Dp and other minimalities

TL;DR

This paper explores the concept of dp-minimality across various algebraic structures, establishing conditions under which these structures are dp-minimal and describing their unary definable sets, with a focus on ordered groups and expansions of the rationals.

Contribution

It extends the understanding of dp-minimality to algebraic closures of finite fields, p-adic fields, and ordered abelian groups, providing a unified framework and characterizations.

Findings

Dp-minimality coincides with o-minimality in first order expansions of real numbers.

Unary definable sets are characterized in dp-minimal expansions of ordered abelian groups and integers.

A correspondence between dp-minimal expansions of rationals and o-minimal expansions of reals is established.

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".