Necessary and Sufficient Conditions for The Representation of A Max- T -transitive Fuzzy Preference by A Utility Function on An Uncountable Set

摘要

Fono and Andjiga [8] established necessary and sufficient conditions under which a given max- T -transitive fuzzy binary relation is representable by a utility function on a countable universe (set of alternatives). In this paper, we study the uncountable case. Specifically, we show that in an uncountable universe, these conditions are necessary but not sufficient. We introduce the condition of "separability of a fuzzy binary relation" and show that this new condition and the previous ones (those obtained in the countable case) are necessary and sufficient for the representation of a given max- T -transitive fuzzy binary relation by a utility function on an uncountable universe.