Weakly minimal groups with a new predicate

Journal of Mathematical Logic 20 (2):2050011 (2020)
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Abstract

Fix a weakly minimal (i.e. superstable U-rank 1) structure M. Let M∗ be an expansion by constants for an elementary substructure, and let A be an arbitrary subset of the universe M. We show that all formulas in the expansion (M∗,A) are equivalent to bounded formulas, and so (M,A) is stable (or NIP) if and only if the M-induced structure AM on A is stable (or NIP). We then restrict to the case that M is a pure abelian group with a weakly minimal theory, and AM is mutually algebraic (equivalently, weakly minimal with trivial forking). This setting encompasses most of the recent research on stable expansions of (Z,+). Using various characterizations of mutual algebraicity, we give new examples of stable structures of the form (M,A). Most notably, we show that if (G,+) is a weakly minimal additive subgroup of the algebraic numbers, A⊆G is enumerated by a homogeneous linear recurrence relation with algebraic coefficients, and no repeated root of the characteristic polynomial of A is a root of unity, then (G,+,B) is superstable for any B⊆A.

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Citations of this work

Contributions to the theory of f-automatic sets.Christopher Hawthorne - 2022 - Journal of Symbolic Logic 87 (1):127-158.

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References found in this work

Paires de structures Stables.Bruno Poizat - 1983 - Journal of Symbolic Logic 48 (2):239-249.
On Superstable Expansions of Free Abelian Groups.Daniel Palacín & Rizos Sklinos - 2018 - Notre Dame Journal of Formal Logic 59 (2):157-169.
Stable theories with a new predicate.Enrique Casanovas & Martin Ziegler - 2001 - Journal of Symbolic Logic 66 (3):1127-1140.
Quasi-o-minimal structures.Oleg Belegradek, Ya'acov Peterzil & Frank Wagner - 2000 - Journal of Symbolic Logic 65 (3):1115-1132.
Stability in geometric theories.Jerry Gagelman - 2005 - Annals of Pure and Applied Logic 132 (2-3):313-326.

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