Stable reducts of elementary extensions of Presburger arithmetic

Eran Alouf; Antongiulio Fornasiero; Itay Kaplan

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arXiv · 2412.10336

Stable reducts of elementary extensions of Presburger arithmetic

Eran Alouf·Antongiulio Fornasiero·Itay Kaplan

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Abstract

Suppose $N$ is elementarily equivalent to an archimedean ordered abelian group $(G,+,<)$ with small quotients (for all $1 \leq n < \omega$, $[G: nG]$ is finite). Then every stable reduct of $N$ which expands $(G,+)$ (equivalently every reduct that does not add new unary definable sets) is interdefinable with $(G,+)$. This extends previous results on stable reducts of $(\mathbb{Z}, +, <)$ to (stable) reducts of elementary extensions of $\mathbb{Z}$. In particular this holds for $G = \mathbb{Z}$ and $G = \mathbb{Q}$. As a result we answer a question of Conant from 2018. This result is a corollary of a more general statement about expansions of weakly-minimal 1-based expansions of abelian groups with small quotients preserving the algebraic closure operator.

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BibTeX RIS

Eran Alouf, Antongiulio Fornasiero, Itay Kaplan. 2024-12-13. Stable reducts of elementary extensions of Presburger arithmetic. https://arxiv.org/abs/2412.10336

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