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| ˜ -Divisibility of Ultrafilters

Boris Šobot

Annals of Pure and Applied Logic 172 (1):102857 (2021)

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More about divisibility in βN.Boris Šobot - 2021 - Mathematical Logic Quarterly 67 (1):77-87.details We continue the research of an extension of the divisibility relation to the Stone‐Čech compactification. First we prove that ultrafilters we call prime actually possess the algebraic property of primality. Several questions concerning the connection between divisibilities in and nonstandard extensions of are answered, providing a few more equivalent conditions for divisibility in. Results on uncountable chains in are proved and used in a construction of a well‐ordered chain of maximal cardinality. Probably the most interesting result is the existence of (...) Download Export citation Bookmark 2 citations
Multiplicative finite embeddability vs divisibility of ultrafilters.Boris Šobot - 2022 - Archive for Mathematical Logic 61 (3):535-553.details We continue the exploration of various aspects of divisibility of ultrafilters, adding one more relation to the picture: multiplicative finite embeddability. We show that it lies between divisibility relations \ and \. The set of its minimal elements proves to be very rich, and the \-hierarchy is used to get a better intuition of this richness. We find the place of the set of \-maximal ultrafilters among some known families of ultrafilters. Finally, we introduce new notions of largeness of subsets (...) Download Export citation Bookmark 1 citation
Congruence of ultrafilters.Boris Šobot - 2021 - Journal of Symbolic Logic 86 (2):746-761.details We continue the research of the relation $\hspace {1mm}\widetilde {\mid }\hspace {1mm}$ on the set $\beta \mathbb {N}$ of ultrafilters on $\mathbb {N}$, defined as an extension of the divisibility relation. It is a quasiorder, so we see it as an order on the set of $=_{\sim }$ -equivalence classes, where $\mathcal {F}=_{\sim }\mathcal {G}$ means that $\mathcal {F}$ and $\mathcal {G}$ are mutually $\hspace {1mm}\widetilde {\mid }$ -divisible. Here we introduce a new tool: a relation of congruence modulo an (...) Download Export citation Bookmark 1 citation

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