Magalì Zuanon

Isotonies on ordered cones through the concept of a decreasing scale

Abstract 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.

Upper semicontinuous representations of interval orders

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.

Existence of Order-Preserving Functions for Nontotal Fuzzy Preference Relations under Decisiveness

Looking at decisiveness as crucial, we discuss the existence of an order-preserving function for the nontotal crisp preference relation naturally associated to a nontotal fuzzy preference relation. We further present conditions for the existence of an upper semicontinuous order-preserving function for a fuzzy binary relation on a crisp topological space.