Maximal Consistent Subsystems of Max-T Fuzzy Relational Equations

In this article, we study the inconsistency of a system of max−T fuzzy relational equations of the form A max-T x = b, where T is a t-norm among min, the product or Lukasiewicz's t-norm. For an inconsistent max−T system, we directly construct a canonical maximal consistent subsystem (w.r.t the inclusion order). The main tool used to obtain it is the analytical formula which compute the Chebyshev distance ∆ = inf c∈C ∥b − c∥ associated to the inconsistent max−T system, where C is the set of second members of consistent systems defined with the same matrix A. Based on the same analytical formula, we give, for an inconsistent max-min system, an efficient method to obtain all its consistent subsystems, and we show how to iteratively get all its maximal consistent subsystems.