Sets of solution-set-invariant coefficient matrices of simple fuzzy relation equations (Q1105600)

Let x, b be two fuzzy sets and A, A' be two fuzzy relations such that the finite fuzzy equations \(x\circ A=b\) and \(x\circ A'=b\) hold, where ``\(\circ ''\) denotes the max-min composition, A, A' are assigned and x is unknown. If the membership values of b are ordered in strictly decreasing sense, then the above equations are called simple. Let X and X', respectively, be the sets of the solutions of the above equations. Using results of \textit{Wang Peizhuang}, \textit{S. Sessa}, the reviewer, and \textit{W. Pedrycz} [Busefal 18, 67-74 (1984; Zbl 0581.04001)], the authors give an iff condition in order to have \(X=X'\). Further results on the number of the distinct elements of X are established.