On definable Skolem functions and trichotomy

TL;DR

This paper characterizes o-minimal structures with definable Skolem functions, showing they are unions of trivial points and open intervals with definable group-intervals after naming finitely many elements.

Contribution

It provides an explicit description of o-minimal structures with definable Skolem functions, clarifying their structure after finitely naming elements from the prime model.

Findings

Structures are unions of trivial points and open intervals.

Open intervals are unions of definable group-intervals.

Characterization holds after finitely naming elements from the prime model.

Abstract

In this paper we give an explicit characterization of o-minimal structures with definable Skolem functions/definable choice. Such structures are, after naming finitely many elements from the prime model, a union of finitely many trivial points each defined over $\emptyset $ and finitely many open intervals each a union of a $\emptyset $-definable family of group-intervals with fixed positive elements.