The ultrasimplicial property for simple dimension groups with unique state, the image of which has rank one

TL;DR

This paper investigates conditions under which certain ordered abelian groups, constructed as direct sums of rank-one and finite-rank torsion-free groups, can be represented as inductive limits with injective maps, focusing on their ultrasimplicial property.

Contribution

It provides necessary and sufficient conditions for these groups to have an inductive limit representation with injective maps, advancing understanding of their structural properties.

Findings

Identifies conditions for ultrasimplicial property in specific ordered groups.

Establishes criteria for inductive limit representations with injective maps.

Enhances classification of simple dimension groups with unique states.

Abstract

Let $G$ be an ordered group that is a direct sum of a rank-one torsion-free abelian group and a finite-rank torsion-free abelian group, with order structure arising from the natural order on the first summand. A necessary condition and a sufficient condition are given for $G$ to have an ordered-group inductive limit representation using injective maps.