi, G.J. Klir, Resolution of finite fuzzy relation equations, Fuzzy Sets and Systems 13 (1984) 65-82. [2] H. Imai, M. Miyakoshi, T. Da-te, Some properties of minimal solutions for a fuzzy relation equation, Fuzzy Sets and Systems, to appear. [3] C.Z. Luo, Reachable solution set of a fuzzy relation equation, J. Math. Anal. Appl. 103 (1984) 524-532. [4] M. Miyakoshi, H. Imai, T. Da-te, Properties of a set of minimal solutions for a fuzzy relation equation, Bull. Faculty Eng. 167 (1994) (in Japanese). [5] M. Miyakoshi, M. Shimbo, Sets of solution-set-invariant coefficient matrices of simple fuzzy relation equations, Fuzzy Sets and Systems 21 (1987) 59-83. [6] M. Miyakoshi, M. Shimbo, Sets of solution-set equivalent coefficient matrices of fuzzy relation equations, Fuzzy Sets and Systems 35 (1990) 357-387. [7] H.T. Nguyen, A note on the extension principle for fuzzy sets, J. Math. Anal. Appl. 64 (1978) 369-380. [8] E. Sanchez, Resolution of composite fuzzy relation equation, Inform. Control 30 (1976) 38-48. [9] M. Wagenknecht, K. Hartmann, On the existence of minimal solutions for fuzzy equations with tolerances, Fuzzy Sets and Systems 34 (1990) 237-244.