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.

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.