Fuzzy Preference Structures With The Strong De Morgan Triplets

Preference structure, which is the elementary tool for preference modeling theory, consists of strict preference relation, indeference relation and incomparability relation. In this research, we systematically discuss the definition and basic properties of fuzzy preference structure, some particular fuzzy preference structures. The major research aspects and results are as follows.Firstly, we review the definitions existing in the literature including the axiomatic definition, De Baets's definition, Bufardi's definition, Llamazares's defintition and Van de Walle's definition one by one, and point out relationships among them. Prom this, we know that the axiomatic definition is most general, because almost all the other definitions can be regard as its special cases. Meanwhile, Van de Walle's definition of fuzzy preference structure has solid theoretical foundation. Therefore, in this research, we choose Van de Walle's definition and study one special case of the axiomatic definition, i.e. the third case.Secondly, based on the third case, we carry out a detailed investigation into basic properties of fuzzy preference structures.. We present concrete mathematical expressions concerning relationships of P, I, J and R, a sufficient and necessary condition of J = 0, and relationships among transitivity, negative transitivity, semitransitivity, Ferrers properties between R and P and I with W as the t-norm. Some desired results are obtained.Finally, we fuzzify three crisp preference structures. We propose the definitions of fuzzy total interval order structure, fuzzy total semiorder structure and fuzzy partial order structure. Based on the definitions, we systematically discuss properties of every structure and give some equivalent statements of the three fuzzy preference structures. At the same time, some fuzzy orders are defined so that we can describe the three preference strutures concisely. For example, we point out that (P, /, J) is a fuzzy total semiorder structure iff R is W-W* total semiorder relation iff P is strict W- W* total semiorder relation. Some results are satisfatory.The research can serve as a guidance to understand the merits and demerits of every definition of fuzzy preference structures and thus to know the research status in fuzzy preference structures' definitions. In addition, the definition of three usual fuzzy preference structuresis given, which provides a method to fuzzify crisp preference structures.