On the existence of minimal solutions for fuzzy equations with tolerances (Q1263536)

The problem of determining solutions to fuzzy relational equations with tolerances is studied. Let X and Y be two universes of discourse. All fuzzy sets defined in them will be denoted by F(X) and F(Y), respectively. The generic type of equation is described as \[ (1)\quad A\circ R=B \] where \(A\in F(X)\), \(B\in F(Y)\), \(R\in F(X\times Y)\) whereas ``\(\circ ''\) stands for max-t composition with continuous t-norm involved. On the basis of (1) two main problems with tolerances are addressed. (1) A so-called modelling problem: determine fuzzy relations R satisfying the system of equations, Ḇ\({}_ i\leq A_ i\circ R\leq \bar B_ i\) \((i=1,2,...,N)\) where \(A_ i\), Ḇ\({}_ i\), \(\bar B_ i\) are given. Ḇ\({}_ i\) as well as \(\bar B_ i\) are treated as tolerances for a given fuzzy set \(B_ i.\) (2) The inverse problem which calls for a solution of the system Ḇ\({}_ i\leq X\circ R_ i\leq \bar B_ i\) \((i=1,2,...,N)\) with respect to X with all remaining sets and relations specified. The existence of minimal solutions in relevant families of solutions is studied. In comparison to existing results the analysis performed here is not restricted to finite cases of the universes X and Y. It implies an essential role of the semi-continuity of the membership functions of the fuzzy sets involved.