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DP AND OTHER MINIMALITIES

Part of: Model theory

Published online by Cambridge University Press:  11 April 2025

PIERRE SIMON Open the ORCID record for PIERRE SIMON [Opens in a new window]  and
ERIK WALSBERG
PIERRE SIMON
Affiliation:
DEPARTMENT OF MATHEMATICS UC BERKELEY BERKELEY, CA 94720 USA E-mail: pierre.sim85@gmail.com
ERIK WALSBERG*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA IRVINE, IRVINE, CA 92697 USA
*
E-mail: erik.walsberg@gmail.com
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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 particular 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 $(\mathbb {Z},+,S)$, where S 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 ${\mathscr R}$ of $(\mathbb {R},+,<)$ such that $({\mathscr R},\mathbb {Q})$ is a “dense pair.”.

Keywords

dp-minimalmodel theorylogicordered structures

MSC classification

Primary: 03C45: Classification theory, stability and related concepts 03C60: Model-theoretic algebra 03C64: Model theory of ordered structures; o-minimality

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The Journal of Symbolic Logic , First View , pp. 1 - 30
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© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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