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On semibounded expansions of ordered groups

Volume 71 / 2023

Pantelis E. Eleftheriou, Alex Savatovsky Bulletin Polish Acad. Sci. Math. 71 (2023), 97-113 MSC: Primary 03C64; Secondary 22B99. DOI: 10.4064/ba230725-27-9 Published online: 3 November 2023


We explore semibounded expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We show that if $\mathcal R=\langle \mathbb R, \lt , +, \ldots \rangle $ is a semibounded o-minimal structure and $P\subseteq \mathbb R$ is a set satisfying certain tameness conditions, then $\langle \mathcal R, P\rangle $ remains semibounded. Examples include the cases when $\mathcal R=\langle \mathbb R, \lt ,+, (x\mapsto \lambda x)_{\lambda \in \mathbb R}, \cdot _{\upharpoonright [0, 1]^2}\rangle $, and $P= 2^\mathbb Z$ or $P$ is an iteration sequence. As an application, we show that smooth functions definable in such $\langle \mathcal R, P\rangle $ are definable in $\mathcal R$.


  • Pantelis E. EleftheriouSchool of Mathematics
    University of Leeds
    Leeds LS2 9JT, UK
  • Alex SavatovskyDepartment of Mathematics
    University of Haifa
    Haifa, Israel

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