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

Conditions for the Upper Semicontinuous Representability of Preferences with Nontransitive Indifference

“…Further, strong separability occurs, for example, whenever an interval order is representable by means of a pair ( ) , u v of nonnegative positively homogeneous functions on a cone in a topological vector space. This kind of representability, in the more general setting of acyclic binary relations, was studied, for example, by Alcantud et al [14] and in the case of not necessarily total preorders by Bosi et al [15].…”

A Survey on the Mathematical Foundations of Axiomatic Entropy: Representability and Orderings

“…In this sense, we point out that, in the literature on numerical representations of ordered structures (mainly, total preorders) endowed with some compatible algebraic operation, one may encounter, mainly, classical results on ordered groups and semigroups (see, e.g., ). A key reference here is the book by L. Fuchs [105] However, there are also deep studies on lattices, where a famous reference is the book [106] by Garret Birkhoff (see also [107]), ordered vector spaces (see, e.g., [108][109][110][111][112]) and even real algebras (see, e.g., [56]). …”

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

“…On the other hand, it should be noted that there is a large number of contributions to the representation of nontotal binary relations (in particular nontotal preorders) in the crisp case (see e.g., Chapter 5 in Bridges amd Mehta [7], Herden and Levin [8], Herden [9], Mehta [10], Bosi et al [11], Bosi and Mehta [12], and Bosi and Zuanon [13]). This is the case when there are elements x, y ∈ X such that neither x y nor y x.…”