Continuous Order Representability Properties of Topological Spaces and Algebraic Structures

Abstract

In the present paper, we study the relationship between continuous order-representability and the fulfillment of the usual covering properties on topological spaces. We also consider the case of some algebraic structures providing an application of our results to the social choice theory context.

Open Questions in Utility Theory

“…Topologies satisfying SRP were studied in [15] under a different nomenclature. Then they have been analyzed in [22,23,24,27]. Definition 4.16.…”

“…To fix ideas, we will include here some results already launched in [27,39]. In that paper, although one could have begun with some much simpler algebraic structure (e.g.…”

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

“…The preorder is said to be separable in the sense of Debreu (see [35,37,44,54] if there exists a countable subset D ⊆ X such that given x, y ∈ X with x ≺ y, there exists some d ∈ D satisfying that x d y. It is said to be perfectly separable (see, e.g., [56]) if there exists a countable subset D ⊆ X such that given x, y ∈ X with x ≺ y, there exists some d ∈ D satisfying that x ≺ d y.…”

“…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]). …”

Continuous Representability of Interval Orders: The Topological Compatibility Setting

“…[20][21][22][23] The existence of a pair of upper semicontinuous real-valued functions representing an interval order on a topological space has recently been characterized by Bosi and Zuanon. In that direction, such a characterization, which generalizes previous interesting results presented by Bridges, is the most general one, since the authors do not impose conditions either on the topology or on the two functions that make up the representation.…”