Strong theories of ordered abelian groups

TL;DR

This paper explores strong expansions of ordered abelian groups, demonstrating their structural properties and providing diverse examples to illustrate the broad applicability of the concept.

Contribution

It introduces the notion of strength in theories of ordered abelian groups and analyzes its implications for definable sets and structures.

Findings

Strength leads to desirable structural properties.

Examples show a wide variety of strong expansions.

Definable infinite discrete sets are characterized.

Abstract

We consider strong expansions of the theory of ordered abelian groups. We show that the assumption of strength has a multitude of desirable consequences for the structure of definable sets in such theories, in particular as relates to definable infinite discrete sets. We also provide a range of examples of strong expansions of ordered abelian groups which demonstrate the great variety of such theories.