L-Fuzzy Weak Preference, Preference, and Choice Relations

Social choice theory investigates how individual preferences can be combined into a collective decision. An individual preference can be given by a preference relation indicating that the individual definitely prefers one alternative over the other, by a weak preference relation indicating that the individual prefers one alternative over the other or is indifferent between the alternatives, or by a choice relation indicating which alternative the individual would choose from a given set of alternatives. All three approaches are equivalent under certain assumptions on the corresponding relations. In this paper we investigate these relations in the context of L-fuzziness, i.e., the corresponding relations are fuzzy relations where the membership values are taken from a Heyting algebra L. Using the abstract theory of arrow categories we provide the most general setting in which these three approaches are equivalent.