Systems of fuzzy relational equations in the general lattice-valued case

Abstract

The paper provides an in-depth analysis of the fuzzy relational equations where fuzzy relations are defined as arbitrary mappings from the square of a domain set to a complete lattice, which is the most general lattice-valued case. In the investigations up to now, the codomain lattice was usually a special lattice, such as residual, Heyting, Browerian, continuous, etc. Using Tarski’s fixed point theorem for an operator on a complete lattice, we prove the existence of a solution for a wide class of fuzzy relational equations and their systems. Tarski’s theorem also implies the existence of the greatest and the least solution for the mentioned class of equations and their systems, and the fact that their solution sets are complete lattices. Starting from some construction versions of Tarski’s fixed point theorem, we also provide algorithms for the construction of the greatest and the least solution.