Dp and other minimalities

Abstract

A first order expansion of $(\mathbb{R},+,<)$ is dp-minimal if and only if it is o-minimal. We prove analogous results for algebraic closures of finite fields, $p$-adic fields, ordered abelian groups with only finitely many convex subgroups (in articular archimedean ordered abelian groups), and abelian groups equipped with archimedean cyclic group orders. The latter allows us to describe unary definable sets in dp-minimal expansions of $(\mathbb{Z},+,C)$, where $C$ is a cyclic group order. Along the way we describe unary definable sets in dp-minimal expansions of ordered abelian groups. In the last section we give a canonical correspondence between dp-minimal expansions of $(\mathbb{Q},+,<)$ and o-minimal expansions $\mathcal{R}$ of $(\mathbb{R},+,<)$ such that $(\mathcal{R},\mathbb{Q})$ is a "dense pair".