S-Implications on Complete Lattices and the Interval Constructor

Abstract

The aim of this work is to present an approach of interval fuzzy logic based on complete lattices. In particular, we study the extensions of the notions of t-conorms, fuzzy negations and S-implication, from the unit interval to arbitrary complete lattices. Some general properties of S-implications on complete lattices are analyzed. We show that the interval extensions of t-conorms, fuzzy negations and S-implications on complete lattices preserve the optimality property, being the best interval representations of these fuzzy connectives.