Linear extensions of partial orders on Abelian groups

TL;DR

This paper characterizes how to extend partial orders to linear orders on Abelian groups using hyperplanes in real vector spaces, providing a uniform approach across different ranks and constructing archimedean orders for torsion-free groups.

Contribution

It offers a unified method to analyze linear order extensions on Abelian groups via hyperplanes, expanding previous results and including constructions for archimedean orders.

Findings

Provides a full characterization of linear order extensions on Abelian groups.

Introduces a hyperplane-based model for linear orders in real vector spaces.

Constructs archimedean directed orders for torsion-free Abelian groups.

Abstract

Partially ordered groups, also known as po-groups, are groups with a compatible partial order. Results from M.I. Zajceva and H.-H. Teh are combined in order to provide a full characterisation of linear order extensions of a given order on a group. In contrast to Teh this approach provides a method to discuss linear orders of different abelian rank in a uniform manner. This will be achieved by modelling the linear orders using hyperplanes in a real vector spaces. Among some additional remarks a construction of an archimedian directed order is given for every torsion free abelian group.