Dp-minimality: basic facts and examples

TL;DR

This paper explores the concept of dp-minimality in model theory, providing foundational facts, equivalent definitions, and examples including o-minimal theories, ordered Abelian groups, and p-adic fields.

Contribution

It offers new insights into dp-minimality by establishing key properties, equivalences, and examples, including the dp-minimality of p-adic fields and certain ordered groups.

Findings

Weakly o-minimal theories are dp-minimal.

A divisible ordered Abelian group can be dp-minimal without being weakly o-minimal.

The field of p-adic numbers is dp-minimal.

Abstract

We study the notion of dp-minimality, beginning by providing several essential facts, establishing several equivalent definitions, and comparing dp-minimality to other minimality notions. The rest of the paper is dedicated to examples. We establish via a simple proof that any weakly o-minimal theory is dp-minimal and then give an example of a weakly o-minimal group not obtained by adding traces of externally definable sets. Next we give an example of a divisible ordered Abelian group which is dp-minimal and not weakly o-minimal. Finally we establish that the field of p-adic numbers is dp-minimal.