Relational equations in totally ordered lattices and their complete resolution (Q1073027)

Discussed are fuzzy relation equations \(R\circ Q=T\) with max-min composition where R, Q, T are fuzzy relations defined in Cartesian products of finite universes of discourse, namely \(R: {\mathcal Y}\times {\mathcal Z}\to L\), \(Q: {\mathcal X}\times {\mathcal Y}\to L\), \(T: {\mathcal X}\times {\mathcal Z}\to L\), and L stands for a totally ordered lattice. The equation is solved with respect to R for Q and T given. Assuming \({\mathcal R}=\{R|\) \(R\circ Q=T\}\neq \emptyset\), all its lower elements are determined and their characterization (in the sense of their structure) is provided. Moreover, elements of \({\mathcal R}\) with special properties are indicated.