Two-mode weakly linear systems of fuzzy relation equations: Structures of solutions, computation methods, and applications

The problem of aggregating large fuzzy relational structures is of major importance. One of the main issues is seeking a simpler fuzzy relational structure, while keeping the structure properties of the original fuzzy relational structure as much as possible. This paper covers the mathematical foundations of the described problem, and deals with the solvability issues of two-mode weakly linear systems over complete residuated lattices. Within these systems, our objective is to identify a pair of unknown fuzzy relations that are present on both sides of the equation or inequality. We provide a characterization of the set of all pairs of fuzzy relations that are solutions to two-mode weakly linear systems to a desired degree, and develop an algorithm to compute the greatest such pair. We also present an algorithm to determine a pair that solves a two-mode weakly linear system to a given extent, such that each component is a fuzzy preorder (resp. fuzzy equivalence). Ultimately, we demonstrate the application aspects of aggregating fuzzy relational structures.