A generalization of Rader's utility representation theorem

Rader's utility representation theorem guarantees the existence of an upper semicontinuous utility function for any upper semicontinuous total preorder on a second countable topological space. In this paper we present a generalization of Rader's theorem to not necessarily total preorders that are weakly upper semicontinuous.Weakly upper semicontinuous preorder; utility function

Weak continuity of preferences with nontransitive indifference

We characterize weak continuity of an interval order on a topological space by using the concept of a scale in a topological space.Weakly continuous interval order; continuous numerical representation

Sublinear and continuous order-preserving functions for noncomplete preorders

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 page

Existence of continuous utility functions for arbitrary binary relations: some sufficient conditions

We present new sufficient conditions for the existence of a continuous utility function for an arbitrary binary relation on a topological space. Such conditions are basically obtained by using both the concept of a weakly continuous binary relation on a topological space and the concept of a countable network weight. In particular, we are concerned with suitable topological notions which generalize the concept of compactness and do not imply second countability or local compactness.hereditarily Lindeloef space; weakly continuous binary relation; countable network weight; hemicompactness; submetrizability

Characterization of Useful Topologies in Mathematical Utility Theory by Countable Chain Conditions

Under the additional assumption of complete regularity, we furnish a simple characterization of all the topologies such that every continuous total preorder is representable by a continuous utility function. In particular, we prove that a completely regular topology satisfies such property if, and only if, it is separable and every linearly ordered collection of clopen sets is countable. Since it is not restrictive to refer to completely regular topologies when dealing with this kind of problem, this is, as far as we are concerned, the simplest characterization of this sort available in the literature. All the famous utility representation theorems are corollaries of our result

A selection of maximal elements under non-transitive indifferences

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

Continuous Order-Preserving Functions on a Preordered Completely Regular Topological Space

A necessary and suffi{}cient condition is presented for the existence of a real continuous order-preserving function f on a topological preordered space $\left(X,\tau,\preceq\right)$under a resonable continuity assumption concerning the preorder $\preceq$ , called \textquotedblleft{}quasi ICcontinuity\textquotedblright{}. Under the same continuity hypotheses, a suffi{}cient condition is provided for the existence of a real continuous order-preserving function in case that $\left(X,\tau\right)$ is a completely regular space

On continuous multi-utility representations of semi-closed and closed preorders

On the basis of the classical continuous multi-utility representation theorem of Levin on locally compact and $\sigma$-compact Hausdorff spaces, we present necessary and sufficient conditions on a topological space $(X,t)$ under which every semi-closed and closed preorder respectively admits a continuous multi-utility representation. This discussion provides the fundaments of a mainly topological theory that systematically combines topological and order theoretic aspects of the continuous multi-utility representation problem

Lifting Theorems for Continuous Order-Preserving Functions and Continuous Multi-Utility

We present some lifting theorems for continuous order-preserving functions on locally and σ -compact Hausdorff preordered topological spaces. In particular, we show that a preorder on a locally and σ -compact Hausdorff topological space has a continuous multi-utility representation if, and only if, for every compact subspace, every continuous order-preserving function can be lifted to the entire space. Such a characterization is also presented by introducing a lifting property of ≾-C-compatible continuous order-preserving functions on closed subspaces. The assumption of paracompactness is also used in connection to lifting conditions

A selection of maximal elements under non-transitive indifferences

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.Maximal element; Selection of maximals; Acyclicity; Interval order; Semiorder