Sup-T Equations: State of the Art

Abstract

The study of fuzzy relational equations is one of the most appealing subjects in fuzzy set theory, both from a mathematical and a systems modelling point of view. The basic fuzzy relational equations are the sup-T equations, with T a t-norm. In this overview paper, we deal with these equations in a general lattice-theoretic framework. We consider t-norms on bounded ordered sets, and in particular on complete lattices. We then solve sup-T equations on distributive, complete lattices of which all elements are either join-irreducible or join-decomposable. Solution sets are represented by means of root systems. Some additional necessary and sufficient solvability conditions are listed. Also systems of sup-T equations are discussed. The theoretical results presented are applied to the real unit interval and to the real unit hypercube. In the latter case, particular attention is paid to pointwise extensions of t-norms defined on the real unit interval and the corresponding residual operators.