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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2401.02302 |
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| _version_ | 1866910288144498688 |
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| author | Šobot, Boris |
| author_facet | Šobot, Boris |
| contents | An extension of the divisibility relation on $\mathbb{N}$ to the set $β\mathbb{N}$ of ultrafilters on $\mathbb{N}$ was defined and investigated in several papers during the last ten years. Here we make a survey of results obtained so far, adding several results connecting the themes of different stages of the research. The highlights include: separation of $β\mathbb{N}$ into the lower part $L$ (with its division into levels) and the upper part; identifying basic ingredients (powers of primes) and fragmentation of each ultrafilter into them; finding the corresponding upward closed sets belonging to an ultrafilter with given basic ingredients; existence and number of successors and predecessors of a given divisibility class; extending the congruence relation (in two ways) and checking properties of the obtained relations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02302 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A survey on divisibility of ultrafilters Šobot, Boris Logic 11U10, 54D35, 54D80, 03H15 An extension of the divisibility relation on $\mathbb{N}$ to the set $β\mathbb{N}$ of ultrafilters on $\mathbb{N}$ was defined and investigated in several papers during the last ten years. Here we make a survey of results obtained so far, adding several results connecting the themes of different stages of the research. The highlights include: separation of $β\mathbb{N}$ into the lower part $L$ (with its division into levels) and the upper part; identifying basic ingredients (powers of primes) and fragmentation of each ultrafilter into them; finding the corresponding upward closed sets belonging to an ultrafilter with given basic ingredients; existence and number of successors and predecessors of a given divisibility class; extending the congruence relation (in two ways) and checking properties of the obtained relations. |
| title | A survey on divisibility of ultrafilters |
| topic | Logic 11U10, 54D35, 54D80, 03H15 |
| url | https://arxiv.org/abs/2401.02302 |