Dp and other minimalities
Pierre Simon, Erik Walsberg

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 is dp-minimal if and only if it is o-minimal. We prove analogous results for algebraic closures of finite fields, -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 , where 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 and o-minimal expansions of such that is a "dense pair".
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Taxonomy
TopicsAdvanced Topology and Set Theory · Rings, Modules, and Algebras · Mathematical and Theoretical Analysis
