Minimal definitions of classical and fuzzy preference structures

Preference structures are fundamental tools in the theory of preference modelling. They consist of three basic relations: the strict preference relation, the indifference relation and the incomparability relation. The generalization of these crisp structures to the fuzzy case, and consequently the enlargement of the sphere of these structures to more realistic decision-analytic settings, has received considerable attention in recent years. This body of research has in particular led to the cornerstone definition of a one-parameter family of fuzzy preference structures, the φ-fuzzy preference structures. We choose to follow a different route, leading to equivalent, yet more compact, formulations of the same definition. Firstly, we re-examine the definition of a crisp preference structure and eliminate its redundant components. This reduction leads to four minimal sets of conditions, each of which is equivalent to the definition of a preference structure. Secondly, we generalize these minimal definitions to the fuzzy case in a straightforward way. The major result of this paper is that each of these four sets of fuzzified conditions is equivalent to the definition of a φ-fuzzy preference structure. Hence, they are indeed minimal definitions of a φ-fuzzy preference structure. The minimality of these definitions alleviates the mathematical difficulties in manipulating fuzzy preference structures, an important asset in real-world preference modelling