Stable reducts of elementary extensions of Presburger arithmetic
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916697585221632 |
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| author | Alouf, Eran Fornasiero, Antongiulio Kaplan, Itay |
| author_facet | Alouf, Eran Fornasiero, Antongiulio Kaplan, Itay |
| contents | Suppose $N$ is elementarily equivalent to an archimedean ordered abelian group $(G,+,<)$ with small quotients (for all $1 \leq n < ω$, $[G: nG]$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_10336 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stable reducts of elementary extensions of Presburger arithmetic Alouf, Eran Fornasiero, Antongiulio Kaplan, Itay Logic 03C45, 03C64 (Primary) 03C07, 06F20 (Secondary) Suppose $N$ is elementarily equivalent to an archimedean ordered abelian group $(G,+,<)$ with small quotients (for all $1 \leq n < ω$, $[G: nG]$ 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. |
| title | Stable reducts of elementary extensions of Presburger arithmetic |
| topic | Logic 03C45, 03C64 (Primary) 03C07, 06F20 (Secondary) |
| url | https://arxiv.org/abs/2412.10336 |