Weak and strong fuzzy interval orders

The purpose of this paper is to introduce and investigate the fuzzification of the classical interval order, one of the most interesting classical preference structures without incomparability. In this study, we consider fuzzy preference structures as defined by De Baets et al. Fuzzy preference structures without incomparability receive special attention: their fuzzy preference and fuzzy large preference relations share certain types of the (script J sign, script N sign)-Ferrers property. Two special types of the (script J sign, script N sign)-Ferrers property are introduced: the φ-weak-Ferrers and strong-Ferrers properties. The classical interval order is briefly reviewed, script J sign-fuzzy interval orders are introduced and it is shown that their fuzzy preference relation is sup-script J sign transitive. Two special types of script J sign-fuzzy interval orders are considered: weak and strong fuzzy interval orders, corresponding to φ-transforms of W and to M. The particular intermediate role of the φ-weak-Ferrers property of the fuzzy preference relation of a FPS without incomparability π_φ is demonstrated: on the one hand it is a necessary condition for the FPS to be a strong fuzzy interval order, while on the other hand it is a sufficient condition for this structure to be a weak fuzzy interval order. Finally, the concept of an α-cut of a FPS is introduced, leading to an interesting characterization of the strong-Ferrers property of a FPS without incomparability.