Finding the Complete Set of Minimal Solutions for Fuzzy Max-Archimedean t-Norm Relational Equations

This study considers the problem of generating all minimal solutions of a system of fuzzy relational equations (FREs) with max-Archimedean t-norm composition. It defines the binding matrix of a system of FREs, and then shows that an irredundant covering of the binding matrix corresponds to a minimal solution of the FREs. Consequently, the problem of finding all minimal solutions of the FREs can be transformed into the problem of finding all irredundant coverings of the binding matrix. We propose an algorithm to solve the transformed problem. This algorithm builds all irredundant coverings in a bottom-up manner, i.e., all irredundant coverings of the binding matrix is built from the irredundant coverings of the submatrices of the binding matrix. Once all irredundant coverings have been found, they can be easily converted into the minimal solutions of original FREs.