# Definable sets up to definable bijections in Presburger groups

## TL;DR

This paper provides a complete classification of definable sets up to definable bijections in Presburger groups, offering insights into the structure of definable families of bounded sets within ordered abelian groups.

## Contribution

It introduces a comprehensive classification of definable sets in Presburger groups, advancing understanding of their structure and relationships.

## Key findings

- Complete classification of definable sets up to bijections in Presburger groups
- Characterization of definable families of bounded definable sets
- Insights into the structure of ordered abelian groups

## Abstract

We entirely classify definable sets up to definable bijections in $\mathbb{Z}$-groups, where the language is the one of ordered abelian groups. From this, we deduce, among others, a classification of definable families of bounded definable sets.