Dp-minimality: invariant types and dp-rank

TL;DR

This paper explores properties of dp-minimal theories, proving key results about invariant types and dp-rank, including their structure, definability, and geometric interpretation, especially in ordered abelian groups.

Contribution

It establishes that invariant dp-minimal types are either finitely satisfiable or definable and provides a geometric characterization of dp-rank in such theories.

Findings

Invariant dp-minimal types are either finitely satisfiable or definable.

Dp-rank is continuous, definable, and geometrically characterized.

In ordered abelian groups, dp-rank equals the order dimension.

Abstract

This paper has two parts. In the first one, we prove that an invariant dp-minimal type is either finitely satisfiable or definable. We also prove that a definable version of the (p,q)-theorem holds in dp-minimal theories of small or medium directionality. In the second part, we study dp-rank in dp-minimal theories and show that it enjoys many nice properties. It is continuous, definable in families and it can be characterised geometrically with no mention of indiscernible sequences. In particular, if the structure expands a divisible ordered abelian group, then dp-rank coincides with the dimension coming from the order.