# An Order-Theorethic Approach to Solving Sup-T Equations

• Bernard De Baets
Chapter
Part of the International Series in Intelligent Technologies book series (ISIT, volume 4)

## 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.

## Keywords

Root System Minimal Element Partial Mapping Complete Lattice Minimal Solution
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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