Some results in the setting of fuzzy relation equations theory (Q802555)

This paper is a continuation of two preceding papers of the author and of the reviewer [Fuzzy Sets Syst. 9, 275-285 (1983; Zbl 0514.94027) and Busefal 11, 14-23 (1982; Zbl 0516.94034)]. This paper contains seven Sections. Sections 1 and 2 include basic preliminaries. Let L be a linear lattice and let \(F(X)=\{A:X\to L\}\) and \(F(Y)=\{B:Y\to L\}\) be the classes of all the fuzzy sets defined on two finite sets X and Y, respectively. The author considers the fuzzy equation (1) \(Q\circ A=B\), where Q is an unknown fuzzy relation, \(Q\in F(X\times Y)=\{Q:X\times Y\to L\}\) and ''\(\circ ''\) stands for the max-min composition. Several results about equation (1) are established in Section 3, whose main result is Theorem 2: Equation (1) has a unique solution iff \(A(x)>0\) for all \(x\in X\) and \(B(y)=0\) for all \(y\in Y\). Further the author considers the composite fuzzy equation (2) \(R\circ Q=T\), introduced by \textit{E. Sanchez} [Inf. Control 30, 38-48 (1976; Zbl 0326.02048)], where \(T\in F(X\times Z)\) and \(R\in F(Y\times Z)\), Z being a third finite set and R unknown. The main result of Section 4 is Theorem 4: If the equation (2) has solutions, then the set \(\Gamma (x,z)=\{y\in Y:\) Q(x,y)\(\geq T(x,z)\}\) is nonempty for all \(x\in X\), \(z\in Z\). In sections 5, 6 and 7 some algebraic properties of a particular solution of the equation (2) are discussed and related examples are also given.