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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2302.08040 |
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| _version_ | 1866915760820977664 |
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| author | Goto, Tatsuya |
| author_facet | Goto, Tatsuya |
| contents | We introduce new cardinal invariants of a poset, called the comparability number and the incomparability number. We determine their value for well-known posets, such as $ω^ω$, $\mathcal{P}(ω)/\mathrm{fin}$, the Turing degrees $\mathcal{D}$, the quotient algebra $\mathsf{Borel}(2^ω)/\mathsf{null}$, the ideals $\mathsf{meager}$ and $\mathsf{null}$. Moreover, we consider these invariants for the Rudin-Keisler ordering of the nonprincipal ultrafilters on $ω$. We also consider these invariants for ideals on $ω$ and on $ω_1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_08040 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The comparability numbers and the incomparability numbers Goto, Tatsuya Logic We introduce new cardinal invariants of a poset, called the comparability number and the incomparability number. We determine their value for well-known posets, such as $ω^ω$, $\mathcal{P}(ω)/\mathrm{fin}$, the Turing degrees $\mathcal{D}$, the quotient algebra $\mathsf{Borel}(2^ω)/\mathsf{null}$, the ideals $\mathsf{meager}$ and $\mathsf{null}$. Moreover, we consider these invariants for the Rudin-Keisler ordering of the nonprincipal ultrafilters on $ω$. We also consider these invariants for ideals on $ω$ and on $ω_1$. |
| title | The comparability numbers and the incomparability numbers |
| topic | Logic |
| url | https://arxiv.org/abs/2302.08040 |