Dp and other minimalities
Pierre Simon, Erik Walsberg
TL;DR
The paper develops a unifying framework for dp-minimality across key algebraic and ordered structures, showing that dp-minimality often coincides with or implies tame variants of classical tameness (e.g., o-minimality or weak o-minimality). It establishes a general dp-minimal criterion for ordered abelian groups with finitely many convex subgroups, describes unary definable sets via cnc decompositions, and connects dp-minimal expansions of the rationals to dense-pair o-minimal expansions of the reals. It then extends the analysis to algebraic closures of finite fields, discretely valued fields (including $\mathbb{Q}_p$ and local fields), and cyclically ordered groups using universal covers and dense-pair correspondences, with p-adic and local-field analogues. The results yield a broad taxonomy of dp-minimal expansions, showing either weak o-minimality in non-valuational cases or definable valuations in valuational cases, and provide a canonical bridge between dp-minimal structures on $\mathbb{Q}$ and o-minimal structures on $\mathbb{R}$. Overall, the work contributes a cohesive set of tools for classifying dp-minimal expansions and clarifies how tameness properties transfer across real, p-adic, and cyclic-ordered settings.
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".