Transitivity and negative transitivity in the fuzzy setting

Abstract

A (crisp) binary relation is transitive if and only if its dual relation is negatively transitive. In preference modelling, if a weak preference relation is complete, the associated strict preference relation is its dual relation. It follows from here this well-known result: given a complete weak preference relation, it is transitive if and only if its strict preference relation is negatively transitive. In the context of fuzzy relations, transitivity is traditionally defined by a t-norm and negative transitivity, by a t-conorm. In this setting, it is also well known that a (valued) binary relation is T-transitive if and only if its dual relation is negatively S-transitive where S stands for the dual t-conorm of the t-norm T. However, in this context there are several proposals to get the strict preference relation from the weak preference relation. Also, there are different definitions of completeness. In this contribution we depart from a reflexive fuzzy relation. We assume that this relation is transitive with respect to a conjunctor (a generalization of t-norms). We consider almost all the possible generators and therefore all the possible strict preference relations obtained from the reflexive relation and we provide a general expression for the negative transitivity that those relations satisfy. © 2011 Springer-Verlag Berlin Heidelberg.