On the fuzzification of the classical definition of preference structure

Abstract

This paper is divided into two main parts. In the first part, we propose an appropriate definition for fuzzy preference structures. The definition is based on two main features. The first feature consists in using a strong de Morgan triplet as a model for the classical de Morgan triplet. The second feature consists in choosing a particular completeness condition such that when it is combined with some other conditions, all the generalized completeness conditions are satisfied. These two features of our definition are justified by a set of remarkable theorems and propositions. Proving some equivalencies, we can write this definition with only three minimal and independent conditions. In the second part, we use the new definition to analyse two methods of construction of fuzzy strict preference, indifference and incomparability relations from a given reflexive fuzzy relation. On the one hand, we prove that the triplets of fuzzy strict preference, indifference and incomparability relations constructed from a reflexive fuzzy relation by using a same t-norm, are not necessarily fuzzy preference structures. On the other hand, we prove that those generated by the method of Fodor and Roubens are fuzzy preference structures according to the proposed definition.