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Stable groups and expansions of $(\mathbb Z,+,0)$

Volume 242 / 2018

Gabriel Conant, Anand Pillay Fundamenta Mathematicae 242 (2018), 267-279 MSC: Primary 03C45; Secondary 03C64. DOI: 10.4064/fm403-8-2017 Published online: 26 February 2018

Abstract

We prove that if $G$ is a sufficiently saturated stable group of finite weight with no infinite, infinite-index, chains of definable subgroups, then $G$ is superstable of finite $U$-rank. Combined with recent work of Palacín and Sklinos, this shows that $(\mathbb Z,+,0)$ has no proper stable expansions of finite weight. A corollary is that if $P\subseteq \mathbb Z^n$ is definable in a finite dp-rank expansion of $(\mathbb Z,+,0)$, and $(\mathbb Z,+,0,P)$ is stable, then $P$ is definable in $(\mathbb Z,+,0)$. In particular, this answers a question of Marker on stable expansions of the group of integers by sets definable in Presburger arithmetic.

Authors

  • Gabriel ConantDepartment of Mathematics
    University of Notre Dame
    Notre Dame, IN 46556, U.S.A.
    e-mail
  • Anand PillayDepartment of Mathematics
    University of Notre Dame
    Notre Dame, IN 46556, U.S.A.
    e-mail

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