Definable choice for a class of weakly o-minimal theories

TL;DR

This paper investigates conditions under which expansions of o-minimal structures with a group operation, involving convex subsets, have definable Skolem functions and explores the implications for definable choice.

Contribution

It characterizes when expanded o-minimal structures with convex subsets possess definable Skolem functions, linking this to the valuational property of the expansion.

Findings

Definable Skolem functions exist iff the expansion is valuational.

Such expansions do not satisfy definable choice.

Provides an elementary proof related to definable choice in these structures.

Abstract

Given an o-minimal structure ${\mathcal M}$ with a group operation, we show that for a properly convex subset $U$, the theory of the expanded structure ${\mathcal M}'=({\mathcal M},U)$ has definable Skolem functions precisely when ${\mathcal M}'$ is valuational. As a corollary, we get an elementary proof that the theory of any such ${\mathcal M}'$ does not satisfy definable choice.