A classification of finite rank dimension groups by their representations in ordered real vector spaces

TL;DR

This paper classifies finite rank dimension groups by their representations in finite dimensional ordered real vector spaces with Riesz interpolation, providing explicit models and a canonical embedding characterization.

Contribution

It offers a complete classification of finite rank dimension groups through explicit models and describes which subgroups have Riesz interpolation.

Findings

Finitely many ordered real vector spaces of each dimension have Riesz interpolation.

Every finite rank dimension group can be embedded into such a vector space.

Characterization of subgroups with Riesz interpolation within these spaces.

Abstract

This paper systematically studies finite rank dimension groups, as well as finite dimensional ordered real vector spaces with Riesz interpolation. We provide an explicit description and classification of finite rank dimension groups, in the following sense. We show that for each n, there are (up to isomorphism) finitely many ordered real vector spaces of dimension n that have Riesz interpolation, and we give an explicit model for each of them in terms of combinatorial data. We show that every finite rank dimension group can be realized as a subgroup of a finite dimensional ordered real vector space with Riesz interpolation via a canonical embedding. We then characterize which of the subgroups of a finite dimensional ordered real vector space have Riesz interpolation (and are therefore dimension groups).