# On O-Minimal Expansions of $(\mathbb{Q},<,+,0)$

## TL;DR

This paper proves that any function definable in an o-minimal expansion of the rational ordered additive structure is eventually linear, and this property holds in all elementarily equivalent structures.

## Contribution

It establishes that all definable functions in such o-minimal structures are eventually linear, extending the result to all elementarily equivalent models.

## Key findings

- Definable functions are eventually linear.
- The property holds in all elementary equivalent structures.
- Provides a classification of definable functions in these structures.

## Abstract

Let $f:\mathbb{Q}\to \mathbb{Q}$ be a function definable in an o-minimal expansion of $(\mathbb{Q},<,+,0)$. We show that $f$ is eventually linear. In addition, we show that this holds in every elementary equivalent structure.

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Source: https://tomesphere.com/paper/1704.03050