A Note on Fuzzy Relational Equations With Min-Implication Composition

The fuzzy relational equations with min-implication composition are considered, where the implication operation involved is the residuum with respect to the minimum operation. The solution set of such a system of equations, whenever nonempty, can be characterized by a minimum solution and finitely many maximal solutions. It is shown that a system of fuzzy relational equations with min-implication composition can be reformulated in polynomial time into a system of 0-1 mixed integer linear inequalities, and consequently, the structure of its solution set can be revealed in a succinct manner. Besides, it is shown that the determination of all maximal solutions is polynomially reducible to the transversal hypergraph generation problem, of which some practically well-performed algorithms are available.