On the solvability of weakly linear systems of fuzzy relation equations
This work tackles a specific mathematical problem in fuzzy logic, offering incremental improvements by extending solvability criteria and computational methods for weakly linear systems.
The paper addresses the problem of solving weakly linear systems of fuzzy relation equations, where an unknown fuzzy relation appears on both sides, by describing the set of fuzzy relations that solve these systems to a certain degree and providing algorithms to compute them, with examples from fuzzy network aggregation.
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.