Finite dimensional ordered vector spaces with Riesz interpolation and Effros-Shen's unimodularity conjecture

TL;DR

This paper demonstrates that finite-dimensional ordered vector spaces over fields within R with Riesz interpolation can be expressed as inductive limits, clarifying the status of Effros-Shen's unimodularity conjecture.

Contribution

It proves that the conjecture holds over fields within R but is false over the integers, and that it becomes true after tensoring with Q.

Findings

Ordered vector spaces over fields in R have inductive limit structures.

Effros-Shen's conjecture is false over Z but true over fields in R.

Tensoring with Q restores the conjecture's validity.

Abstract

It is shown that, for any field F \subseteq R, any ordered vector space structure of F^n with Riesz interpolation is given by an inductive limit of sequence with finite stages (F^n,\F_{>= 0}^n) (where n does not change). This relates to a conjecture of Effros and Shen, since disproven, which is given by the same statement, except with F replaced by the integers, Z. Indeed, it shows that although Effros and Shen's conjecture is false, it is true after tensoring with Q.