AbstractThis work studies the problem of solving a sup-T composite finite fuzzy relation equation, where T is an infinitely distributive strong pseudo-t-norm. A criterion for the equation to have a solution is given. It is proved that if the equation is solvable then its solution set is determined by the greatest solution and a finite number of minimal solutions. A necessary and sufficient condition for the equation to have a unique solution is obtained. Also an algorithm for finding the solution set of the equation is presented

Han, Song-Chol

Li, Hong-Xing

Wang, Jia-Yin

Elsevier - Publisher Connector 

Resolution of finite fuzzy relation equations based on strong pseudo-t-norms 

Applied Mathematics Letters 19 (2006) 752–757
www.elsevier.com/locate/aml
Resolution of finite fuzzy relation equations based on strong
pseudo-t-norms✩
Song-Chol Hana,b, Hong-Xing Lia,∗, Jia-Yin Wanga
a School of Mathematical Sciences, Beijing Normal University, Beijing 100875, PR China
b Department of Mathematics and Mechanics, Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea
Received 30 December 2004; received in revised form 25 October 2005; accepted 10 November 2005
Abstract
This work studies the problem of solving a sup-T composite finite fuzzy relation equation, where T is an infinitely distributive
strong pseudo-t-norm. A criterion for the equation to have a solution is given. It is proved that if the equation is solvable then its
solution set is determined by the greatest solution and a finite number of minimal solutions. A necessary and sufficient condition
for the equation to have a unique solution is obtained. Also an algorithm for finding the solution set of the equation is presented.
c© 2005 Elsevier Ltd. All rights reserved.
Keywords: Fuzzy relation equation; Strong pseudo-t-norm; Infinitely distributive strong pseudo-t-norm
1. Introduction
The resolution of fuzzy relation equations is one of the most important and widely studied problems in the field of
fuzzy sets and fuzzy systems. The majority of fuzzy inference systems can be implemented by using the fuzzy relation
equations [11]. Fuzzy relation equations can also be used for processes of compression/decompression of images and
videos [8].
The sup–inf composite fuzzy relation equation was first proposed by Sanchez in 1976, and since then different
kinds of fuzzy relation equations have been studied by many researchers [2–5,8–13,18]. Recently, Wang and Yu [15]
introduced the notion of pseudo-t-norms. Building on this, Dai and Wang [1,14] considered the fuzzy relation
equations with pseudo-t-norms. Meanwhile, Han and Li [7] introduced the concept of a strong pseudo-t-norm to
correct some incorrect main results in [1,14–16].
In this work, we study in detail the resolution problem of a sup-T composite finite fuzzy relation equation, where
T is an infinitely distributive strong pseudo-t-norm.
Throughout this work, L denotes the real unit interval [0, 1] and J always stands for any nonempty set of subscripts.
✩ Supported by the National Natural Science Foundation of China (60474023), Research Fund for the Doctoral Program of Higher Education
(20020027013), Science and Technology Key Project Fund of the Ministry of Education (03184) and Major State Basic Research Development
Program of China (2002CB312200).
∗ Corresponding author.
E-mail address: lhxqx@bnu.edu.cn (H.-X. Li).
0893-9659/$ - see front matter c© 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.aml.2005.11.001
S.-C. Han et al. / Applied Mathematics Letters 19 (2006) 752–757 753
2. Strong pseudo-t-norm
In this section, we give some definitions.
A binary operation T on L is called a pseudo-t-norm [15] if it satisfies the following conditions:
(T1) T (1, a) = a and T (0, a) = 0 for all a ∈ L,
(T2) a, b, c ∈ L and b  c ⇒ T (a, b)  T (a, c).
A pseudo-t-norm T on L is said to be infinitely ∨-distributive [15] if it satisfies the following condition:
(T∨) a, b j ∈ L( j ∈ J ) ⇒ T (a,∨ j∈J b j ) = ∨ j∈J T (a, b j ).
A pseudo-t-norm T on L is said to be infinitely ∧-distributive [16] if it satisfies the following condition:
(T∧) a, b j ∈ L( j ∈ J ) ⇒ T (a,∧ j∈J b j ) = ∧ j∈J T (a, b j ).
A pseudo-t-norm T on L is said to be infinitely distributive [16] if it is both infinitely ∨-distributive and infinitely
∧-distributive.
Let A ∈ L L×L . Define I (A), T (A) ∈ L L×L as follows:
I (A)(a, b) := ∨{u ∈ L | A(a, u)  b},
T (A)(a, b) := ∧{u ∈ L | A(a, u)  b},
where a, b ∈ L. It is tacitly assumed that ∨∅ = 0 and ∧∅ = 1.
Theorem 2.1. If T is an infinitely ∨-distributive pseudo-t-norm on L, then the following conditions are equivalent:
(1) T (a, c)  b ⇔ c  I (T )(a, b) for all a, b, c ∈ L;
(2) T (a, 0) = 0 for all a ∈ L.
Proof. (1) ⇒ (2) For any a ∈ L, we have 0  I (T )(a, 0). Using (1), we obtain T (a, 0)  0, i.e., T (a, 0) = 0.
(2) ⇒ (1) Let a, b, c ∈ L. If T (a, c)  b, then I (T )(a, b) = ∨{u ∈ L | T (a, u)  b}  c. Conversely, suppose
I (T )(a, b)  c. Using (2), 0 ∈ {u ∈ L | T (a, u)  b} 	= ∅. By (T2) and (T∨), T (a, c)  T (a, I (T )(a, b)) =
T (a,∨{u ∈ L | T (a, u)  b}) = ∨{T (a, u) | T (a, u)  b}  b. 
A pseudo-t-norm T on L is said to be strong [7] if it satisfies the following condition:
(T3) T (a, 0) = 0 for all a ∈ L.
It is obvious that t-norms and weak t-norms [6] are the particular cases of strong pseudo-t-norms. And there exists
an infinitely ∨-distributive pseudo-t-norm that is not strong (see the pseudo-t-norm TM in [15], for instance).
Example 2.1. Put
T (a, b) =


b, a = 1,
0, a = 0,
0, 0 < a < 1, b = 0,
1, 0 < a < 1, b > 0,
where a, b ∈ L. Then T is an infinitely ∨-distributive strong pseudo-t-norm on L. However, it is not infinitely ∧-
distributive on L, since T (a,∧{b ∈ L | b > 0}) = T (a, 0) = 0 	= 1 = ∧{T (a, b) | b > 0} for 0 < a < 1.
Example 2.2 (Wang and Yu [15]). Let
TW (a, b) =
{
b, a = 1,
0, otherwise,
where a, b ∈ L. Then TW is an infinitely distributive strong pseudo-t-norm on L.
754 S.-C. Han et al. / Applied Mathematics Letters 19 (2006) 752–757
Example 2.3 (Yager [17]). Let
TY (a, b) =
{
b1/a, a · b > 0,
0, a · b = 0,
where a, b ∈ L. Then TY is also an infinitely distributive strong pseudo-t-norm on L. Moreover,
I (TY )(a, b) =


1, a = 0,
0, a > 0, b = 0,
ba, a > 0, b > 0,
T (TY )(a, b) =
{
0, b = 0,
ba, b > 0,
where a, b ∈ L. Here we notice that TY (a, 1) ≡ a does not hold.
From now on, T denotes any given infinitely distributive strong pseudo-t-norm on L.
3. Solvability of equation
In this section, we first give a criterion for the existence of the solution.
We denote by Lm×n and Ln the set of all m × n matrices over L and the set of all column vectors of order n over
L, respectively. For any positive integer n, n always indicates the set {1, 2, . . . , n}.
Given two matrices A = (aik) ∈ Lm×n and B = (bkj ) ∈ Ln×l , the sup-T composition A ◦T B ∈ Lm×l of A and B
is defined by
A ◦T B := (∨k∈n T (aik, bkj )).
Lemma 3.1. If A ∈ Lm×n and C1,C2 ∈ Ln×l with C1  C2, then A ◦T C1  A ◦T C2.
Proof. It follows immediately from the definitions. 
The present work deals with the fuzzy relation equation A ◦T X = B , i.e.,
∨ j∈n T (ai j , x j ) = bi , i ∈ m, (1)
where A = (ai j ) ∈ Lm×n and B = (bi ) ∈ Lm are given, but X = (x j ) ∈ Ln is unknown. We denote by X the set of
all solutions of Eq. (1). Eq. (1) is said to be solvable in L when X 	= ∅.
Put X = (x j ) ∈ Ln , where
x j = ∧i∈m I (T )(ai j , bi ), j ∈ n. (2)
Theorem 3.1. If Eq. (1) is solvable in L, then X is the greatest solution of it.
Proof. If X 	= ∅ and X = (x j ) ∈ X , then for any i ∈ m and j ∈ n we have T (ai j , x j )  bi . Hence,
x j  I (T )(ai j , bi ), and so x j  ∧i∈m I (T )(ai j , bi ) = x j . By Theorem 2.1, T (ai j , x j )  T (ai j , x j ) 
T (ai j , I (T )(ai j , bi ))  bi . Thus bi = ∨ j∈n T (ai j , x j )  ∨ j∈n T (ai j , x j )  bi , i.e., ∨ j∈n T (ai j , x j ) = bi . Therefore,
X is the greatest element of X . 
Theorem 3.1 gives a necessary and sufficient condition for the existence of the solution of Eq. (1). And Theorem 3.1
also holds for any infinitely ∨-distributive strong pseudo-t-norm T .
We next consider the solution set of Eq. (1) in the case that m = n = 1.
Theorem 3.2. Let a, b ∈ L. If the equation T (a, x) = b is solvable in L, then its solution set is the interval
[T (T )(a, b), I (T )(a, b)] in L.
Proof. By Theorem 3.1, I (T )(a, b) is the greatest solution. If x ∈ L is a solution, then x ∈ {u ∈ L | T (a, u) 
b} 	= ∅. Hence, T (T )(a, b)  x . By (T∧), b = T (a, x)  T (a, T (T )(a, b)) = T (a,∧{u ∈ L | T (a, u)  b}) =
∧{T (a, u) ∈ L | T (a, u)  b}  b, i.e., T (a, T (T )(a, b)) = b. Thus T (T )(a, b) is the least solution. By Lemma 3.1,
we can see that the interval [T (T )(a, b), I (T )(a, b)] in L is the solution set of the equation. 
S.-C. Han et al. / Applied Mathematics Letters 19 (2006) 752–757 755
4. Solution set of equation
In this section, we assume that Eq. (1) is solvable, and show that the solution set is represented by the greatest
solution and a finite number of minimal solutions.
Given a solution X = (x j ) ∈ X of Eq. (1), we put
Ji (X) := { j ∈ n | T (ai j , x j ) = bi }, i ∈ m, (3)
J (X) := J1(X) × J2(X) × · · · × Jm(X). (4)
Then Ji (X) 	= ∅ for all i ∈ m, and so J (X) 	= ∅.
For every f = ( fi ) ∈ J (X) and for every j ∈ n, we put
I ( f )j := {i ∈ m | fi = j}, (5)
x
( f )
j :=


∨i∈I ( f )j T (T )(ai j , bi ), I
( f )
j 	= ∅,
0, I ( f )j = ∅.
(6)
Thus we can make a vector X ( f ) := (x ( f )j ) ∈ Ln . Finally, we construct a finite poset
F(X) := {X ( f ) | f ∈ J (X)}. (7)
Lemma 4.1. If X ∈ X and f ∈ J (X), then X ( f )  X.
Proof. Suppose X ( f )  X . Then there exists a j ∈ n such that x ( f )j  x j . Hence, x ( f )j 	= 0 and so
∨i∈I ( f )j T (T )(ai j , bi )  x j with I
( f )
j 	= ∅. There is an i ∈ I ( f )j such that T (T )(ai j , bi )  x j . On the other hand,
i ∈ I ( f )j implies that j ∈ Ji (X) and T (ai j , x j ) = bi . Therefore, T (T )(ai j , bi )  x j , which is a contradiction. 
Lemma 4.2. If X ∈ X , then F(X) ⊆ X .
Proof. Let f ∈ J (X). By Lemmas 3.1 and 4.1, A ◦T X ( f )  A ◦T X = B . It suffices to verify A ◦T X ( f ) 
B . Assume A ◦T X ( f )  B . Then there is a k ∈ m such that ∨ j∈n T (akj , x ( f )j )  bk . On the other hand,
∨ j∈n T (akj , x j ) = bk and T (akl , xl) = bk for l = fk . Hence, k ∈ I ( f )l 	= ∅. Since x ( f )l = ∨i∈I ( f )l T (T )(ail , bi ) 
T (T )(akl , bk), by Theorem 3.2 we have T (akl , x ( f )l )  T (akl , T (T )(akl , bk)) = bk . Thus ∨ j∈n T (akj , x ( f )j )  bk ,
which is a contradiction. 
Lemma 4.3. If X,Y ∈ X with X  Y and X ( f ) ∈ F(X), then X ( f ) = Y ( f ) ∈ F(Y ).
Proof. Let Y = (y j ). If f = ( fi ) ∈ J (X), then for any i ∈ m, bi = T (ai fi , x fi )  T (ai fi , y fi )  bi , i.e.,
T (ai fi , y fi ) = bi . Hence, fi ∈ Ji (Y ) and f = ( fi ) ∈ J (Y ). Therefore, X ( f ) = Y ( f ) ∈ F(Y ). 
For any poset P , we denote by P0 the set of all minimal elements in P .
Lemma 4.4 ([12, Lemma 2.2]). Let Q be a nonempty subset of a poset P. If for every p ∈ P there exists a q ∈ Q
with q  p, then P0 = Q0.
Theorem 4.1. The solution set X of Eq. (1) and the finite poset F(X) contain the same minimal elements, i.e.,
X0 = F(X)0 	= ∅.
Proof. By Lemma 4.2, F(X) ⊆ X . If X ∈ X , then X  X by Theorem 3.1. By Lemma 4.3, X ( f ) ∈ F(X) implies
X ( f ) = X ( f ) ∈ F(X). Again by Lemma 4.1, X ( f )  X . Hence, X0 = F(X)0 by Lemma 4.4. Moreover,F(X)0 	= ∅
since F(X) is a nonempty finite set. 
With respect to the solution set X of Eq. (1), we obtain the following representation theorem.
756 S.-C. Han et al. / Applied Mathematics Letters 19 (2006) 752–757
Theorem 4.2. The solution set X of Eq. (1) is the union of a finite number of intervals in Ln:
X =
⋃
X∧∈F(X)0
[X∧, X ], (8)
where [X∧, X ] = {X ∈ Ln | X∧  X  X } for any X∧ ∈ F(X)0.
Proof. If X ∈ X , then it follows from the proof of Theorem 4.1 that there exists a Y ∈ F(X) with Y  X . Since
F(X) is a finite poset, there is an X∧ ∈ F(X)0 with X∧  Y . Hence, X∧  Y  X  X by Theorem 3.1, and so
X ∈ [X∧, X]. Therefore,X ⊆⋃X∧∈F(X)0[X∧, X]. The converse inclusion follows from Lemmas 4.2 and 3.1. 
Now we can derive a criterion for the uniqueness of the solution of Eq. (1).
Theorem 4.3. Eq. (1) has a unique solution if and only if F(X) = {X }.
Proof. The necessity follows from Theorem 3.1 and Lemma 4.2. The sufficiency follows from Theorem 4.2. 
5. An algorithm for resolution
We first present an algorithm for solving Eq. (1) that is ensured by the preceding theorems:
Step 1. Compute X using (2) and check by the substitution whether X is a solution of Eq. (1). If no, then put X = ∅
and stop. Otherwise, go to Step 2.
Step 2. Construct J (X) using (3) and (4).
Step 3. Determine F(X) using (5)–(7).
Step 4. Find F(X)0 by pairwise comparison.
Step 5. Construct X using (8).
We next give a numerical example to illustrate the algorithm presented above.
Example 5.1. Let TY be the infinitely distributive strong pseudo-t-norm on L as in Example 2.3. We consider a fuzzy
relation equation A ◦TY X = B , where
A =

0.4 0.5 10.8 0.5 0.6
0 0.125 0.5

 , B =

 0.20.16
0.04

 .
We will use I (TY ) and T (TY ) in Example 2.3. Using (2), we compute X = (0.160.8, 0.4, 0.2)T. Since A ◦TY X = B ,
the equation is solvable and X is the greatest solution. Using (3) and (4), we construct J (X) : J1(X) = {3}, J2(X) =
{1, 2}, J3(X) = {3}; J (X) = {(3, 1, 3), (3, 2, 3)}. Using (5)–(7), we determine F(X). For f = ( f1, f2, f3) =
(3, 1, 3), we have
I ( f )1 = {2}, x ( f )1 = T (TY )(0.8, 0.16) = 0.160.8;
I ( f )2 = ∅, x ( f )2 = 0;
I ( f )3 = {1, 3}, x ( f )3 = T (TY )(1, 0.2)∨ T (TY )(0.5, 0.04) = 0.21 ∨ 0.040.5 = 0.2;
X ( f ) = (0.160.8, 0, 0.2)T.
For f = ( f1, f2, f3) = (3, 2, 3), we have
I ( f )1 = ∅, x ( f )1 = 0;
I ( f )2 = {2}, x ( f )2 = T (TY )(0.5, 0.16) = 0.160.5 = 0.4;
I ( f )3 = {1, 3}, x ( f )3 = T (TY )(1, 0.2)∨ T (TY )(0.5, 0.04) = 0.21 ∨ 0.040.5 = 0.2;
X ( f ) = (0, 0.4, 0.2)T.
Thus F(X) = {(0.160.8, 0, 0.2)T, (0, 0.4, 0.2)T}. The set of minimal solutions is F(X)0 = F(X). Using (8), we
obtain the solution set X = {(0.160.8, [0, 0.4], 0.2)T, ([0, 0.160.8], 0.4, 0.2)T}.
S.-C. Han et al. / Applied Mathematics Letters 19 (2006) 752–757 757
References
[1] F.M. Dai, Z.D. Wang, Fuzzy relations and fuzzy relation equations, Journal of Yangzhou University 5 (2) (2002) 8–10 (in Chinese).
[2] B. De Baets, An order-theoretic approach to solving sup-T equations, in: D. Ruan (Ed.), Fuzzy Set Theory and Advanced Mathematical
Applications, Kluwer Academic Publishers, Dordrecht, 1995, pp. 67–87.
[3] B. De Baets, Analytical solution methods for fuzzy relational equations, in: D. Dubois, H. Prade (Eds.), Fundamentals of Fuzzy Sets, in: The
Handbooks of Fuzzy Sets Series, vol. 1, Kluwer Academic Publishers, Dordrecht, 2000, pp. 291–340.
[4] A. Di Nola, On solving relational equations in Brouwerian lattices, Fuzzy Sets and Systems 34 (1990) 365–376.
[5] A. Di Nola, S. Sessa, W. Pedrycz, E. Sanchez, Fuzzy Relation Equations and Their Applications to Knowledge Engineering, Kluwer Academic
Publishers, New York, 1989.
[6] J.C. Fodor, Strict preference relations based on weak t-norms, Fuzzy Sets and Systems 43 (1991) 327–336.
[7] S.C. Han, H.X. Li, Notes on “Pseudo-t-norms and implication operators on a complete Brouwerian lattice” and “Pseudo-t-norms and
implication operators: direct products and direct product decompositions”, Fuzzy Sets and Systems 153 (2005) 289–294.
[8] V. Loia, S. Sessa, Fuzzy relation equations for coding/decoding processes of images and videos, Information Sciences 171 (2005) 145–172.
[9] M. Miyakoshi, M. Shimbo, Solutions of composite fuzzy relational equations with triangular norms, Fuzzy Sets and Systems 16 (1985) 53–63.
[10] E. Sanchez, Resolution of composite fuzzy relation equations, Information and Control 30 (1976) 38–48.
[11] G.B. Stamou, S.G. Tzafestas, Resolution of composite fuzzy relation equations based on Archimedean triangular norms, Fuzzy Sets and
Systems 120 (2001) 395–407.
[12] M. Wagenknecht, K. Hartmann, On direct and inverse problems for fuzzy equation systems with tolerances, Fuzzy Sets and Systems 24 (1987)
93–102.
[13] X.P. Wang, Method of solution to fuzzy relation equations in a complete Brouwerian lattice, Fuzzy Sets and Systems 120 (2001) 409–414.
[14] Z.D. Wang, F.M. Dai, The sets of solutions of L-relation equations T (a, x) = b and I (a, x) = b, Journal of Yangzhou University 6 (3) (2003)
8–10 (in Chinese).
[15] Z.D. Wang, Y.D. Yu, Pseudo-t-norms and implication operators on a complete Brouwerian lattice, Fuzzy Sets and Systems 132 (2002)
113–124.
[16] Z.D. Wang, Y.D. Yu, Pseudo-t-norms and implication operators: direct products and direct product decompositions, Fuzzy Sets and Systems
139 (2003) 673–683.
[17] R. Yager, An approach to inference in approximate reasoning, International Journal of Man–Machine Studies 13 (1980) 323–338.
[18] C.K. Zhao, On matrix equations in a class of complete and completely distributive lattices, Fuzzy Sets and Systems 22 (1987) 303–320.


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Resolution of finite fuzzy relation equations based on strong pseudo-t-norms

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