Divisibility classes of ultrafilters and their patterns
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908387334160384 |
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| author | Šobot, Boris |
| author_facet | Šobot, Boris |
| contents | A divisibility relation on ultrafilters on the set $\mathbb{N}$ of natural numbers is defined as follows: ${\cal F}\hspace{1mm}\widetilde{\mid}\hspace{1mm}{\cal G}$ if and only if every set in $\cal F$ upward closed for divisibility also belongs to $\cal G$. Previously we isolated basic classes: powers of prime ultrafilters, and described the pattern of an ultrafilter, measuring the quantity of members of each basic class dividing a given ultrafilter. In this paper we define a topology on the set of basic classes which will allow us to calculate the pattern of the limit of a $\widetilde{\mid}$-increasing chain of ultrafilters. Using this we characterize which patterns can actually appear as patterns of an ultrafilter. Defining the $=_\sim$-divisibility classes by identifying mutually divisible ultrafilters, in the respective quotient order $(β\mathbb{N}/=_\sim,\widetilde{\mid})$ we identify singleton classes and consider their patterns. Finally, we give a sufficient condition for a $=_\sim$-divisibility class to have an immediate predecessor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19753 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Divisibility classes of ultrafilters and their patterns Šobot, Boris Logic 03H15, 11U10, 54D35, 54D80 A divisibility relation on ultrafilters on the set $\mathbb{N}$ of natural numbers is defined as follows: ${\cal F}\hspace{1mm}\widetilde{\mid}\hspace{1mm}{\cal G}$ if and only if every set in $\cal F$ upward closed for divisibility also belongs to $\cal G$. Previously we isolated basic classes: powers of prime ultrafilters, and described the pattern of an ultrafilter, measuring the quantity of members of each basic class dividing a given ultrafilter. In this paper we define a topology on the set of basic classes which will allow us to calculate the pattern of the limit of a $\widetilde{\mid}$-increasing chain of ultrafilters. Using this we characterize which patterns can actually appear as patterns of an ultrafilter. Defining the $=_\sim$-divisibility classes by identifying mutually divisible ultrafilters, in the respective quotient order $(β\mathbb{N}/=_\sim,\widetilde{\mid})$ we identify singleton classes and consider their patterns. Finally, we give a sufficient condition for a $=_\sim$-divisibility class to have an immediate predecessor. |
| title | Divisibility classes of ultrafilters and their patterns |
| topic | Logic 03H15, 11U10, 54D35, 54D80 |
| url | https://arxiv.org/abs/2412.19753 |