Some results on fuzzy relation equations provided with one solution (Q1073028)

Let \(F(A)=\{f| f:A\to L\}\) denote the set of fuzzy subsets of A, where L is a totally ordered set with 0 and 1. Let X, Y, Z be finite non- empty sets, \(Q\in F(X\times Y)\), \(T\in F(X\times Z)\) and consider the equation \(R\circ Q=T\) in the unknown \(R\in F(Y\times Z)\), where \((R\circ Q)(x,z)=\bigvee_{y}[Q(x,y)\wedge R(y,z)]\). In this paper the authors consider the case when card X\(>card Y\) and Z is a singleton, and determine necessary and sufficient conditions for the equation to have a unique solution, in each of the subcases when T is a bijection and when it is not. The case when card \(X\leq card Y\) was treated in another paper of the authors [ibid. 13, 83-94 (1984; Zbl 0553.04004)].