On dp-minimal expansions of the integers II
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914683176353792 |
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| author | Alouf, Eran |
| author_facet | Alouf, Eran |
| contents | We first prove that if $\mathcal{Z}$ is a dp-minimal expansion of $\left(\mathbb{Z},+,0,1\right)$ which is not interdefinable with $\left(\mathbb{Z},+,0,1,<\right)$, then every infinite subset of $\mathbb{Z}$ definable in $\mathcal{Z}$ is generic in $\mathbb{Z}$. Using this, we prove that if $\mathcal{Z}$ is a dp-minimal expansion of $\left(\mathbb{Z},+,0,1\right)$ with monster model $G$ such that $G^{00}\neq G^{0}$, then for some $α\in\mathbb{R}\backslash\mathbb{Q}$, the cyclic order on $\mathbb{Z}$ induced by the embedding $n\mapsto nα+\mathbb{Z}$ of $\mathbb{Z}$ in $\mathbb{R}\big/\mathbb{Z}$ is definable in $\mathcal{Z}$. The proof employs the Gleason-Yamabe theorem for abelian groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11146 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On dp-minimal expansions of the integers II Alouf, Eran Logic 03C45, 03C65 We first prove that if $\mathcal{Z}$ is a dp-minimal expansion of $\left(\mathbb{Z},+,0,1\right)$ which is not interdefinable with $\left(\mathbb{Z},+,0,1,<\right)$, then every infinite subset of $\mathbb{Z}$ definable in $\mathcal{Z}$ is generic in $\mathbb{Z}$. Using this, we prove that if $\mathcal{Z}$ is a dp-minimal expansion of $\left(\mathbb{Z},+,0,1\right)$ with monster model $G$ such that $G^{00}\neq G^{0}$, then for some $α\in\mathbb{R}\backslash\mathbb{Q}$, the cyclic order on $\mathbb{Z}$ induced by the embedding $n\mapsto nα+\mathbb{Z}$ of $\mathbb{Z}$ in $\mathbb{R}\big/\mathbb{Z}$ is definable in $\mathcal{Z}$. The proof employs the Gleason-Yamabe theorem for abelian groups. |
| title | On dp-minimal expansions of the integers II |
| topic | Logic 03C45, 03C65 |
| url | https://arxiv.org/abs/2402.11146 |