A structure theorem for abelian quasi-ordered groups

TL;DR

This paper characterizes the structure of compatible quasi-ordered abelian groups, revealing their complexity beyond the field case and connecting quasi-order-minimality with C-minimality.

Contribution

It provides a comprehensive structure theorem for compatible quasi-ordered abelian groups and introduces the concept of quasi-order-minimality, extending prior results from fields to groups.

Findings

Compatible quasi-ordered abelian groups have a complex structure.

A connection between quasi-order-minimality and C-minimality is established.

The group case differs significantly from the field case in structure.

Abstract

We introduce a notion of compatible quasi-ordered groups which unifies valued and ordered abelian groups. It was proved in a paper by Fakhruddin that a compatible quasi-order on a field is always either an order or a valuation. We show here that the group case is more complicated than the field case and describe the general structure of a compatible quasi-ordered abelian group. We also develop a notion of quasi-order-minimality and establish a connection with C-minimality, thus answering a question of F.Delon.