Tame extension of almost o-minimal structure

TL;DR

This paper proves that in almost o-minimal structures, certain definable sets in tame extensions remain definable within the original structure, enhancing understanding of their stability under extensions.

Contribution

It establishes that definable sets with parameters in tame extensions are already definable in the original almost o-minimal structure, extending known properties of o-minimality.

Findings

Definability is preserved in tame extensions for sets with parameters.

Introduces corollaries related to definability in almost o-minimal structures.

Enhances understanding of structure stability under tame extensions.

Abstract

We consider an almost o-minimal expansion of an ordered group $\mathcal M=(M,<,+,0,\ldots)$ and its tame extension $\mathcal N=(N,<,+,0,\ldots)$. We demonstrate that the subset $\{x \in M^n\;|\; \mathcal N \models \Phi(x,a)\}$ of $M^n$ defined by a formula $\Phi(x,y)$ with $\mathcal M$-bounded parameters $a$ in $\mathcal N$ is $\mathcal M$-definable. We also introduce its corollaries.