The existence of a Richter-Peleg multi-utility representation of a preorder by means of upper semicontinuous or continuous functions is discussed in connection with the existence of a Richter-Peleg utility representation.\ud We give several applications that include the analysis of countable Richter-Peleg multi-utility representations

In this paper, we present a new simple axiomatization of useful topologies, i.e. topologies on an arbitrary set, with respect to which every con- Communicated by Juan-Enrique Martinez LegazThis paper is dedicated to the memory of Professor Gerhard Herden, who passed away on January 30, 2019. He was a friend and an exceptionally clever mathematician. We are deeply indebted to him.

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 [7]). 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.

We introduce the concept of quasi upper semicontinuity of a not necessarily\ud total preorder on a topological space and we prove that there exists a maximal\ud element for a preorder on a compact topological space provided that it is quasi upper\ud semicontinuous. In this way, we generalize many classical and well known results in\ud the literature. We compare the concept of quasi upper semicontinuity with the other\ud semicontinuity concepts to arrive at the conclusion that our definition can be viewed\ud as the most appropriate and natural when dealing with maximal elements of preorders\ud on compact spaces

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