Fuzzy correspondence inequations and equations

Abstract

The paper deals with fuzzy correspondences, i.e., with mappings from a direct product of sets into a complete lattice. Fuzzy control problems connected with fuzzy correspondence inequations and equations are considered. A fuzzy correspondence is fixed, while the solutions are investigated for the input and output fuzzy sets which are unknown. First we prove that these solutions can be analyzed and formulated in crisp framework, solving the corresponding cut problems. Further, the space of solutions of an inequation is proved to be a complete lattice; the same holds for the space of solutions of the corresponding equation. In case the membership values belong to an infinitely distributive lattice (frame), the solutions can be found in specified intervals; moreover, we were able to locate in such intervals minimal and maximal solutions of these problems.