Order representability in groups and vector spaces (Q435157)

This paper presents a systematic study of the numerical representation problem for total preorders on groups and real vector spaces, with a focus on the following two aspects:NEWLINENEWLINE\noindent (1) the theory of order-preserving real-valued functions that satisfy additional topological and algebraic properties;NEWLINENEWLINE\noindent (2) foundational aspects related to social choice theory.NEWLINENEWLINEAfter an introduction and a section with basic definitions and notations, Section 3 introduces the (ordinal) representation problem in an algebraic framework and presents solutions to the problem that one may find in the literature. Section 4 is devoted to introduce the continuous representation problem for total preorders, and the continuous representability property for groups and real vector spaces equipped with a topology (in particular, topological groups and topological real vector spaces). The continuous representation problem has to do with the existence of a continuous order-preserving function for an arbitrary total preorder defined on a topological space, while the latter provides topological conditions that ensure the continuous representability of every continuous total preorder defined on the given topological space. The algebraic extensions of these two situations are considered in Section 5. Here the authors look for continuous numerical representations of total preorders that are, in addition, algebraic homomorphisms. Finally, the last section offers an application of the algebraic and topological approach of Section 5 to social choice theory, namely to the characterization of the utilitarian functional in the context of utility theory.