Real dimension groups

TL;DR

This paper explores the properties of real dimension groups, characterizing their structure through limits of simplicial vector spaces and examining the Riesz interpolation property in polynomial algebras.

Contribution

It provides a characterization of real dimension groups analogous to dimension groups and extends results on Riesz interpolation in polynomial algebras.

Findings

Sequential limits of simplicial vector spaces require strong assumptions.

Real polynomial algebras with pointwise order satisfy Riesz interpolation.

Characterization of real dimension groups analogous to classical dimension groups.

Abstract

We show the characterization analogous to dimension groups of partially ordered real vector spaces with interpolation works, but sequential direct limits of simplicial vector spaces only under strong assumptions. We also provide and generalize a proof of a result of Fuchs asserting that the real polynomial algebra with pointwise ordering coming from an interval satisfies Riesz interpolation