On the solvability of weakly linear systems of fuzzy relation equations

TL;DR

This paper explores the solvability of weakly linear fuzzy relation systems, extending the understanding from exact solutions to approximate solutions, and develops algorithms for computing fuzzy preorders and equivalences.

Contribution

It introduces methods to compute approximate solutions to weakly linear fuzzy relation systems and develops algorithms for fuzzy preorders and equivalences.

Findings

Characterization of approximate solutions to weakly linear systems

Algorithms for computing fuzzy preorders and equivalences

Application to aggregation of fuzzy networks

Abstract

Systems of fuzzy relation equations and inequalities in which an unknown fuzzy relation is on the one side of the equation or inequality are linear systems. They are the most studied ones, and a vast literature on linear systems focuses on finding solutions and solvability criteria for such systems. The situation is quite different with the so-called weakly linear systems, in which an unknown fuzzy relation is on both sides of the equation or inequality. Precisely, the scholars have only given the characterization of the set of exact solutions to such systems. This paper describes the set of fuzzy relations that solve weakly linear systems to a certain degree and provides ways to compute them. We pay special attention to developing the algorithms for computing fuzzy preorders and fuzzy equivalences that are solutions to some extent to weakly linear systems. We establish additional properties for the set of such approximate solutions over some particular types of complete residuated lattices. We demonstrate the advantage of this approach via many examples that arise from the problem of aggregation of fuzzy networks.