Expansion of Presburger arithmetic with the exchange property

TL;DR

This paper characterizes when an expanded Presburger arithmetic model remains minimal by linking the exchange property and bounded definable sets having maxima.

Contribution

It establishes a precise criterion involving the exchange property and bounded sets for the minimality of theories in expanded Presburger structures.

Findings

The theory is $ ext{L}_{Pres}$-minimal iff it has the exchange property.

Bounded definable sets in the theory have maxima.

Provides a characterization of minimality in expanded Presburger models.

Abstract

Let $G$ be a model of Presburger arithmetic. Let $\mathcal{L}$ be an expansion of the language of Presburger $\mathcal{L}_{Pres}$. In this paper we prove that the $\mathcal{L}$-theory of $G$ is $\mathcal{L}_{Pres}$-minimal iff it has the exchange property and any bounded definable set has a maximum.