A theory of pairs for non-valuational structures

TL;DR

This paper develops a theory for pairs of weakly o-minimal structures and their completions, showing how their theories relate and establishing near model completeness for the pair structure.

Contribution

It introduces a canonical language for the o-minimal completion and proves that the theory of the weakly o-minimal structure determines the completion's theory, extending the theory of dense pairs.

Findings

The theory of the pair is near model complete.

Every definable open subset in the pair is already definable in the completion.

An example structure interprets the completion but is not elementarily equivalent to any o-minimal trace.

Abstract

Given a weakly o-minimal structure $\mathcal M$ and its o-minimal completion $\bar {\mathcal M}$, we first associate to $\bar {\mathcal M}$ a canonical language and then prove that $Th(\mathcal M)$ determines $Th(\bar {\mathcal M})$. We then investigate the theory of the pair $\mathcal M^P=(\bar {\mathcal M};M)$ in the spirit of the theory of dense pairs of o-minimal structures, and prove, among other results, that it is near model complete, and every $\mathcal M^P$-definable open subset of $\bar M^n$ is already definable in $\bar {\mathcal M}$. We give an example of a weakly o-minimal structure which interprets $\bar {\mathcal M}$ and show that it is not elementarily equivalent to any reduct of an o-minimal trace.