Self-divisible ultrafilters and congruences in $β\mathbb{Z}$
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866912674325987328 |
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| author | Di Nasso, Mauro Baglini, Lorenzo Luperi Mennuni, Rosario Pierobon, Moreno Ragosta, Mariaclara |
| author_facet | Di Nasso, Mauro Baglini, Lorenzo Luperi Mennuni, Rosario Pierobon, Moreno Ragosta, Mariaclara |
| contents | We introduce self-divisible ultrafilters, which we prove to be precisely those $w$ such that the weak congruence relation $\equiv_w$ introduced by Šobot is an equivalence relation on $β\mathbb{Z}$. We provide several examples and additional characterisations; notably we show that $w$ is self-divisible if and only if $\equiv_w$ coincides with the strong congruence relation $\equiv^{\mathrm{s}}_{w}$, if and only if the quotient $(β\mathbb{Z},\oplus)/\mathord{\equiv^{\mathrm{s}}_w}$ is a profinite group. We also construct an ultrafilter $w$ such that $\equiv_w$ fails to be symmetric, and describe the interaction between the aforementioned quotient and the profinite completion $\hat{\mathbb{Z}}$ of the integers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_09983 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Self-divisible ultrafilters and congruences in $β\mathbb{Z}$ Di Nasso, Mauro Baglini, Lorenzo Luperi Mennuni, Rosario Pierobon, Moreno Ragosta, Mariaclara Logic 54D80, 11A07 (Primary). 03H15, 03C62, 20E18 (Secondary) We introduce self-divisible ultrafilters, which we prove to be precisely those $w$ such that the weak congruence relation $\equiv_w$ introduced by Šobot is an equivalence relation on $β\mathbb{Z}$. We provide several examples and additional characterisations; notably we show that $w$ is self-divisible if and only if $\equiv_w$ coincides with the strong congruence relation $\equiv^{\mathrm{s}}_{w}$, if and only if the quotient $(β\mathbb{Z},\oplus)/\mathord{\equiv^{\mathrm{s}}_w}$ is a profinite group. We also construct an ultrafilter $w$ such that $\equiv_w$ fails to be symmetric, and describe the interaction between the aforementioned quotient and the profinite completion $\hat{\mathbb{Z}}$ of the integers. |
| title | Self-divisible ultrafilters and congruences in $β\mathbb{Z}$ |
| topic | Logic 54D80, 11A07 (Primary). 03H15, 03C62, 20E18 (Secondary) |
| url | https://arxiv.org/abs/2302.09983 |