Cardinality of the sets of dimension functions in ordered structures
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912889477005312 |
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| author | Fujita, Masato |
| author_facet | Fujita, Masato |
| contents | We compute the cardinality $\mathfrak n_{\dim}(\mathcal M)$ of the sets of dimension functions on the ordered structures $\mathcal M$. The inequality $\mathfrak n_{\dim}(\mathcal M) \leq 1$ holds if $\mathcal M$ is a d-minimal expansion of an ordered group. If $\mathcal M$ is o-minimal and $\mathfrak n_{\dim}(\mathcal M)<\infty$, there exists a positive integer $m$ such that $\mathfrak n_{\dim}(\mathcal M)=2^m-1$. For every positive integer $m$, there exists a weakly o-minimal expansion $\mathcal M$ of an ordered divisible Abelian group such that $\mathfrak n_{\dim}(\mathcal M)=m$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10154 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cardinality of the sets of dimension functions in ordered structures Fujita, Masato Logic Primary 03C64, Secondary 05C45, 54F45 We compute the cardinality $\mathfrak n_{\dim}(\mathcal M)$ of the sets of dimension functions on the ordered structures $\mathcal M$. The inequality $\mathfrak n_{\dim}(\mathcal M) \leq 1$ holds if $\mathcal M$ is a d-minimal expansion of an ordered group. If $\mathcal M$ is o-minimal and $\mathfrak n_{\dim}(\mathcal M)<\infty$, there exists a positive integer $m$ such that $\mathfrak n_{\dim}(\mathcal M)=2^m-1$. For every positive integer $m$, there exists a weakly o-minimal expansion $\mathcal M$ of an ordered divisible Abelian group such that $\mathfrak n_{\dim}(\mathcal M)=m$. |
| title | Cardinality of the sets of dimension functions in ordered structures |
| topic | Logic Primary 03C64, Secondary 05C45, 54F45 |
| url | https://arxiv.org/abs/2512.10154 |