WEAK AND STRONG FUZZY INTERVAL ORDERS

The purpose of this paper is to introduce and investigate the fuzzific ation of the classical interval order, one of the most interesting cla ssical preference structures without incomparability. In this study, w e consider fuzzy preference structures as defined by De Baets et al. F uzzy preference structures without incomparability receive special att ention: their fuzzy preference and fuzzy large preference relations sh are certain types of the (T, N)-Ferrers property. Two special types of the (T, N)-Ferrers property are introduced: the phi-weak-Ferrers and strong-Ferrers properties. The classical interval order is briefly rev iewed, T-fuzzy interval orders are introduced and it is shown that the ir fuzzy preference relation is sup-T transitive. Two special types of T-fuzzy interval orders are considered: weak and strong fuzzy interva l orders, corresponding to phi-transforms of W and to M. The particula r intermediate role of the phi-weak-Ferrers property of the fuzzy pref erence relation of a FPS without incomparability Pi(phi) is demonstrat ed: on the one hand it is a necessary condition for the FPS to be a st rong fuzzy interval order, while on the other hand it is a sufficient condition for this structure to be a weak fuzzy interval order. Finall y, the concept of an alpha-cut of a FPS is introduced, leading to an i nteresting characterization of the strong-Ferrers property of a FPS wi thout incomparability.