Systems of Fuzzy Relation Equations in a Space with Fuzzy Preorder

In this paper, we consider the problem of solving systems of fuzzy relation equations in a space with fuzzy preorder. Two types of these systems with different compositions are considered. New solvability criteria are proposed for systems of both types. The new criteria are weaker than all the known ones that are based on the assumption that fuzzy sets on the left-hand side of a system establish a fuzzy partition of a respective universe. Keywords— system of fuzzy relation equations, fuzzy preorder, criterion of solvability 1 Preliminaries Let throughout this contribution L = 〈L,∨,∧, ∗,→, 0, 1〉 be an integral, residuated, commutative l-monoid (a residuated lattice), X a non-empty set and L a set of L-valued functions on X . Fuzzy subsets of X are identified with L-valued functions on X (membership functions). Let X and Y be two universes, not necessary different, Ai ∈ L , Bi ∈ L arbitrarily chosen fuzzy subsets of respective universes, and R ∈ LX×Y a fuzzy subset of X × Y . The latter is called a fuzzy relation. Lattice operations ∨ and ∧ induce respective union and intersection of fuzzy sets. Two other binary operations ∗,→ of L are used for compositions binary operations on LX×Y . We will consider two of them: sup-*-composition that is usually denoted by ◦, and inf-→composition that is denoted by . The first one has been introduced by L. Zadeh [16] and the second one by W. Bandler and L. Kohout [1]. We will demonstrate definitions of both compositions on particular examples of set-relation compositions A ◦R and A R where A ∈ L and R ∈ LX×Y : (A ◦R)(y) = ∨ x∈X (A(x) ∗R(x, y)), (A R)(y) = ∨ x∈X (A(x) → R(x, y)). Remark 1 Let us remark that both compositions can be considered as set-set compositions where R is assumed to be replaced by a fuzzy set. In this reduced form they are used in instances of systems of fuzzy relation equations below. By a system of fuzzy relation equations with sup-*composition (SFRE∗), we mean the following system of equations Ai ◦R = Bi, 1 ≤ i ≤ n, (1) that is considered with respect to unknown fuzzy relation R ∈ LX×Y . Its counterpart is a system of fuzzy relation equations with inf-→composition (SFRE→) Ai R = Bi, 1 ≤ i ≤ n, (2) that is considered with respect to unknown R ∈ LX×Y also. System (1) and its potential solutions are well investigated in the literature (see e.g. [3, 2, 4, 5, 12, 8, 13, 15]). On the other hand, investigation of solvability of (2) is not so intensive (see [2, 10]). Both systems of fuzzy relation equations arise when a system of fuzzy IF-THEN rules is modeled by a fuzzy relation (below in (3) it is denoted by R), and continuity of the model [9] is requested. In order to explain this request, we recall that in relation models, a computation of an output value (B) which relates to a given input A ∈ L is performed with the help of sup-*, respectively inf-→ composition: B = A ◦R or B = A R. (3) (3) is a computational realization of the Generalized Modus Ponens inference scheme in fuzzy logic (in a broader sense). It is often welcome if thus constructed model is continuous in the sense that when (input) fuzzy sets A′, A′′ ∈ L are close to each other (in some space) so do output fuzzy sets A′ ◦ R and A′′ ◦R (respectively, A′ R and A′′ R). We proved in [9] that this is possible if and only if R solves the respective system (1) or (2) of fuzzy relation equations. This fact gives additional importance to the problem of solvability of systems of fuzzy relation equations. In general, solutions of (1) or (2) may not exist. Therefore, investigation of necessary and sufficient conditions for solvability (or at least, sufficient conditions) is needed. This problem has been widely studied in the literature, and some nice theoretical results have been obtained in the cited above papers. If the universes of discourse X and Y are infinite then the complexity of verification of necessary and sufficient conditions is comparable with the direct checking of solvability. Therefore, the problem of discovering easy-to-check solvability conditions or criteria is still open (see [11, 14] for some results). Besides other, this paper is a contribution to this topic. We recall basic facts concerning solvability of system (1) of fuzzy relation equations (similar conditions are known for system (2), so that we will not recall theme (see e.g. [2, 10])). Theorem 1 (a) [13] If system (1) with respect to unknown fuzzy relation R is solvable then relation