An Order-Theorethic Approach to Solving Sup-T Equations

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 [8]. The most fundamental fuzzy relational equations are the sup-T equations, with T a triangular norm. It is well-known how to solve these equations on the real unit interval [4]. Zhao has shown how to solve sup-⌢equations on complete Brouwerian lattices of which all elements have an irredundant finite decomposition in join-irreducible elements [14]. In this paper, we generalize various aspects of these results. We consider triangular norms on bounded ordered sets, and in particular on complete lattices. We then solve sup-Tinequalities and sup-T equations on distributive, complete lattices of which all elements are either join-irreducible or join-decomposable. The morphism behaviour of the partial mappings of the triangular norm T plays an important role in this study. We also introduce the notion of a maximally surjective triangular norm and show how the solution procedures can be simplified for such triangular norms. The results in these papers are obtained in a systematic way using the new order-theoretic concept root system. We conclude the paper with a discussion of the product lattice ([0, l]ⁿ, ≤) : we show how to construct triangular norms on it, and how to calculate the corresponding residual operators. By means of extensive examples we illustrate and comment on the solution procedures.