Convexly orderable groups and valued fields

TL;DR

This paper explores convex orderability in model theory, establishing its relation to VC-minimality, classifying VC-minimal theories of ordered and abelian groups, and examining implications for valued fields like the p-adics.

Contribution

It introduces convex orderability as a notion between VC-minimality and dp-minimality, and classifies VC-minimal theories in specific algebraic structures.

Findings

Convex orderability and VC-minimality are equivalent in some algebraic theories.

Complete classification of VC-minimal theories of ordered and abelian groups.

p-adic fields are not quasi-VC-minimal.

Abstract

We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also considered, including a necessary condition for a theory of valued fields to be quasi-VC-minimal. For example, the p-adics are not quasi-VC-minimal.