Numerical Representability of Ordered Topological Spaces with Compatible Algebraic Structure

Abstract

We analyze the numerical representability of total preorders defined on semitopological real algebras through continuous order-preserving real-valued functions that are also additive and multiplicative. The results obtained are used to interpret important concepts arising in Social Choice theory.

Open Questions in Utility Theory

“…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.…”

“…Needless to say, on most of them they some partial results have already appeared (See e.g. [32,39,40,42,72]). Problem 5.6.…”

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]). …”

Continuous Order Representability Properties of Topological Spaces and Algebraic Structures

“…(i) τ is connected and separable (see Eilenberg [19]), (ii) τ is separably connected and satisfies CCC (see Campión et al [4]), (iii) τ is separably connected and compact, (iv) τ is separably connected and σ-compact, (v) τ is separably connected and satisfies the Lindelöf property, (vi) τ is second countable (see Debreu [17]), (vii) τ is locally connected and separable (see Candeal et al [15]).…”

“…Before presenting the application some definitions and notations are needed. Other insights regarding the potential use of algebraic techniques in social choice theory can be seen in Candeal, Induráin and Molina [15].…”