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We characterize weak continuity of an interval order on a topological space (X, τ) by using the concept of a scale in a topological space.

Rader’s utility representation theorem guarantees the existence of an upper semicontinuous utility function u for any upper semicontinuous total preorder on a second countable topological space (X, t). In this paper we present a generalization of Rader’s theorem to not necessarily total preorders that are weakly upper semicontinuous.

We present necessary and sufficient conditions for the representabi lity of a coherent risk measure on $L^{\infty}_{+}(\Omega , {\cal F}, \Bbb P )$ as the Choquet integral with respect to a concave probability distortion.

We characterize the existence of a nonnegative, sublinear and continuous order-preserving function for a not necessarily complete preorder on a real convex cone in an arbitrary topological real vector space. As a corollary of the main result, we present necessary and sufficient conditions for the existence of such an order-preserving function for a complete preorder., Comment: 8 pages

We discuss the existence of an upper semicontinuous multi-utility representation of a preorder on a topological space. We then prove that every weakly upper semicontinuous preorder is extended by an upper semicontinuous preorder and use this fact in order to show that every weakly upper semicontinuous preorder on a compact topological space admits a maximal element.

We characterize the existence of a nonnegative, sublinear and continuous order-preserving function for a not necessarily complete preorder on a real convex cone in an arbitrary topological real vector space. As a corollary of the main result, we present necessary and sufficient conditions for the existence of such an order-preserving function for a complete preorder.

Given an interval order on a topological space, we characterize its representability by means of a pair of upper semicontinuous real-valued functions. This characterization is only based on separability and continuity conditions related to both the interval order and one of its two traces. As a corollary, we obtain the classical Rader’s theorem concerning the existence of an upper semicontinuous representation for an upper semicontinuous total preorder on a second countable topological space.

We present different conditions for the existence of a pair of upper semicontinuous functions representing an interval order on a topological space without imposing any restrictive assumptions neither on the topological space nor on the representing functions. The particular case of second countable topological spaces, which is particularly interesting and frequent in economics, is carefully considered. Some final considerations concerning semiorders finish the paper.

In this work we are concerned with maximality issues under intransitivity of the indifference. Our approach relies on the analysis of "undominated maximals" (cf., Peris and Subiza, J Math Psychology 2002). Provided that an agent's binary relation is acyclic, this is a selection of its maximal elements that can always be done when the set of alternatives is finite. In the case of semiorders, proceeding in this way is the same as using Luce's selected maximals. We put forward a sufficient condition for the existence of undominated maximals for interval orders without any cardinality restriction. Its application to certain type of continuous semiorders is very intuitive and accommodates the well-known "sugar example" by Luce.

Using techniques based on decreasing scales, necessary and sufficient conditions are presented for the existence of a continuous and homogeneous of degree one real-valued function representing a (not necessarily complete) preorder defined on a cone of a real vector space. Applications to measure theory and expected utility are given as consequences.