There is only one class of fuzzy preference structures without incomparability

In this paper we deal with the study of properties of fuzzy preference structures without incomparability in the general case where the fuzzy relation of strict preference is only asymmetrical with respect to the ϕ-transform of the Lukasiewicz t-norm where ϕ denotes the auto-morphism of the unit interval that characterizes the strong De Morgan triplet generalizing the classical De Morgan triplet. We prove that a ϕ-preference structure without incomparability is always characterizable by its large preference relation R = P∪W′ϕ I in such a way that P = R^d and I = R ∩Wϕ R^–1. We also show that some properties satisfied by a fuzzy preference structure without incomparability whose strict preference relation is min-asymmetrical are not preserved by a general fuzzy preference structure without incomparability whose strict preference relation is only Wϕ-asymmetrical.