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Some properties of infinite fuzzy relational equations with sup–inf composition
Information Sciences 252 (2013) 32–41
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Some properties of infinite fuzzy relational equations with sup–inf composition q Qing-quan Xiong, Xue-ping Wang ⇑ College of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan 610066, PR China
a r t i c l e
i n f o
Article history: Available online 2 August 2011 Keywords: Fuzzy relation Fuzzy relational equation Minimal solution Solution set
a b s t r a c t This paper deals with infinite fuzzy relational equations with sup–inf composition. First, some properties of them are investigated. Then some necessary and sufficient conditions for the existence of attainable solutions (resp. unattainable solutions and partially attainable ones) are given if the solution set is nonempty. Finally, the set of attainable solutions is described. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The study of fuzzy relational equations is one of the most appealing subjects in the field of fuzzy sets and fuzzy logic [27,28], both from a mathematical and a systems modeling point of view [6]. Sanchez [15] was the first one who studied fuzzy relational equations with max–min composition. Since then, several authors further enlarged the theory with different composite operators over various lattices, see [1,2,4–10,12–26]. A system of fuzzy relational equations is of the form
AX ¼B
ð1Þ
or
_ ðaij ^ xj Þ ¼ bi for all i 2 I; j2J
where is a sup–inf composition, X = (xj)j2J is unknown, A = (aij)IJ and B = (bi)i2I are known with aij, xj, bi 2 [0, 1]. The system of fuzzy relational equations is denoted by Eq. (1) and its solution set is defined as X ¼ fX ¼ ðxj Þj2J : A X ¼ Bg. A special case of Eq. (1) is as follows:
AX ¼b
ð2Þ
or
_ ðaj ^ xj Þ ¼ b; j2J
q Supported by National Natural Science Foundation of China (No. 10671138) and Doctoral Fund of Ministry of Education of China (No. 20105134110002). ⇑ Corresponding author. E-mail addresses: [email protected] (Q.-q. Xiong), [email protected] (X.-p. Wang).
0020-0255/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2011.07.041
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Q.-q. Xiong, X.-p. Wang / Information Sciences 252 (2013) 32–41
where A = (aj)j2J is a row vector with aj 2 [0, 1], b 2 [0, 1]. The equation is signed by Eq. (2) whose solution set is defined by X2 ¼ fX ¼ ðxj Þj2J : A X ¼ bg. In order to completely describe X, Wagenknecht and Hartmann [20] gave some conditions for the existence of minimal solutions when J is a compact metric space. Miyakoshi, Imai and Da-te [10] introduced the notions of attainable solution and unattainable one, and got a necessary and sufficient condition for the existence of the attainable solution. Imai et al. [8,7] further showed a necessary and sufficient condition for the existence of a minimal solution without any topological assumption, moreover, they also obtained some sufficient conditions for the existence of a partially attainable solution (resp. the unattainable solution). Wang and Qu described X2 in both cases that b is a join-irreducible element and b has an irredundant continuous join-decomposition, see [14,22], respectively. Wang and Xia [23] dealt with fuzzy relational equations with supconjunctor composition on complete distributive lattices, they checked a necessary and sufficient condition for the existence of the attainable solution (resp. the unattainable solution) under the condition that either the right-hand side of equations is a continuous join-irreducible element or it has an irredundant continuous join-decomposition. However, they did not discuss the system of equations. Shieh [17] extended the results of Xiong and Wang [25] to a more general condition and investigated some properties of the solutions, while he described the solution sets which just include the attainable solutions and the unattainable ones. Indeed, Infinite fuzzy relational equations occur in practice. For instance, in fuzzy reasoning [11], antecedents and consequences are usually defined on infinite sets. Consider an inference which contains a fuzzy conditional proposition: Ant 1: If x is A, then y is B, Ant 2: y is B0 , Cons: x is A0 . This inference can be viewed as a generalized modus tollens, where the Ant 1 ‘‘If x is A, then y is B’’ represents a certain relationship between A and B (the inference rule), and the Ant 2 ‘‘y is B0 ’’ is a known consequence. Now the problem of determining antecedents A0 (the Cons ‘‘x is A0 ’’) can be interpreted as the form of fuzzy relational equation A0 R = B0 , where A0 , B0 and R are fuzzy sets and fuzzy relation, respectively, with B0 and R known. Thus, its results can be applied to the problem as the universal set of A0 is an infinite set and ‘‘’’ is a composition operator (see e.g. [11,17] for detail). This paper will investigate Eq. (1) on [0, 1] without any topological assumption. Its rest is organized as follows. For the sake of convenience, some notions and previous results are summarized in Section 2. Some necessary and sufficient conditions which describe the attainable solutions (resp. the unattainable solutions and the partially attainable ones) are shown, and the set of attainable solutions is obtained in Section 3. Conclusions are drawn in Section 4. 2. Previous results In the whole paper, unless otherwise stated, we assume that I is finite, J is countably infinite, A = (aij)IJ is a matrix, X = (xj)j2J and B = (bi)i2I are two column vectors with aij, xj, bi 2 [0, 1]. In what follows, we shall recall some basic concepts and important results for the sake of convenience.
Definition 2.1 [3]. Let (P, 6) be a partially ordered set and X # P. An element p 2 X is minimal if and only if for all x 2 X if x 6 p then p = x. The greatest element of X is an element g 2 X such that x 6 g for all x 2 X. Definition 2.2 [3]. Let a, b 2 [0, 1]. Define aab = max{x 2 [0, 1] : a ^ x 6 b}. Let Ai = (aij)j2J be the ith row vector of A; Xi2 ¼ fX ¼ ðxj Þj2J : Ai X ¼ bi g for each i 2 I. Denote the set of all minimal elements of X; X2 and X0 ; X02 and X0i2 , respectively. a _ b = sup{a, b}, a ^ b = inf{a, b}. V Xi2 by T T T Ai abi ¼ ðaij abi Þj2J ; A B ¼ , where A represents the transpose of A. Then we have: i2I aij abi j2J
Lemma 2.1 [21]. (1) X – ; if and only if \i2I Xi2 – ;. Moreover, X ¼ \i2I Xi2 ; (2) If X – ;, then X⁄ = solution of Eq. (1). Lemma 2.2 [15]. X – ; if and only if AT Lemma 2.3 [25]. X2 – ; if and only if
W
B 2 X. Further, X 6 AT j2Jaj
V
abi)T is the greatest
i2I(Ai
B for all X 2 X.
P b.
Definition 2.3 [8]. A solution X = (xj)j2J of Eq. (1) is called attainable if for every i 2 I there exists an index j0 2 J such that aij0 ^ xj0 ¼ bi , the set of them is denoted by XðþÞ . A solution X = (xj)j2J of Eq. (1) is called unattainable if aij ^ xj < bi for all i 2 I and j 2 J, the set of them is set as XðÞ . A solution X of Eq. (1) which is neither attainable nor unattainable is called partially attainable whose set is defined by XðÞ .
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Example 2.1. Consider the system of equations
8 þ1 þ1 1 1 W 1 W 1 > > > ¼ 13 ; ^ x ^ x ^ x ^ x _ _ _ 1 2 2n3 2n2 > 4 3 nþ1 < 3 n¼3 n¼3 þ1 þ1 > 1 1 W 1 W 1 > > > _ ^ x ^ x ^ x ^ x _ _ ¼ 12 : 1 2 2n3 2n2 : 2 3 nþ1 2 n¼3
n¼3
T It is obvious that X 1 ¼ 12 ; 0; . . . ; 0; . . . is attainable, X2 = (0, 0, x3, x4, . . .)T is unattainable with x2k3 < 13 ; x2k2 < 12 for T Wþ1 Wþ1 1 k = 3, 4, . . . , k¼3 x2k3 ¼ 3 and k¼3 x2k2 ¼ 12, and X 3 ¼ 13 ; 0; x3 ; x4 ; . . . is partially attainable with xk < 12 for k = 3, 4, . . . , and Wþ1 1 k¼3 xk ¼ 2. From Definition 2.3, we know that X ¼ XðþÞ [ XðÞ [ XðÞ and XðþÞ \ XðÞ ¼ XðþÞ \ XðÞ ¼ XðÞ \ XðÞ ¼ ;. Therefore, the solution set X is exactly divided into three separate parts. Lemma 2.4 [8]. Let X ¼ ðxj Þj2J 2 XðÞ . Then j{j 2 J : xj > 0}j = +1, where j{ }j denotes the cardinality of set { }. 3. Resolution of the system of equations Let I0 = {i 2 I : bi = 0} and J I0 ¼ fj 2 J : aij > 0 for some i 2 I0}. Then every solution X = (xj)j2J of Eq. (1) is that xj = 0 for all j 2 J I0 . Thus, it is possible to omit the equations with indices from I0 and the columns of A with indices from J I0 (cfr. [26]). Therefore, in what follows, we assume that bi > 0 for all i 2 I and 1 < jIj = m < +1, jJj = +1. Denote G1(bi) = {j 2 J : aij > bi}, G2(bi) = {j 2 J : aij = bi} and G(bi) = G1(bi) [ G2(bi) for all i 2 I. Using the results in [17,25], we know that the known coefficients of the equation Ai X = bi with i 2 I have only four cases as follows: (i) (ii) (iii) (iv)
W Gðbi Þ – ;; j2JnGðbi Þ aij < bi ; W G(bi) = ;, j2Jaij = bi; W Gðbi Þ – ;; j2JnGðbi Þ aij ¼ bi ; W G(bi) = ;, j2Jaij < bi.
By Lemmas 2.1 and 2.3, if A has a row vector Ai satisfying the case (iv), then the system of equations has no solutions. In W fact, if there exists an i0 2 I such that Gðbi0 Þ ¼ ; and j2J ai0 j < bi0 , then Xi0 2 ¼ ; by Lemma 2.3. Therefore, X ¼ ; by Lemma 2.1. Furthermore, if Eq. (1) has solutions, then every row vector of A must satisfy one of cases (i), (ii) and (iii). In the rest of the paper, we assume that Xi2 – ; for all i 2 I and b1 6 b2 6 6 bm with m , jIj < +1.
Lemma 3.1 [25]. If aj < b for all j 2 J and Lemma 3.2. If G(bi) = ; and
W
j2Jaij
W
j2Jaj
= b, then X2 – ;:
= bi for all i 2 I, then X – ;:
Proof. X = (1)j2J is a solution of Eq. (1).
h
Lemma 3.3 [15]. Let X 1 ; X 2 2 X and X such that X1 6 X 6 X2. Then X 2 X. Lemma 3.4 [25]. If Xi2 – ; and G(bi) = ;, then Lemma 3.5 [25]. If X ¼ ðxj Þj2J 2 X2 , then Lemma 3.6. If X ¼ ðxj Þj2J 2 X, then
W
j2Jxj
W
W
j2Jxj
P
j2J aij
ðÞ
¼ bi ; Xi2 ¼ Xi2 and X0i2 ¼ ;:
P b.
W
i2Ibi.
Theorem 3.1. If X – ; and there exists an i0 2 I such that Gðbi0 Þ ¼ ;, then XðþÞ ¼ ;: ðÞ
Proof. Let X ¼ ðxj Þj2J 2 X. Then Ai0 X ¼ bi0 . By Lemma 3.4 and Gðbi0 Þ ¼ ;, we have X 2 Xi0 2 . Therefore, XðþÞ ¼ ; by Definition 2.3. h Theorem 3.2. If X – ;, then
W
j2Jaij
P bi for all i 2 I.
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W
Proof. Let X ¼ ðxj Þj2J 2 X. Then bi =
j2J(aij
^ xj) 6
W
j2Jaij
for all i 2 I. h
Remark 3.1. The inverse of Theorem 3.2 is not true when jIj P 2. Example 3.1. Consider the system of equations
8 þ1 1 W 1 > 1 > ¼ 12 ; ^ x 1 _ ^ x > 1 n < 2 4 n n¼2 þ1 W 1 > > 1 > ð1 ^ x1 Þ _ 1 ^ x ¼ 13 : : n 4 n n¼2
W
Then j2Jaij P bi for all i 2 I. However, the system of equations has no solution. Define J 1 ¼ Gðb1 Þ; J i ¼ Gðbi Þ n [i1 k¼1;bk –bi G1 ðbk Þ , and J2i = {j 2 Ji : j 2 G(bk) for some k 2 {1, 2, . . . , i 1}} for all i 2 In{1}. Then the following characterization holds: Theorem 3.3. XðþÞ – ; if and only if Ji – ; for all i 2 I. Proof. In the first part, suppose that there exists an i0 2 I such that J i0 ¼ ;. Since X ¼ ðxj Þj2J 2 XðþÞ , there exists an index j 2 J such that ai0 j ^ xj ¼ bi0 , which implies that xj P bi0 and ai0 j P bi0 , i.e., Gðbi0 Þ – ;. If i0 = 1, then from Lemma 3.4 we have that ðÞ X 2 Xi0 2 since J i0 ¼ Gðbi0 Þ ¼ ;, which contradicts the fact that X 2 XðþÞ . Therefore i0 > 1. By the definition of Ji and Gðbi0 Þ – ;, we can assert that there must exist an index k 2 {1, 2, . . . , i0 1} such that j 2 G1(bk) since J i0 ¼ ;, that is to say, akj > bk and W bi0 – bk . Therefore, bi0 > bk and akj ^ xj P akj ^ bi0 > bk . The last inequality implies that j2J(akj ^ xj) > bk, i.e., X R Xk2 , which is in the opposition to Lemma 2.1 (1). Therefore, Ji – ; for all i 2 I. In the converse implication, let Ji – ; for all i 2 I. Then. (i) If \i2IJi – ;, then by the definition of Ji we can define X = (xj)j2J with
xj ¼
bm ; j 2 \i2I J i ; 0;
otherwise:
We first note that for j 2 \i2IJi, aij = bi 6 bm if i – m, and aij P bi if i = m. Therefore, if i – m then
2
Ai X ¼ 4
3
_
2
ðaij ^ bm Þ5 _ 4
j2\i2I J i
3
_
ðaij ^ 0Þ5 ¼
j2Jnð\i2I J i Þ
_
ðaij ^ bm Þ ¼ aij ¼ bi
j2\i2I J i
and if i = m then
2 Ai X ¼ 4
3
_
2
ðaij ^ bm Þ5 _ 4
j2\i2I J i
3
_
ðaij ^ 0Þ5 ¼
j2Jnð\i2I J i Þ
_
ðaij ^ bm Þ ¼ bm :
j2\i2I J i
Thus X 2 X. Moreover, for every j 2 \i2IJi, we know that aij ^ bm = aij = bi for i – m and aij ^ bm = bm for i = m, i.e., X 2 XðþÞ . (ii) If \i2IJi = ; and Js \ Jt = ; for every s, t 2 I with s – t, then we define X = (xj)j2J with
xj ¼
bi ; if j 2 J i with i ¼ 1; 2; . . . ; m; 0;
otherwise:
Thus
2 " # " # " # _ _ _ _ 6 ðaij ^ xj Þ ¼ ðaij ^ xj Þ _ _ ðaij ^ xj Þ _ _ ðaij ^ xj Þ _ 4 j2J
j2J 1
" ¼
_ j2J 1
" ¼
_ j2J 1
j2J i
j2J m
j2Jnð[m J k¼1 k Þ
2 " # " # _ _ 6 ðaij ^ b1 Þ _ _ ðaij ^ bi Þ _ _ ðaij ^ bm Þ _ 4 #
j2J i
# ðaij ^ b1 Þ _ _ bi _ _
j2J m
"
_
#
_
_
3 7 ðaij ^ xj Þ5 3
7 ðaij ^ 0Þ5 j2Jnð[m J k¼1 k Þ
ðaij ^ bm Þ
j2J m
W for all i 2 I. In the last equality, if k 6 i then j2Jk ðaij ^ bk Þ 6 bk 6 bi , otherwise, by the definition of Jk we have aij 6 bi 6 bk, W W W hence j2Jk ðaij ^ bk Þ 6 j2Jk aij 6 bi . Therefore, j2J(aij ^ xj) = bi for every i 2 I, i.e., X 2 X. Since aij ^ xj = aij ^ bi = bi for every i 2 I and j 2 Ji, we have X 2 XðþÞ :
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Q.-q. Xiong, X.-p. Wang / Information Sciences 252 (2013) 32–41
(iii) If \i2IJi = ; and Js \ Jt – ; for some s, t 2 I and s – t, then we define X = (xj)j2J with
8 > < bs _ bt ; j 2 J s \ J t ; xj ¼ b i ; j 2 J i with i ¼ 1; 2; . . . ; m and i – s; t; > : 0; otherwise: The fact that X 2 XðþÞ is immediate from the proofs of (i) and (ii). Finally, with (i), (ii) and (iii) we have that XðþÞ – ;: Theorem 3.4. If G(bi) = ; and
W
j2Jaij
h
= bi for all i 2 I, then X ¼ XðÞ and X0 ¼ ;:
Proof. Y ¼ ð1Þj2J 2 X implies that X – ;. Now, let X ¼ ðxj Þj2J 2 X. Then aij ^ xj 6 aij < bi for all i 2 I and j 2 J, i.e., X 2 XðÞ . Therefore, X ¼ XðÞ . It follows from Lemma 2.4 that j{j 2 J : xj > 0}j = +1. For every j0 2 {j 2 J : xj > 0}, define X⁄ = (xj⁄)j 2J with
xj ¼
0;
if j ¼ j0 ;
xj ; if j – j0 :
It is easy to show that X 2 X; X 6 X and X⁄ – X, therefore, X0 ¼ ;: By Theorem 3.4, we have:
h
Theorem 3.5 [17]. If X0 – ;, then G(bi) – ; for all i 2 I. Note that Theorem 3.5 is not invertible even if X – ; and G(bi) – ; for all i 2 I, as is proved in the next example: Example 3.2. Consider the system of equations
8 þ1 þ1 1 W 1 W 1 > > ^ x ^ x ^ x ¼ 14 ; _ _ > 1 2n2 2n1 < 3 4 nþ1 n¼2 n¼2 þ1 W > > n > 1 ^ x1 _ 2 ^ x2 _ ^ x ¼ 12 : : n 2 5 2nþ1 n¼3
It is obvious that the system has no minimal solutions. However, we have the following theorem: Theorem 3.6. X0 – ; if and only if Ji – ; for all i 2 I. Proof. For proving the first part, let Ji – ; for all i 2 I. Then XðþÞ – ; by Theorem 3.3. Let X ¼ ðxj Þj2J 2 XðþÞ . Then W j2J(aij ^ xj) = bi for all i 2 I, and for every i 2 I there exists a ji 2 J such that aiji ^ xji ¼ bi . Then: (i) If \i2IJi – ;, then we can choose one index j0 2 \i2IJi by the definition of Ji, and define X⁄ = (xj⁄)j2J with
xj ¼
b m ; j ¼ j0 ; 0;
otherwise:
Then we easily show that X 2 X0 , i.e., X0 – ;: (ii) If \i2IJi = ; and Js \ Jt = ; for all s, t 2 I and s – t. Then let J0 = {j1, j2, . . . , jm} with jk 2 Jk and define X⁄ = (xj⁄)j2J with
xj ¼
b i ; j ¼ ji 2 J 0 ; 0;
otherwise:
Then it is easy to see that X 2 X0 , i.e., X0 – ;. (iii) If \i2IJi = ; and Js \ Jt – ; for some s, t 2 I and s – t. Then we construct X⁄ as follows: Choose j1 2 Jm, let I1 = {k 2 I : j1 2 Jk}. Then jI1j < m since \i2IJi = ;. Choose j2 2 J r2 with r2 = max(InI1), let I2 = {k 2 I : j2 2 Jk}. If jI1 [ I2j < m then .. . Choose jp 2 J rp with r p ¼ max I n [p1 Ik , let Ip = {k 2 I : j2 2 Jk}. k¼1 P Continue as above till pk¼1 jIk j ¼ m. The above proceeding implies that I1, I2, . . . , Ip is a partition of I. Define X⁄ = (xj⁄)j2J with
xj ¼
8W < bi ; j ¼ jk and Ik – ;; :
k ¼ 1; 2; ; p:
i2Ik
0;
otherwise:
Then for all i0 2 I there exists exactly one q 2 {1, 2, . . . , p} such that i0 2 Iq. Therefore, with the structure of X⁄ we have
Q.-q. Xiong, X.-p. Wang / Information Sciences 252 (2013) 32–41
2 _ ðai0 j ^ xj Þ ¼ ðai0 j1 ^ xj1 Þ _ _ ðai0 jp ^ xjp Þ _ 4 j2J
¼
_
ai0 j1 ^
bk
j2Jnfj1 ;j2 ;;jp g
0
!
_
_
_ _ @ai0 jq ^
1
3 ðai0 j ^ xj Þ5 ¼ ðai0 j1 ^ xj1 Þ _ _ ðai0 jp ^ xjp Þ 0
bi A _ _ @ai0 jp ^
i2Iq
k2I1
37
_
1 b k A:
k2Ip
W W In the last equality, if i0 = maxIq, then ai0 jq P i2Iq bi ¼ bi0 . If i0 < maxIq, then ai0 jq ¼ bi0 6 i2Iq bi . Consequently, W W W ai0 jq ^ i2Iq bi ¼ bi0 . Moreover, ai0 jk ^ s2Ik bs 6 bi0 for all k 2 {1, 2, . . . , p}n{q}. Indeed, if ai0 jk ^ s2Ik bs > bi0 , then ai0 jk > bi0 and W W s2Ik bs > bi0 . Let t = maxIk. Then bt > bi0 , so that jk 2 Jt, which contradicts the definition of Jt. Therefore, j2J ðai0 j ^ xj Þ ¼ bi0 for all i0 2 I, i.e., X 2 X. Now, suppose that X 2 X satisfies that X < X⁄. Without loss of generality, we assume that W W xjk < xjk ¼ i2Ik bi for Ik – ;. Then for the tth equation, with analogous proof to that of X 2 X we have that j2J(atj ^ xj) < bt. 0 0 Therefore, X 2 X , this implies that X – ;. W The converse implication, let X ¼ ðxj Þj2J 2 X0 . Then j2J(aij ^ xj) = bi for all i 2 I. We claim that there must exist an index ji 2 J such that aiji ^ xji ¼ bi for all i 2 I. Otherwise, if there exists an index i 2 I such that aij ^ xj < bi for all j 2 J, then j{j 2 J : xj > 0}j = +1 by Lemma 2.4. Choose an index j0 2 {j 2 J : xj > 0}, and define X⁄ = (xj⁄)j2J with
xj ¼
0;
j ¼ j0 ;
xj ; j – j 0 :
Then X 6 X; X 2 X and X⁄ – X, which contradicts the fact of X 2 X0 . Therefore, X 2 XðþÞ : Further, with Theorem 3.3 we have that Ji – ; for all i 2 I. h From the proof of Theorem 3.6, the following three corollaries hold: Corollary 3.1. If X ¼ ðxj Þj2J 2 X0 , then j{j 2 J : xj⁄ > 0}j 6 m. Corollary 3.2 [17]. X0 # XðþÞ . Corollary 3.3. XðþÞ – ; if and only if X0 – ;. With Theorems 3.3, 3.6 and Corollary 3.3, we can easily obtain Theorem 5 in [8]. Theorem 3.7. Suppose that XðþÞ – ; and J2i = ; for all i 2 In{1}. Then the following three statements hold: (1) Let j1 2 J1, j2 2 J2nJ1, j3 2 J3n(J1 [ J2), . . . , jm 2 Jmn(J1 [ [ Jm1), and define X = (xj)j2J with
xj ¼
bk ; j ¼ jk with k 2 f1; 2; . . . ; mg; 0;
ð3Þ
otherwise:
Then X is a minimal element of X. (2) All minimal elements of X have the form of (3). Q (3) jX0 j ¼ i2I jJ i j:
Proof (1) It is easy to identify that X 2 X. Now, let X 0 ¼ ðx0j Þj2J 2 X and X0 6 X. For every j 2 J, if j – jk with k 2 {1, 2, . . . , m}, then xj = 0, which implies that x0j ¼ 0. Therefore, in order to prove that X 2 X0 , we just need to prove that if j = jk with k 2 {1, 2, . . . , m} then xjk ¼ x0jk . In fact, for all i 2 I
bi ¼
m m _ _ _ _ ðaijk ^ x0j Þ ¼ ðaijk ^ x0jk Þ 6 ðaijk ^ xjk Þ 6 ðaijk ^ xj Þ j2J
k¼1
¼ ðaij1 ^ b1 Þ _ _ ðaijm ^ bm Þ _
j2J
k¼1
"
_
#
ðaij ^ xj Þ :
ð4Þ
j2Jnfj1 ;j2 ;...;jm g
Since aijk ^ bk 6 aijk ; jk 2 J k and jk R Ji with k > i, we have that aijk 6 bi . Otherwise, we have jk 2 G1(bi), this follows that jk R Jk r1 since J k ¼ Gðbk Þ n [k¼1;b G1 ðbr Þ , a contradiction. Therefore, aijk ^ bk 6 aijk 6 bi , this with inequality (4) deduces that r –bk m _ k¼1
ðaijk ^ x0jk Þ ¼
m _
ðaijk ^ xjk Þ ¼ bi :
k¼1
ð5Þ
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Q.-q. Xiong, X.-p. Wang / Information Sciences 252 (2013) 32–41
We assert that there must be x0jk ¼ xjk for all k 2 {1, 2, . . . , m}. Otherwise, if there exists an index k0 2 {1, 2, . . . , m} such that x0jk < xjk , then there are two cases: 0
0
(i) If i < k0, then ak0 ji ^ bi 6 bi . In this case, if ak0 ji ^ bi < bi , then ak0 ji ^ bi < bi 6 bk0 . If ak0 ji ^ bi ¼ bi , then there must be bi < bk0 . Otherwise, if bi ¼ bk0 , then ak0 ji ^ bi ¼ bi ¼ bk0 , which implies that ak0 ji P bi ¼ bk0 , thus ji 2 Ji and ji 2 J k0 , a contradiction since J2i = ; for all i 2 In{1}. Therefore, we always have that ak0 ji ^ bi 6 bi < bk0 . (ii) If i > k0, then ak0 ji ^ bi 6 ak0 ji for all ji 2 Ji. Again, from formula (3), if xji ¼ bi then ji 2 Ji, hence ji R J k0 since ji 2 Jin(J1 [ [ Ji1) and J2i = ; for all i 2 In{1}. Therefore, ak0 ji ^ bi 6 ak0 ji < bk0 . Consequently, with Cases (i) and (ii) we have that
ðak0 j1 ^ x0j1 Þ _ _ ðak0 jk ^ x0jk Þ _ _ ðak0 jm ^ x0jm Þ 6 ðak0 j1 ^ b1 Þ _ _ ðak0 jk ^ x0jk Þ _ _ ðak0 jm ^ bm Þ 0 0 0 0 ! ! k_ m 0 1 _ 6 bi _ x0jk _ ak0 ji < bk0 ; 0
i¼1
i¼k0 þ1
which contradicts the equality (5). Therefore, ¼ xjk , it results X = X, i.e., X 2 X0 : W (2) Let X ¼ ðxj Þj2J 2 X0 . Then bi = j2J(aij ^ xj), and Ji – ; for all i 2 I by Theorem 3.3. If there exists an index k 2 Ji \ Jj for i – j, without loss of generality, let i < j, then k 2 J2j. This is a contradiction since J2i = ; for every i 2 In{1}. Therefore, Ji \ Jj = ; for all i, j 2 I and i – j. We assert that for every i 2 I there must be an index ji 2 Ji such that xji P bi : Otherwise, if there exists an index i0 2 I such that xji < bi0 for all ji0 2 J i0 , then x0jk
0
0
3 " # 2 _ _ _ _ 4 5 ðai0 j ^ xj Þ ¼ ðai0 j ^ xj Þ _ _ ðai0 j ^ xj Þ _ _ ðai0 j ^ xj Þ _ 4 "
j2J
#
2
j2J 1
j2J i
j2Jm
0
_
3
ðai0 j ^ xj Þ5:
j2Jnð[m J k Þ k¼1
Analogously to the proof of Theorem 3.7 (1), we have
"
_
#
2
ðai0 j ^ xj Þ _ _ 4
j2J 1
_ j2J i
3 ðai0 j ^ xj Þ5 _ _
"
_
# ðai0 j ^ xj Þ < bi0 :
j2J m
0
W W In view of J2i = ; for all i 2 In{1}, we have that X 2 X0 and j2Jnð[m Jk Þ ðai0 j ^ xj Þ < bi0 , therefore j2J ðai0 j ^xj Þ< bi0 , a contradick¼1 tion since X 2 X. Therefore, for every k 2 I there exists an index jk 2 Jk such that xjk P bk : Define X 0 ¼ x0j with
x0j
¼
j2J
bk ; j ¼ jk with k 2 f1; 2; . . . ; mg; 0;
otherwise:
0
Obviously, X 2 X and X0 6 X, the minimality of X implies that X = X0 , therefore, all minimal elements of X have the form of (3). Q (3) From the proofs of (1) and (2), we have jX0 j ¼ i2I jJ i j: h
Remark 3.2. In Theorem 3.7, the assumption of J2i = ; for all i 2 In{1} cannot be dropped. For instance,
þ1 8 W 1 1 > > ^ x1 _ 14 ^ x2 _ 15 ^ x3 _ ^ xn ¼ 13 ; > 3 n > > n¼4 > > þ1 > > 1 1 1 W 1 > > ^ xn ¼ 12 ; > < 2 ^ x1 _ 2 ^ x2 _ 2 ^ x3 _ n n¼4 þ1 1 2 1 W 1 > > > ^ x ^ x ^ x ^ x _ _ _ ¼ 23 ; > 1 2 3 n 4 3 5 n > > n¼4 > > þ1 > > W 1 > > : 14 ^ x1 _ 14 ^ x2 _ ð1 ^ x3 Þ _ ^ xn ¼ 1: n n¼4
Then J1 = {1}, J2 = {1, 2, 3}, J3 = {2}, J4 = {3}, J22 = {1} – ;, J23 = {2} – ; and J24 = {3} – ;, hence jJ1j jJ2j jJ3j jJ4j = 3. However, T the system of equations has only one minimal solution X ¼ 13 ; 23 ; 1; 0; . . . . However, we have the following theorem: Theorem 3.8. Let XðþÞ – ;. Then X 2 XðþÞ if and only if there exists an X 2 X0 such that X⁄ 6 X. Proof. In the first part, let X ¼ ðxj Þj2J 2 XðþÞ . By Definition 2.3, there exists an index jk 2 J such that akjk ^ xjk ¼ bk for every k 2 {1, 2, . . . , m}, therefore bk 6 xjk and bk 6 akjk . Let x0jk ¼ inffx : akjk ^ x ¼ bk g for each k 2 I. Obviously, xjk 2 fx : akjk ^ x ¼ bk g, and akjk ^ x0jk ¼ bk since jIj < +1, this deduces that x0jk 6 xjk for all k 2 I. Now, let J0 be the set of all jks such that {k : jk 2 J0} = I and {k : jk 2 J0 } – I for any J0 $ J0. Define X⁄ = (xj⁄)j2J with
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Q.-q. Xiong, X.-p. Wang / Information Sciences 252 (2013) 32–41
( xj ¼
x0jk ; if jk 2 J 0 ; 0;
otherwise:
Then X 2 X0 and X⁄ 6 X. In the converse implication, let X ¼ ðxj Þj2J 2 X. If there exists an X ¼ ðxj Þj2J 2 X0 such that X⁄ 6 X, then Ji – ; for all i 2 I. By Corollary 3.2, X 2 XðþÞ . This follows that for every i 2 I there exists an index ji 2 J such that aiji ^ xji ¼ bi . Since bi 6 aiji ^ xji 6 aiji ^ xji 6 bi for all i 2 I, we have that for every i 2 I there exists an index ji 2 J such that aiji ^ xji ¼ bi , i.e., X 2 XðþÞ : h Theorem 3.9. If G(bi) = ; for all i 2 I, then XðÞ ¼ ;. Proof. Suppose that XðÞ – ; and X ¼ ðxj Þj2J 2 XðÞ . Then from Definition 2.3 there exist two nonempty sets I1, I2 # I with I1 [ I2 = I and I1 \ I2 = ; such that aij ^ xj < bi for all i 2 I1, j 2 J, and for every k 2 I2 there exists an index jk 2 J satisfying akjk ^ xjk ¼ bk , i.e., akjk P bk , a contradiction since G(bk) = ;. Therefore, XðÞ ¼ ;. h Corollary 3.4. If XðÞ – ;, then there exists an index i0 2 I such that Gðbi0 Þ – ;. W
Theorem 3.10. Let X – ;; Gðbi Þ – ;; an i 2 I such that jJij = +1.
j2JnGðbi Þ aij
< bi for all i 2 I and J2i = ; for all i 2 In{1}. Then XðÞ – ; if and only if there exists
Proof. Let us assume that X ¼ ðxj Þj2J 2 XðÞ . Then from Definition 2.3 there are two nonempty subsets I1 and I2 of I with I1 [ I2 = I and I1 \ I2 = ; which satisfy that there exists an index j 2 J such that aij ^ xj = bi for all i 2 I1, and aij ^ xj < bi for all i 2 I2, j 2 J. For i 2 I2, there are two cases: hW i hW i W W ða1j ^ xj Þ ¼ ða1j ^ xj Þ _ ða1j ^ xj Þ ¼ b1 , therefore since (i) If i = 1, then j2J j2Gðb Þ j2JnGðb Þ j2Gðb1 Þ ða1j ^ xj Þ ¼ b1 1 1 W W j2JnGðb1 Þ a1j < b1 . Since a1j P b1 for all j 2 G(b1) and a1j ^ xj < b1 for all j 2 J, we have that j2Gðb1 Þ xj ¼ b1 and xj < b1, which implies that jG(b1)j = jJ1j = +1.h i hW i W W W W (ii) If i > 1, then ðaij ^ xj Þ ¼ ðaij ^ xj Þ _ ðaij ^ xj Þ ¼ bi : Note that j2J j2Gðb Þ j2JnGðb Þ j2JnGðbi Þ ðaij ^ xj Þ 6 j2JnGðbi Þ aij < bi . i i W ða ^ x Þ ¼ b and jG(b )j = +1. If jJ j = +1, then we come to the conclusion. Otherwise, by Hence ij j i i i j2Gðb Þ i i1 i1 J i ,Gðbi Þ n [k¼1;bk – bi G1 ðbk Þ , we have [k¼1;bk – bi G1 ðbk Þ ¼ þ1 which implies that there exists an index r 2 {1, 2, . . . , i 1} such that br – bi and jG1(br)j = +1. Hence, jG(br)j = +1. If jJrj = +1, then the conclusion is obtained. Otherwise, continue as above. Conversely, suppose that there exists an index i0 2 I such that jJ i0 j ¼ þ1. Then we can choose j1 2 J1, j2 2 J2nJ1, W j3 2 J3n(J2 [ J1), . . . , jm 2 Jmn(J1 [ [ Jm1) since X – ;; Gðbi Þ – ;; j2JnGðbi Þ aij < bi for all i 2 I and J2i = ; for all i 2 In{1}. Define X = (xj)j2J with
8 > < bk ; if j ¼ jk with k 2 f1; 2; . . . ; mg n fi0 g; xj ¼ xj ; if j 2 J i0 ; > : 0; else; W
in which xj satisfies that xj < bi0 and
j2Ji
0
xj ¼ bi0 . For every i 2 I, there are two cases as follows:
(i) If i = i0, then
2 Ai0 X ¼ ðai0 j1 ^ b1 Þ _ _ 4
_
3
¼ ðai0 j1
^ b1 Þ _ _ @
0
_
1
3
_
ðai0 j ^ xj Þ5 _ _ ðai0 jm ^ bm Þ _ 4
j2J i
0
2
ðai0 ^ 0Þ5
j2JnðJ i [fj1 ;j2 ;...;jm gnfji gÞ 0
0
xj A _ _ ðai0 jm ^ bm Þ:
j2J i
0
Note that in the last equality, if k < i0, then ai0 jk ^ bk 6 bk 6 bi0 . If k > i0, then ai0 jk ^ bk 6 ai0 jk . Again, from the structure of X, if xjk ¼ bk then jk 2 Jk. Thus jk R J i0 since jk 2 Jkn(J1 [ [ Jk1) and J2i = ; for all i 2 In{1}. Hence, in the case k > i0, we have that ai0 jk ^ bk 6 ai0 jk 6 bi0 . Therefore, Ai0 X ¼ bi0 .
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Q.-q. Xiong, X.-p. Wang / Information Sciences 252 (2013) 32–41
(ii) If i – i0, then
2
Ai X ¼ ðaij1 ^ b1 Þ _ _ 4
_
3
¼ ðaij1 ^ b1 Þ _ _ 4
0
_
3
_
ðaij ^ xj Þ5 _ _ ðaijm ^ bm Þ _ 4
j2J i
2
2
ðaij ^ 0Þ5
j2JnðJ i [fj1 ;j2 ;...;jm gnfji gÞ 0
3
0
ðaij ^ xj Þ5 _ _ ðaijm ^ bm Þ:
ð6Þ
j2J i
0
For the index j of every aij in equality (6), distinguishing two situations, we can have: Case (a) . It is jk with k 2 {1, 2, . . . , m}n{i0}. This includes three subcases as follows. – If k < i, then aijk ^ bk 6 bk 6 bi . – If k > i, then aijk ^ bk 6 aijk 6 bi since J2i = ; for all i 2 In{1}. – If k = i, then aijk ^ bk ¼ bk ¼ bi . W W Case (b) . j 2 J i0 . In this case, if bi0 > bi , then i0 > i and j R Ji, therefore j2Ji ðaij ^ xj Þ 6 j2Ji aij 6 bi since J2i = ; for all i 2 In{1}. W W 0 0 Otherwise, if bi0 6 bi , then j2Ji ðaij ^ xj Þ 6 j2Ji xj 6 bi0 6 bi . 0
0
Cases (a) and (b) imply that Ai X = bi. Finally, with (i) and (ii) we have that X 2 X. Again, since aiji ^ xji ¼ aiji ^ bi ¼ bi if ðþÞ T ðÞ i – i0, and aij ^ xj < bi for all j 2 J if i = i0, we know that X 2 Xi2 Xi0 2 , that is to say X 2 XðÞ : h Note that Theorem 3.10 has no topological assumption, and its condition J2i = ; for all i 2 In{1} cannot be moved. For example, Example 3.3. Consider the system of equations
If the system has a solution, then x3 P 34 by Eq. (9), a23 ^ x3 = b2 and a33 ^ x3 = b3. Moreover, x1 P 13 or x2 ¼ 13 by Eq. (7), therefore a12 ^ x2 = b1 or a11 ^ x1 = b1. Consequently, all the solutions of the system are attainable. However, J22 – ; and J23 – ;. The system has partially attainable solutions
X1 ¼
T 1 3 ; 0; ; x4 ; x5 ; x6 ; ; 3 4
with xj < 12 for j = 4, 5, . . . , and
Wþ1
j¼4 xj
X2 ¼
1 3 0; ; ; x4 ; x5 ; x6 ; 3 4
T
¼ 12.
Theorem 3.11. Let XðþÞ – ;. Then XðþÞ ¼
S
T X 2X0 ½X ; A
B.
Proof. Let X ¼ ðxj Þj2J 2 X. If for every i 2 I there exists an index j 2 J such that aij ^ xj = bi, then there is an X 2 X0 such that S X⁄ 6 X by Theorem 3.8. This follows from Lemmas 2.2 and 3.3 that X 2 X 2X0 ½X ; AT B. h 4. Conclusions This paper discusses some properties of infinite sup–inf fuzzy relational equations on [0, 1]. First, we get some necessary and sufficient conditions for the existence of the attainable solutions (resp. the unattainable solutions and the partially attainable ones) when X – ;. Then we show the set of attainable solutions XðþÞ . However, we know that the solution set X is exactly divided into three separate parts, i.e., XðþÞ ; XðÞ and XðÞ , respectively, with X ¼ XðþÞ [ XðÞ [ XðÞ and XðþÞ \ XðÞ ¼ XðþÞ \ XðÞ ¼ XðÞ \ XðÞ ¼ ;. As one can see that we just have obtained the structure of the first one in this paper, consequently, in order to completely describe X, we have to investigate the other parts of X. Unfortunately, from the results as above mentioned together with those in [25,17] we know that the study of both solution sets XðÞ and XðÞ remain open when jIj P 2. Acknowledgement The authors thank the Editor-in-Chief Professor W. Pedrycz and the anonymous referees for their valuable comments.
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41
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A. Abbasi Molai, Resolution of a system of the max-product fuzzy relation equations using LU-factorization, Information Sciences, 234 (2011) 86–96. A. Abbasi Molai, E. Khorram, An algorithm for solving fuzzy relation equations with max-T composition operator, Inform. Sci. 178 (2008) 1293–1308. G. Birkhoff, Lattice theory, 3rd ed., vol, XXV, American Mathematical Society Colloquium Publications, Providence, R.I., 1979. B. De Baets, Analytical solution methods for fuzzy relational equations, in: D. Dubois, H. Prade (Eds.), The Handbooks of Fuzzy Sets Series, vol. 1, Kluwer Academic Publishers., Dordrecht, 2000, pp. 291–340. A. Di Nola, S. Sessa, W. Pedrycz, M. Higashi, Minimal and maximal solutions of a decomposition problem of fuzzy relations, Int. J. General Syst. 11 (1985) 103–116. A. Di Nola, S. Sessa, W. Pedrycz, E. Sanchez, Fuzzy Relation Equations and Their Applications to Knowledge Engineering, Kluwer Academic Publishers, Dordrecht, 1989. H. Imai, K. Kikuchi, M. Miyakoshi, Unattinable solutions of a fuzzy relation equation, Fuzzy Sets Syst. 99 (1998) 193–196. H. Imai, M. Miyakoshi, T. Date, Some properties of minimal solutions for a fuzzy equation, Fuzzy Sets Syst. 90 (1997) 335–340. J.L. Lin, Y.K. Wu, S.M. Guu, On fuzzy relational equations and the covering problem, Information Sciences, 181 (2011) 2951–2963. M. Miyakoshi, H. Imai, T. Da-te, Properties of a set of minimal solutions for a fuzzy relation equation, Bull. Fac. Enging. Hokkaido Univ 167 (1994) 1–10. in Japanese. M. Mizumoto, H.J. Zimmermann, Comparison of fuzzy reasoning methods, Fuzzy Sets Syst. 8 (1982) 253–283. K. Peeva, Resolution of Fuzzy Relational Equations-Method, Algorithm and Software with Applications, Information Sciences, 234 (2011) 44–63. I. Perfilieva, L. Nosková, System of fuzzy relation equations with inf-? composition: complete set of solutions, Fuzzy Sets Syst. 159 (2008) 2256–2271. X.B. Qu, X.P. Wang, Some properties of infinite fuzzy relational equations on complete Brouwerian lattices, Fuzzy Sets Syst. 158 (2007) 1327–1339. E. Sanchez, Resolution of composite fuzzy relation equations, Inform. Control 30 (1976) 38–48. B.S. Shieh, Solutions of fuzzy relation equations based on continuous t-norms, Inform. Sci. 177 (2007) 4208–4215. B.S. Shieh, Infinite fuzzy relation equations with continuous t-norms, Inform. Sci. 178 (2008) 1961–1967. B.S. Shieh, Deriving minimal solutions for fuzzy relation equations with max-product composition, Inform. Sci. 178 (2008) 3766–3774. B.S. Shieh, Minimizing a linear objective function under a fuzzy max-t norm relation equation constraint, Inform. Sci. 181 (2011) 832–841. M. Wagenknecht, K. Hartmann, On the existence of minimal solutions for fuzzy equations with tolerances, Fuzzy Sets Syst. 34 (1990) 237–244. X.P. Wang, Method of solution to fuzzy relation equations in a complete Brouwerian lattice, Fuzzy Sets Syst. 120 (2001) 409–414. X.P. Wang, X.B. Qu, Continuous join-irreducible elements and their applications to describe the solution set of fuzzy relational equations, Acta Math. Sinica, Chinese Series 49 (2006) 1171–1180. in Chinese. X.P. Wang, C. Xia, The solution sets of infinite fuzzy relational equations with sup-conjunctor composition complete distributive lattices, Fuzzy Sets Syst. 160 (2009) 2989–3006. X.P. Wang, S. Zhao, Solution sets of finite fuzzy relation equations with sup–inf composition over bounded Brouwerian lattices, Information Sciences, 234 (2011) 80–85. Q.Q. Xiong, X.P. Wang, Some properties of sup-min fuzzy relational equations on infinite domains, Fuzzy Sets Syst. 151 (2005) 393–402. C.T. Yeh, On the minimal solutions of max–min fuzzy relational equations, Fuzzy Sets Syst. 159 (2008) 23–39. L.A. Zadeh, Toward a generalized theory of uncertainty (GTU) – an outline, Inform. Sci. 172 (2005) 1–40. L.A. Zadeh, Is there a need for fuzzy logic?, Inform Sci. 178 (2008) 2751–2779.
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Information Sciences journal homepage: www.elsevier.com/locate/ins
Some properties of infinite fuzzy relational equations with sup–inf composition q Qing-quan Xiong, Xue-ping Wang ⇑ College of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan 610066, PR China
a r t i c l e
i n f o
Article history: Available online 2 August 2011 Keywords: Fuzzy relation Fuzzy relational equation Minimal solution Solution set
a b s t r a c t This paper deals with infinite fuzzy relational equations with sup–inf composition. First, some properties of them are investigated. Then some necessary and sufficient conditions for the existence of attainable solutions (resp. unattainable solutions and partially attainable ones) are given if the solution set is nonempty. Finally, the set of attainable solutions is described. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The study of fuzzy relational equations is one of the most appealing subjects in the field of fuzzy sets and fuzzy logic [27,28], both from a mathematical and a systems modeling point of view [6]. Sanchez [15] was the first one who studied fuzzy relational equations with max–min composition. Since then, several authors further enlarged the theory with different composite operators over various lattices, see [1,2,4–10,12–26]. A system of fuzzy relational equations is of the form
AX ¼B
ð1Þ
or
_ ðaij ^ xj Þ ¼ bi for all i 2 I; j2J
where is a sup–inf composition, X = (xj)j2J is unknown, A = (aij)IJ and B = (bi)i2I are known with aij, xj, bi 2 [0, 1]. The system of fuzzy relational equations is denoted by Eq. (1) and its solution set is defined as X ¼ fX ¼ ðxj Þj2J : A X ¼ Bg. A special case of Eq. (1) is as follows:
AX ¼b
ð2Þ
or
_ ðaj ^ xj Þ ¼ b; j2J
q Supported by National Natural Science Foundation of China (No. 10671138) and Doctoral Fund of Ministry of Education of China (No. 20105134110002). ⇑ Corresponding author. E-mail addresses: [email protected] (Q.-q. Xiong), [email protected] (X.-p. Wang).
0020-0255/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2011.07.041
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Q.-q. Xiong, X.-p. Wang / Information Sciences 252 (2013) 32–41
where A = (aj)j2J is a row vector with aj 2 [0, 1], b 2 [0, 1]. The equation is signed by Eq. (2) whose solution set is defined by X2 ¼ fX ¼ ðxj Þj2J : A X ¼ bg. In order to completely describe X, Wagenknecht and Hartmann [20] gave some conditions for the existence of minimal solutions when J is a compact metric space. Miyakoshi, Imai and Da-te [10] introduced the notions of attainable solution and unattainable one, and got a necessary and sufficient condition for the existence of the attainable solution. Imai et al. [8,7] further showed a necessary and sufficient condition for the existence of a minimal solution without any topological assumption, moreover, they also obtained some sufficient conditions for the existence of a partially attainable solution (resp. the unattainable solution). Wang and Qu described X2 in both cases that b is a join-irreducible element and b has an irredundant continuous join-decomposition, see [14,22], respectively. Wang and Xia [23] dealt with fuzzy relational equations with supconjunctor composition on complete distributive lattices, they checked a necessary and sufficient condition for the existence of the attainable solution (resp. the unattainable solution) under the condition that either the right-hand side of equations is a continuous join-irreducible element or it has an irredundant continuous join-decomposition. However, they did not discuss the system of equations. Shieh [17] extended the results of Xiong and Wang [25] to a more general condition and investigated some properties of the solutions, while he described the solution sets which just include the attainable solutions and the unattainable ones. Indeed, Infinite fuzzy relational equations occur in practice. For instance, in fuzzy reasoning [11], antecedents and consequences are usually defined on infinite sets. Consider an inference which contains a fuzzy conditional proposition: Ant 1: If x is A, then y is B, Ant 2: y is B0 , Cons: x is A0 . This inference can be viewed as a generalized modus tollens, where the Ant 1 ‘‘If x is A, then y is B’’ represents a certain relationship between A and B (the inference rule), and the Ant 2 ‘‘y is B0 ’’ is a known consequence. Now the problem of determining antecedents A0 (the Cons ‘‘x is A0 ’’) can be interpreted as the form of fuzzy relational equation A0 R = B0 , where A0 , B0 and R are fuzzy sets and fuzzy relation, respectively, with B0 and R known. Thus, its results can be applied to the problem as the universal set of A0 is an infinite set and ‘‘’’ is a composition operator (see e.g. [11,17] for detail). This paper will investigate Eq. (1) on [0, 1] without any topological assumption. Its rest is organized as follows. For the sake of convenience, some notions and previous results are summarized in Section 2. Some necessary and sufficient conditions which describe the attainable solutions (resp. the unattainable solutions and the partially attainable ones) are shown, and the set of attainable solutions is obtained in Section 3. Conclusions are drawn in Section 4. 2. Previous results In the whole paper, unless otherwise stated, we assume that I is finite, J is countably infinite, A = (aij)IJ is a matrix, X = (xj)j2J and B = (bi)i2I are two column vectors with aij, xj, bi 2 [0, 1]. In what follows, we shall recall some basic concepts and important results for the sake of convenience.
Definition 2.1 [3]. Let (P, 6) be a partially ordered set and X # P. An element p 2 X is minimal if and only if for all x 2 X if x 6 p then p = x. The greatest element of X is an element g 2 X such that x 6 g for all x 2 X. Definition 2.2 [3]. Let a, b 2 [0, 1]. Define aab = max{x 2 [0, 1] : a ^ x 6 b}. Let Ai = (aij)j2J be the ith row vector of A; Xi2 ¼ fX ¼ ðxj Þj2J : Ai X ¼ bi g for each i 2 I. Denote the set of all minimal elements of X; X2 and X0 ; X02 and X0i2 , respectively. a _ b = sup{a, b}, a ^ b = inf{a, b}. V Xi2 by T T T Ai abi ¼ ðaij abi Þj2J ; A B ¼ , where A represents the transpose of A. Then we have: i2I aij abi j2J
Lemma 2.1 [21]. (1) X – ; if and only if \i2I Xi2 – ;. Moreover, X ¼ \i2I Xi2 ; (2) If X – ;, then X⁄ = solution of Eq. (1). Lemma 2.2 [15]. X – ; if and only if AT Lemma 2.3 [25]. X2 – ; if and only if
W
B 2 X. Further, X 6 AT j2Jaj
V
abi)T is the greatest
i2I(Ai
B for all X 2 X.
P b.
Definition 2.3 [8]. A solution X = (xj)j2J of Eq. (1) is called attainable if for every i 2 I there exists an index j0 2 J such that aij0 ^ xj0 ¼ bi , the set of them is denoted by XðþÞ . A solution X = (xj)j2J of Eq. (1) is called unattainable if aij ^ xj < bi for all i 2 I and j 2 J, the set of them is set as XðÞ . A solution X of Eq. (1) which is neither attainable nor unattainable is called partially attainable whose set is defined by XðÞ .
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Q.-q. Xiong, X.-p. Wang / Information Sciences 252 (2013) 32–41
Example 2.1. Consider the system of equations
8 þ1 þ1 1 1 W 1 W 1 > > > ¼ 13 ; ^ x ^ x ^ x ^ x _ _ _ 1 2 2n3 2n2 > 4 3 nþ1 < 3 n¼3 n¼3 þ1 þ1 > 1 1 W 1 W 1 > > > _ ^ x ^ x ^ x ^ x _ _ ¼ 12 : 1 2 2n3 2n2 : 2 3 nþ1 2 n¼3
n¼3
T It is obvious that X 1 ¼ 12 ; 0; . . . ; 0; . . . is attainable, X2 = (0, 0, x3, x4, . . .)T is unattainable with x2k3 < 13 ; x2k2 < 12 for T Wþ1 Wþ1 1 k = 3, 4, . . . , k¼3 x2k3 ¼ 3 and k¼3 x2k2 ¼ 12, and X 3 ¼ 13 ; 0; x3 ; x4 ; . . . is partially attainable with xk < 12 for k = 3, 4, . . . , and Wþ1 1 k¼3 xk ¼ 2. From Definition 2.3, we know that X ¼ XðþÞ [ XðÞ [ XðÞ and XðþÞ \ XðÞ ¼ XðþÞ \ XðÞ ¼ XðÞ \ XðÞ ¼ ;. Therefore, the solution set X is exactly divided into three separate parts. Lemma 2.4 [8]. Let X ¼ ðxj Þj2J 2 XðÞ . Then j{j 2 J : xj > 0}j = +1, where j{ }j denotes the cardinality of set { }. 3. Resolution of the system of equations Let I0 = {i 2 I : bi = 0} and J I0 ¼ fj 2 J : aij > 0 for some i 2 I0}. Then every solution X = (xj)j2J of Eq. (1) is that xj = 0 for all j 2 J I0 . Thus, it is possible to omit the equations with indices from I0 and the columns of A with indices from J I0 (cfr. [26]). Therefore, in what follows, we assume that bi > 0 for all i 2 I and 1 < jIj = m < +1, jJj = +1. Denote G1(bi) = {j 2 J : aij > bi}, G2(bi) = {j 2 J : aij = bi} and G(bi) = G1(bi) [ G2(bi) for all i 2 I. Using the results in [17,25], we know that the known coefficients of the equation Ai X = bi with i 2 I have only four cases as follows: (i) (ii) (iii) (iv)
W Gðbi Þ – ;; j2JnGðbi Þ aij < bi ; W G(bi) = ;, j2Jaij = bi; W Gðbi Þ – ;; j2JnGðbi Þ aij ¼ bi ; W G(bi) = ;, j2Jaij < bi.
By Lemmas 2.1 and 2.3, if A has a row vector Ai satisfying the case (iv), then the system of equations has no solutions. In W fact, if there exists an i0 2 I such that Gðbi0 Þ ¼ ; and j2J ai0 j < bi0 , then Xi0 2 ¼ ; by Lemma 2.3. Therefore, X ¼ ; by Lemma 2.1. Furthermore, if Eq. (1) has solutions, then every row vector of A must satisfy one of cases (i), (ii) and (iii). In the rest of the paper, we assume that Xi2 – ; for all i 2 I and b1 6 b2 6 6 bm with m , jIj < +1.
Lemma 3.1 [25]. If aj < b for all j 2 J and Lemma 3.2. If G(bi) = ; and
W
j2Jaij
W
j2Jaj
= b, then X2 – ;:
= bi for all i 2 I, then X – ;:
Proof. X = (1)j2J is a solution of Eq. (1).
h
Lemma 3.3 [15]. Let X 1 ; X 2 2 X and X such that X1 6 X 6 X2. Then X 2 X. Lemma 3.4 [25]. If Xi2 – ; and G(bi) = ;, then Lemma 3.5 [25]. If X ¼ ðxj Þj2J 2 X2 , then Lemma 3.6. If X ¼ ðxj Þj2J 2 X, then
W
j2Jxj
W
W
j2Jxj
P
j2J aij
ðÞ
¼ bi ; Xi2 ¼ Xi2 and X0i2 ¼ ;:
P b.
W
i2Ibi.
Theorem 3.1. If X – ; and there exists an i0 2 I such that Gðbi0 Þ ¼ ;, then XðþÞ ¼ ;: ðÞ
Proof. Let X ¼ ðxj Þj2J 2 X. Then Ai0 X ¼ bi0 . By Lemma 3.4 and Gðbi0 Þ ¼ ;, we have X 2 Xi0 2 . Therefore, XðþÞ ¼ ; by Definition 2.3. h Theorem 3.2. If X – ;, then
W
j2Jaij
P bi for all i 2 I.
35
Q.-q. Xiong, X.-p. Wang / Information Sciences 252 (2013) 32–41
W
Proof. Let X ¼ ðxj Þj2J 2 X. Then bi =
j2J(aij
^ xj) 6
W
j2Jaij
for all i 2 I. h
Remark 3.1. The inverse of Theorem 3.2 is not true when jIj P 2. Example 3.1. Consider the system of equations
8 þ1 1 W 1 > 1 > ¼ 12 ; ^ x 1 _ ^ x > 1 n < 2 4 n n¼2 þ1 W 1 > > 1 > ð1 ^ x1 Þ _ 1 ^ x ¼ 13 : : n 4 n n¼2
W
Then j2Jaij P bi for all i 2 I. However, the system of equations has no solution. Define J 1 ¼ Gðb1 Þ; J i ¼ Gðbi Þ n [i1 k¼1;bk –bi G1 ðbk Þ , and J2i = {j 2 Ji : j 2 G(bk) for some k 2 {1, 2, . . . , i 1}} for all i 2 In{1}. Then the following characterization holds: Theorem 3.3. XðþÞ – ; if and only if Ji – ; for all i 2 I. Proof. In the first part, suppose that there exists an i0 2 I such that J i0 ¼ ;. Since X ¼ ðxj Þj2J 2 XðþÞ , there exists an index j 2 J such that ai0 j ^ xj ¼ bi0 , which implies that xj P bi0 and ai0 j P bi0 , i.e., Gðbi0 Þ – ;. If i0 = 1, then from Lemma 3.4 we have that ðÞ X 2 Xi0 2 since J i0 ¼ Gðbi0 Þ ¼ ;, which contradicts the fact that X 2 XðþÞ . Therefore i0 > 1. By the definition of Ji and Gðbi0 Þ – ;, we can assert that there must exist an index k 2 {1, 2, . . . , i0 1} such that j 2 G1(bk) since J i0 ¼ ;, that is to say, akj > bk and W bi0 – bk . Therefore, bi0 > bk and akj ^ xj P akj ^ bi0 > bk . The last inequality implies that j2J(akj ^ xj) > bk, i.e., X R Xk2 , which is in the opposition to Lemma 2.1 (1). Therefore, Ji – ; for all i 2 I. In the converse implication, let Ji – ; for all i 2 I. Then. (i) If \i2IJi – ;, then by the definition of Ji we can define X = (xj)j2J with
xj ¼
bm ; j 2 \i2I J i ; 0;
otherwise:
We first note that for j 2 \i2IJi, aij = bi 6 bm if i – m, and aij P bi if i = m. Therefore, if i – m then
2
Ai X ¼ 4
3
_
2
ðaij ^ bm Þ5 _ 4
j2\i2I J i
3
_
ðaij ^ 0Þ5 ¼
j2Jnð\i2I J i Þ
_
ðaij ^ bm Þ ¼ aij ¼ bi
j2\i2I J i
and if i = m then
2 Ai X ¼ 4
3
_
2
ðaij ^ bm Þ5 _ 4
j2\i2I J i
3
_
ðaij ^ 0Þ5 ¼
j2Jnð\i2I J i Þ
_
ðaij ^ bm Þ ¼ bm :
j2\i2I J i
Thus X 2 X. Moreover, for every j 2 \i2IJi, we know that aij ^ bm = aij = bi for i – m and aij ^ bm = bm for i = m, i.e., X 2 XðþÞ . (ii) If \i2IJi = ; and Js \ Jt = ; for every s, t 2 I with s – t, then we define X = (xj)j2J with
xj ¼
bi ; if j 2 J i with i ¼ 1; 2; . . . ; m; 0;
otherwise:
Thus
2 " # " # " # _ _ _ _ 6 ðaij ^ xj Þ ¼ ðaij ^ xj Þ _ _ ðaij ^ xj Þ _ _ ðaij ^ xj Þ _ 4 j2J
j2J 1
" ¼
_ j2J 1
" ¼
_ j2J 1
j2J i
j2J m
j2Jnð[m J k¼1 k Þ
2 " # " # _ _ 6 ðaij ^ b1 Þ _ _ ðaij ^ bi Þ _ _ ðaij ^ bm Þ _ 4 #
j2J i
# ðaij ^ b1 Þ _ _ bi _ _
j2J m
"
_
#
_
_
3 7 ðaij ^ xj Þ5 3
7 ðaij ^ 0Þ5 j2Jnð[m J k¼1 k Þ
ðaij ^ bm Þ
j2J m
W for all i 2 I. In the last equality, if k 6 i then j2Jk ðaij ^ bk Þ 6 bk 6 bi , otherwise, by the definition of Jk we have aij 6 bi 6 bk, W W W hence j2Jk ðaij ^ bk Þ 6 j2Jk aij 6 bi . Therefore, j2J(aij ^ xj) = bi for every i 2 I, i.e., X 2 X. Since aij ^ xj = aij ^ bi = bi for every i 2 I and j 2 Ji, we have X 2 XðþÞ :
36
Q.-q. Xiong, X.-p. Wang / Information Sciences 252 (2013) 32–41
(iii) If \i2IJi = ; and Js \ Jt – ; for some s, t 2 I and s – t, then we define X = (xj)j2J with
8 > < bs _ bt ; j 2 J s \ J t ; xj ¼ b i ; j 2 J i with i ¼ 1; 2; . . . ; m and i – s; t; > : 0; otherwise: The fact that X 2 XðþÞ is immediate from the proofs of (i) and (ii). Finally, with (i), (ii) and (iii) we have that XðþÞ – ;: Theorem 3.4. If G(bi) = ; and
W
j2Jaij
h
= bi for all i 2 I, then X ¼ XðÞ and X0 ¼ ;:
Proof. Y ¼ ð1Þj2J 2 X implies that X – ;. Now, let X ¼ ðxj Þj2J 2 X. Then aij ^ xj 6 aij < bi for all i 2 I and j 2 J, i.e., X 2 XðÞ . Therefore, X ¼ XðÞ . It follows from Lemma 2.4 that j{j 2 J : xj > 0}j = +1. For every j0 2 {j 2 J : xj > 0}, define X⁄ = (xj⁄)j 2J with
xj ¼
0;
if j ¼ j0 ;
xj ; if j – j0 :
It is easy to show that X 2 X; X 6 X and X⁄ – X, therefore, X0 ¼ ;: By Theorem 3.4, we have:
h
Theorem 3.5 [17]. If X0 – ;, then G(bi) – ; for all i 2 I. Note that Theorem 3.5 is not invertible even if X – ; and G(bi) – ; for all i 2 I, as is proved in the next example: Example 3.2. Consider the system of equations
8 þ1 þ1 1 W 1 W 1 > > ^ x ^ x ^ x ¼ 14 ; _ _ > 1 2n2 2n1 < 3 4 nþ1 n¼2 n¼2 þ1 W > > n > 1 ^ x1 _ 2 ^ x2 _ ^ x ¼ 12 : : n 2 5 2nþ1 n¼3
It is obvious that the system has no minimal solutions. However, we have the following theorem: Theorem 3.6. X0 – ; if and only if Ji – ; for all i 2 I. Proof. For proving the first part, let Ji – ; for all i 2 I. Then XðþÞ – ; by Theorem 3.3. Let X ¼ ðxj Þj2J 2 XðþÞ . Then W j2J(aij ^ xj) = bi for all i 2 I, and for every i 2 I there exists a ji 2 J such that aiji ^ xji ¼ bi . Then: (i) If \i2IJi – ;, then we can choose one index j0 2 \i2IJi by the definition of Ji, and define X⁄ = (xj⁄)j2J with
xj ¼
b m ; j ¼ j0 ; 0;
otherwise:
Then we easily show that X 2 X0 , i.e., X0 – ;: (ii) If \i2IJi = ; and Js \ Jt = ; for all s, t 2 I and s – t. Then let J0 = {j1, j2, . . . , jm} with jk 2 Jk and define X⁄ = (xj⁄)j2J with
xj ¼
b i ; j ¼ ji 2 J 0 ; 0;
otherwise:
Then it is easy to see that X 2 X0 , i.e., X0 – ;. (iii) If \i2IJi = ; and Js \ Jt – ; for some s, t 2 I and s – t. Then we construct X⁄ as follows: Choose j1 2 Jm, let I1 = {k 2 I : j1 2 Jk}. Then jI1j < m since \i2IJi = ;. Choose j2 2 J r2 with r2 = max(InI1), let I2 = {k 2 I : j2 2 Jk}. If jI1 [ I2j < m then .. . Choose jp 2 J rp with r p ¼ max I n [p1 Ik , let Ip = {k 2 I : j2 2 Jk}. k¼1 P Continue as above till pk¼1 jIk j ¼ m. The above proceeding implies that I1, I2, . . . , Ip is a partition of I. Define X⁄ = (xj⁄)j2J with
xj ¼
8W < bi ; j ¼ jk and Ik – ;; :
k ¼ 1; 2; ; p:
i2Ik
0;
otherwise:
Then for all i0 2 I there exists exactly one q 2 {1, 2, . . . , p} such that i0 2 Iq. Therefore, with the structure of X⁄ we have
Q.-q. Xiong, X.-p. Wang / Information Sciences 252 (2013) 32–41
2 _ ðai0 j ^ xj Þ ¼ ðai0 j1 ^ xj1 Þ _ _ ðai0 jp ^ xjp Þ _ 4 j2J
¼
_
ai0 j1 ^
bk
j2Jnfj1 ;j2 ;;jp g
0
!
_
_
_ _ @ai0 jq ^
1
3 ðai0 j ^ xj Þ5 ¼ ðai0 j1 ^ xj1 Þ _ _ ðai0 jp ^ xjp Þ 0
bi A _ _ @ai0 jp ^
i2Iq
k2I1
37
_
1 b k A:
k2Ip
W W In the last equality, if i0 = maxIq, then ai0 jq P i2Iq bi ¼ bi0 . If i0 < maxIq, then ai0 jq ¼ bi0 6 i2Iq bi . Consequently, W W W ai0 jq ^ i2Iq bi ¼ bi0 . Moreover, ai0 jk ^ s2Ik bs 6 bi0 for all k 2 {1, 2, . . . , p}n{q}. Indeed, if ai0 jk ^ s2Ik bs > bi0 , then ai0 jk > bi0 and W W s2Ik bs > bi0 . Let t = maxIk. Then bt > bi0 , so that jk 2 Jt, which contradicts the definition of Jt. Therefore, j2J ðai0 j ^ xj Þ ¼ bi0 for all i0 2 I, i.e., X 2 X. Now, suppose that X 2 X satisfies that X < X⁄. Without loss of generality, we assume that W W xjk < xjk ¼ i2Ik bi for Ik – ;. Then for the tth equation, with analogous proof to that of X 2 X we have that j2J(atj ^ xj) < bt. 0 0 Therefore, X 2 X , this implies that X – ;. W The converse implication, let X ¼ ðxj Þj2J 2 X0 . Then j2J(aij ^ xj) = bi for all i 2 I. We claim that there must exist an index ji 2 J such that aiji ^ xji ¼ bi for all i 2 I. Otherwise, if there exists an index i 2 I such that aij ^ xj < bi for all j 2 J, then j{j 2 J : xj > 0}j = +1 by Lemma 2.4. Choose an index j0 2 {j 2 J : xj > 0}, and define X⁄ = (xj⁄)j2J with
xj ¼
0;
j ¼ j0 ;
xj ; j – j 0 :
Then X 6 X; X 2 X and X⁄ – X, which contradicts the fact of X 2 X0 . Therefore, X 2 XðþÞ : Further, with Theorem 3.3 we have that Ji – ; for all i 2 I. h From the proof of Theorem 3.6, the following three corollaries hold: Corollary 3.1. If X ¼ ðxj Þj2J 2 X0 , then j{j 2 J : xj⁄ > 0}j 6 m. Corollary 3.2 [17]. X0 # XðþÞ . Corollary 3.3. XðþÞ – ; if and only if X0 – ;. With Theorems 3.3, 3.6 and Corollary 3.3, we can easily obtain Theorem 5 in [8]. Theorem 3.7. Suppose that XðþÞ – ; and J2i = ; for all i 2 In{1}. Then the following three statements hold: (1) Let j1 2 J1, j2 2 J2nJ1, j3 2 J3n(J1 [ J2), . . . , jm 2 Jmn(J1 [ [ Jm1), and define X = (xj)j2J with
xj ¼
bk ; j ¼ jk with k 2 f1; 2; . . . ; mg; 0;
ð3Þ
otherwise:
Then X is a minimal element of X. (2) All minimal elements of X have the form of (3). Q (3) jX0 j ¼ i2I jJ i j:
Proof (1) It is easy to identify that X 2 X. Now, let X 0 ¼ ðx0j Þj2J 2 X and X0 6 X. For every j 2 J, if j – jk with k 2 {1, 2, . . . , m}, then xj = 0, which implies that x0j ¼ 0. Therefore, in order to prove that X 2 X0 , we just need to prove that if j = jk with k 2 {1, 2, . . . , m} then xjk ¼ x0jk . In fact, for all i 2 I
bi ¼
m m _ _ _ _ ðaijk ^ x0j Þ ¼ ðaijk ^ x0jk Þ 6 ðaijk ^ xjk Þ 6 ðaijk ^ xj Þ j2J
k¼1
¼ ðaij1 ^ b1 Þ _ _ ðaijm ^ bm Þ _
j2J
k¼1
"
_
#
ðaij ^ xj Þ :
ð4Þ
j2Jnfj1 ;j2 ;...;jm g
Since aijk ^ bk 6 aijk ; jk 2 J k and jk R Ji with k > i, we have that aijk 6 bi . Otherwise, we have jk 2 G1(bi), this follows that jk R Jk r1 since J k ¼ Gðbk Þ n [k¼1;b G1 ðbr Þ , a contradiction. Therefore, aijk ^ bk 6 aijk 6 bi , this with inequality (4) deduces that r –bk m _ k¼1
ðaijk ^ x0jk Þ ¼
m _
ðaijk ^ xjk Þ ¼ bi :
k¼1
ð5Þ
38
Q.-q. Xiong, X.-p. Wang / Information Sciences 252 (2013) 32–41
We assert that there must be x0jk ¼ xjk for all k 2 {1, 2, . . . , m}. Otherwise, if there exists an index k0 2 {1, 2, . . . , m} such that x0jk < xjk , then there are two cases: 0
0
(i) If i < k0, then ak0 ji ^ bi 6 bi . In this case, if ak0 ji ^ bi < bi , then ak0 ji ^ bi < bi 6 bk0 . If ak0 ji ^ bi ¼ bi , then there must be bi < bk0 . Otherwise, if bi ¼ bk0 , then ak0 ji ^ bi ¼ bi ¼ bk0 , which implies that ak0 ji P bi ¼ bk0 , thus ji 2 Ji and ji 2 J k0 , a contradiction since J2i = ; for all i 2 In{1}. Therefore, we always have that ak0 ji ^ bi 6 bi < bk0 . (ii) If i > k0, then ak0 ji ^ bi 6 ak0 ji for all ji 2 Ji. Again, from formula (3), if xji ¼ bi then ji 2 Ji, hence ji R J k0 since ji 2 Jin(J1 [ [ Ji1) and J2i = ; for all i 2 In{1}. Therefore, ak0 ji ^ bi 6 ak0 ji < bk0 . Consequently, with Cases (i) and (ii) we have that
ðak0 j1 ^ x0j1 Þ _ _ ðak0 jk ^ x0jk Þ _ _ ðak0 jm ^ x0jm Þ 6 ðak0 j1 ^ b1 Þ _ _ ðak0 jk ^ x0jk Þ _ _ ðak0 jm ^ bm Þ 0 0 0 0 ! ! k_ m 0 1 _ 6 bi _ x0jk _ ak0 ji < bk0 ; 0
i¼1
i¼k0 þ1
which contradicts the equality (5). Therefore, ¼ xjk , it results X = X, i.e., X 2 X0 : W (2) Let X ¼ ðxj Þj2J 2 X0 . Then bi = j2J(aij ^ xj), and Ji – ; for all i 2 I by Theorem 3.3. If there exists an index k 2 Ji \ Jj for i – j, without loss of generality, let i < j, then k 2 J2j. This is a contradiction since J2i = ; for every i 2 In{1}. Therefore, Ji \ Jj = ; for all i, j 2 I and i – j. We assert that for every i 2 I there must be an index ji 2 Ji such that xji P bi : Otherwise, if there exists an index i0 2 I such that xji < bi0 for all ji0 2 J i0 , then x0jk
0
0
3 " # 2 _ _ _ _ 4 5 ðai0 j ^ xj Þ ¼ ðai0 j ^ xj Þ _ _ ðai0 j ^ xj Þ _ _ ðai0 j ^ xj Þ _ 4 "
j2J
#
2
j2J 1
j2J i
j2Jm
0
_
3
ðai0 j ^ xj Þ5:
j2Jnð[m J k Þ k¼1
Analogously to the proof of Theorem 3.7 (1), we have
"
_
#
2
ðai0 j ^ xj Þ _ _ 4
j2J 1
_ j2J i
3 ðai0 j ^ xj Þ5 _ _
"
_
# ðai0 j ^ xj Þ < bi0 :
j2J m
0
W W In view of J2i = ; for all i 2 In{1}, we have that X 2 X0 and j2Jnð[m Jk Þ ðai0 j ^ xj Þ < bi0 , therefore j2J ðai0 j ^xj Þ< bi0 , a contradick¼1 tion since X 2 X. Therefore, for every k 2 I there exists an index jk 2 Jk such that xjk P bk : Define X 0 ¼ x0j with
x0j
¼
j2J
bk ; j ¼ jk with k 2 f1; 2; . . . ; mg; 0;
otherwise:
0
Obviously, X 2 X and X0 6 X, the minimality of X implies that X = X0 , therefore, all minimal elements of X have the form of (3). Q (3) From the proofs of (1) and (2), we have jX0 j ¼ i2I jJ i j: h
Remark 3.2. In Theorem 3.7, the assumption of J2i = ; for all i 2 In{1} cannot be dropped. For instance,
þ1 8 W 1 1 > > ^ x1 _ 14 ^ x2 _ 15 ^ x3 _ ^ xn ¼ 13 ; > 3 n > > n¼4 > > þ1 > > 1 1 1 W 1 > > ^ xn ¼ 12 ; > < 2 ^ x1 _ 2 ^ x2 _ 2 ^ x3 _ n n¼4 þ1 1 2 1 W 1 > > > ^ x ^ x ^ x ^ x _ _ _ ¼ 23 ; > 1 2 3 n 4 3 5 n > > n¼4 > > þ1 > > W 1 > > : 14 ^ x1 _ 14 ^ x2 _ ð1 ^ x3 Þ _ ^ xn ¼ 1: n n¼4
Then J1 = {1}, J2 = {1, 2, 3}, J3 = {2}, J4 = {3}, J22 = {1} – ;, J23 = {2} – ; and J24 = {3} – ;, hence jJ1j jJ2j jJ3j jJ4j = 3. However, T the system of equations has only one minimal solution X ¼ 13 ; 23 ; 1; 0; . . . . However, we have the following theorem: Theorem 3.8. Let XðþÞ – ;. Then X 2 XðþÞ if and only if there exists an X 2 X0 such that X⁄ 6 X. Proof. In the first part, let X ¼ ðxj Þj2J 2 XðþÞ . By Definition 2.3, there exists an index jk 2 J such that akjk ^ xjk ¼ bk for every k 2 {1, 2, . . . , m}, therefore bk 6 xjk and bk 6 akjk . Let x0jk ¼ inffx : akjk ^ x ¼ bk g for each k 2 I. Obviously, xjk 2 fx : akjk ^ x ¼ bk g, and akjk ^ x0jk ¼ bk since jIj < +1, this deduces that x0jk 6 xjk for all k 2 I. Now, let J0 be the set of all jks such that {k : jk 2 J0} = I and {k : jk 2 J0 } – I for any J0 $ J0. Define X⁄ = (xj⁄)j2J with
39
Q.-q. Xiong, X.-p. Wang / Information Sciences 252 (2013) 32–41
( xj ¼
x0jk ; if jk 2 J 0 ; 0;
otherwise:
Then X 2 X0 and X⁄ 6 X. In the converse implication, let X ¼ ðxj Þj2J 2 X. If there exists an X ¼ ðxj Þj2J 2 X0 such that X⁄ 6 X, then Ji – ; for all i 2 I. By Corollary 3.2, X 2 XðþÞ . This follows that for every i 2 I there exists an index ji 2 J such that aiji ^ xji ¼ bi . Since bi 6 aiji ^ xji 6 aiji ^ xji 6 bi for all i 2 I, we have that for every i 2 I there exists an index ji 2 J such that aiji ^ xji ¼ bi , i.e., X 2 XðþÞ : h Theorem 3.9. If G(bi) = ; for all i 2 I, then XðÞ ¼ ;. Proof. Suppose that XðÞ – ; and X ¼ ðxj Þj2J 2 XðÞ . Then from Definition 2.3 there exist two nonempty sets I1, I2 # I with I1 [ I2 = I and I1 \ I2 = ; such that aij ^ xj < bi for all i 2 I1, j 2 J, and for every k 2 I2 there exists an index jk 2 J satisfying akjk ^ xjk ¼ bk , i.e., akjk P bk , a contradiction since G(bk) = ;. Therefore, XðÞ ¼ ;. h Corollary 3.4. If XðÞ – ;, then there exists an index i0 2 I such that Gðbi0 Þ – ;. W
Theorem 3.10. Let X – ;; Gðbi Þ – ;; an i 2 I such that jJij = +1.
j2JnGðbi Þ aij
< bi for all i 2 I and J2i = ; for all i 2 In{1}. Then XðÞ – ; if and only if there exists
Proof. Let us assume that X ¼ ðxj Þj2J 2 XðÞ . Then from Definition 2.3 there are two nonempty subsets I1 and I2 of I with I1 [ I2 = I and I1 \ I2 = ; which satisfy that there exists an index j 2 J such that aij ^ xj = bi for all i 2 I1, and aij ^ xj < bi for all i 2 I2, j 2 J. For i 2 I2, there are two cases: hW i hW i W W ða1j ^ xj Þ ¼ ða1j ^ xj Þ _ ða1j ^ xj Þ ¼ b1 , therefore since (i) If i = 1, then j2J j2Gðb Þ j2JnGðb Þ j2Gðb1 Þ ða1j ^ xj Þ ¼ b1 1 1 W W j2JnGðb1 Þ a1j < b1 . Since a1j P b1 for all j 2 G(b1) and a1j ^ xj < b1 for all j 2 J, we have that j2Gðb1 Þ xj ¼ b1 and xj < b1, which implies that jG(b1)j = jJ1j = +1.h i hW i W W W W (ii) If i > 1, then ðaij ^ xj Þ ¼ ðaij ^ xj Þ _ ðaij ^ xj Þ ¼ bi : Note that j2J j2Gðb Þ j2JnGðb Þ j2JnGðbi Þ ðaij ^ xj Þ 6 j2JnGðbi Þ aij < bi . i i W ða ^ x Þ ¼ b and jG(b )j = +1. If jJ j = +1, then we come to the conclusion. Otherwise, by Hence ij j i i i j2Gðb Þ i i1 i1 J i ,Gðbi Þ n [k¼1;bk – bi G1 ðbk Þ , we have [k¼1;bk – bi G1 ðbk Þ ¼ þ1 which implies that there exists an index r 2 {1, 2, . . . , i 1} such that br – bi and jG1(br)j = +1. Hence, jG(br)j = +1. If jJrj = +1, then the conclusion is obtained. Otherwise, continue as above. Conversely, suppose that there exists an index i0 2 I such that jJ i0 j ¼ þ1. Then we can choose j1 2 J1, j2 2 J2nJ1, W j3 2 J3n(J2 [ J1), . . . , jm 2 Jmn(J1 [ [ Jm1) since X – ;; Gðbi Þ – ;; j2JnGðbi Þ aij < bi for all i 2 I and J2i = ; for all i 2 In{1}. Define X = (xj)j2J with
8 > < bk ; if j ¼ jk with k 2 f1; 2; . . . ; mg n fi0 g; xj ¼ xj ; if j 2 J i0 ; > : 0; else; W
in which xj satisfies that xj < bi0 and
j2Ji
0
xj ¼ bi0 . For every i 2 I, there are two cases as follows:
(i) If i = i0, then
2 Ai0 X ¼ ðai0 j1 ^ b1 Þ _ _ 4
_
3
¼ ðai0 j1
^ b1 Þ _ _ @
0
_
1
3
_
ðai0 j ^ xj Þ5 _ _ ðai0 jm ^ bm Þ _ 4
j2J i
0
2
ðai0 ^ 0Þ5
j2JnðJ i [fj1 ;j2 ;...;jm gnfji gÞ 0
0
xj A _ _ ðai0 jm ^ bm Þ:
j2J i
0
Note that in the last equality, if k < i0, then ai0 jk ^ bk 6 bk 6 bi0 . If k > i0, then ai0 jk ^ bk 6 ai0 jk . Again, from the structure of X, if xjk ¼ bk then jk 2 Jk. Thus jk R J i0 since jk 2 Jkn(J1 [ [ Jk1) and J2i = ; for all i 2 In{1}. Hence, in the case k > i0, we have that ai0 jk ^ bk 6 ai0 jk 6 bi0 . Therefore, Ai0 X ¼ bi0 .
40
Q.-q. Xiong, X.-p. Wang / Information Sciences 252 (2013) 32–41
(ii) If i – i0, then
2
Ai X ¼ ðaij1 ^ b1 Þ _ _ 4
_
3
¼ ðaij1 ^ b1 Þ _ _ 4
0
_
3
_
ðaij ^ xj Þ5 _ _ ðaijm ^ bm Þ _ 4
j2J i
2
2
ðaij ^ 0Þ5
j2JnðJ i [fj1 ;j2 ;...;jm gnfji gÞ 0
3
0
ðaij ^ xj Þ5 _ _ ðaijm ^ bm Þ:
ð6Þ
j2J i
0
For the index j of every aij in equality (6), distinguishing two situations, we can have: Case (a) . It is jk with k 2 {1, 2, . . . , m}n{i0}. This includes three subcases as follows. – If k < i, then aijk ^ bk 6 bk 6 bi . – If k > i, then aijk ^ bk 6 aijk 6 bi since J2i = ; for all i 2 In{1}. – If k = i, then aijk ^ bk ¼ bk ¼ bi . W W Case (b) . j 2 J i0 . In this case, if bi0 > bi , then i0 > i and j R Ji, therefore j2Ji ðaij ^ xj Þ 6 j2Ji aij 6 bi since J2i = ; for all i 2 In{1}. W W 0 0 Otherwise, if bi0 6 bi , then j2Ji ðaij ^ xj Þ 6 j2Ji xj 6 bi0 6 bi . 0
0
Cases (a) and (b) imply that Ai X = bi. Finally, with (i) and (ii) we have that X 2 X. Again, since aiji ^ xji ¼ aiji ^ bi ¼ bi if ðþÞ T ðÞ i – i0, and aij ^ xj < bi for all j 2 J if i = i0, we know that X 2 Xi2 Xi0 2 , that is to say X 2 XðÞ : h Note that Theorem 3.10 has no topological assumption, and its condition J2i = ; for all i 2 In{1} cannot be moved. For example, Example 3.3. Consider the system of equations
If the system has a solution, then x3 P 34 by Eq. (9), a23 ^ x3 = b2 and a33 ^ x3 = b3. Moreover, x1 P 13 or x2 ¼ 13 by Eq. (7), therefore a12 ^ x2 = b1 or a11 ^ x1 = b1. Consequently, all the solutions of the system are attainable. However, J22 – ; and J23 – ;. The system has partially attainable solutions
X1 ¼
T 1 3 ; 0; ; x4 ; x5 ; x6 ; ; 3 4
with xj < 12 for j = 4, 5, . . . , and
Wþ1
j¼4 xj
X2 ¼
1 3 0; ; ; x4 ; x5 ; x6 ; 3 4
T
¼ 12.
Theorem 3.11. Let XðþÞ – ;. Then XðþÞ ¼
S
T X 2X0 ½X ; A
B.
Proof. Let X ¼ ðxj Þj2J 2 X. If for every i 2 I there exists an index j 2 J such that aij ^ xj = bi, then there is an X 2 X0 such that S X⁄ 6 X by Theorem 3.8. This follows from Lemmas 2.2 and 3.3 that X 2 X 2X0 ½X ; AT B. h 4. Conclusions This paper discusses some properties of infinite sup–inf fuzzy relational equations on [0, 1]. First, we get some necessary and sufficient conditions for the existence of the attainable solutions (resp. the unattainable solutions and the partially attainable ones) when X – ;. Then we show the set of attainable solutions XðþÞ . However, we know that the solution set X is exactly divided into three separate parts, i.e., XðþÞ ; XðÞ and XðÞ , respectively, with X ¼ XðþÞ [ XðÞ [ XðÞ and XðþÞ \ XðÞ ¼ XðþÞ \ XðÞ ¼ XðÞ \ XðÞ ¼ ;. As one can see that we just have obtained the structure of the first one in this paper, consequently, in order to completely describe X, we have to investigate the other parts of X. Unfortunately, from the results as above mentioned together with those in [25,17] we know that the study of both solution sets XðÞ and XðÞ remain open when jIj P 2. Acknowledgement The authors thank the Editor-in-Chief Professor W. Pedrycz and the anonymous referees for their valuable comments.
Q.-q. Xiong, X.-p. Wang / Information Sciences 252 (2013) 32–41
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