REPRESENTING PREFERENCES WITH NONTRANSITIVE INDIFFERENCE BY A SINGLE REAL-VALUED FUNCTION

Let < be an interval order on a topological space (X, tau), and let x <(similar to) y if and only if [y < z double right arrow x < z], and x <(similar to)* y if and only if [z < x double right arrow z < y]. T hen <(similar to) and <(similar to)** are complete preorders. In the particular case when < is a semiorder, let x <(0)(similar to) y if and only if x <(similar to) y and x <(similar to)** y. Then <(0)(similar to), is a complete preorder, too. We present sufficient conditions fo r the existence of continuous utility functions representing <(similar to), <(similar to)** and <(0)(similar to), by using the notion of st rong separability of a preference relation, which was introduced by Ch ateauneuf (Journal of Mathematical Economics, 1987, 16, 139-146). Fina lly, we discuss the existence of a pair of continuous functions u, u r epresenting a strongly separable interval order < on a measurable topo logical space (X, tau, mu, M).