Nonvaluational ordered Abelian groups of finite burden
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913707530911744 |
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| author | Fujita, Masato |
| author_facet | Fujita, Masato |
| contents | Consider an expansion $\mathcal R=(R,<,+,\ldots)$ of an ordered divisible Abelian group of finite burden defining no nonempty subset $X$ of $R$ which is dense and codense in a definable open subset $U$ of $R$ with $X \subseteq U$. We further assume that $\mathcal R$ is nonvaluational, that is, for every nonempty definable subsets $A,B$ of $R$ with $A <B$ and $A \cup B=R$, $\inf\{b-a\;|\;a \in A, b \in B\}=0$. Then, $\mathcal R$ is $*$-locally weakly o-minimal. We also give a complete description of sets definable in a definably complete expansion of ordered group of burden two if it defines an infinite discrete set. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_18721 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonvaluational ordered Abelian groups of finite burden Fujita, Masato Logic Primary 03C64, Secondary 03C45 Consider an expansion $\mathcal R=(R,<,+,\ldots)$ of an ordered divisible Abelian group of finite burden defining no nonempty subset $X$ of $R$ which is dense and codense in a definable open subset $U$ of $R$ with $X \subseteq U$. We further assume that $\mathcal R$ is nonvaluational, that is, for every nonempty definable subsets $A,B$ of $R$ with $A <B$ and $A \cup B=R$, $\inf\{b-a\;|\;a \in A, b \in B\}=0$. Then, $\mathcal R$ is $*$-locally weakly o-minimal. We also give a complete description of sets definable in a definably complete expansion of ordered group of burden two if it defines an infinite discrete set. |
| title | Nonvaluational ordered Abelian groups of finite burden |
| topic | Logic Primary 03C64, Secondary 03C45 |
| url | https://arxiv.org/abs/2502.18721 |