Geodesic
A survey on divisibility of ultrafilters
Zbornik radova, Tome 22 (2025) no. 30, p. 491

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DOI

An extension of the divisibility relation on $\mathbb{N}$ to the set $\beta\mathbb{N}$ 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 $\beta\mathbb{N}$ 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 direct successors of a given divisibility class; extending the congruence relation (in two ways) and checking properties of the obtained relations.
DOI : 10.64191/zr24040410115
Classification : 54-02, 54D80, 11U10, 54D35, 03H15
Keywords: ultrafilter, Stone-Čech compactification, nonstandard extension, divisibility, congruence 
Boris Šobot. A survey on divisibility of ultrafilters. Zbornik radova, Tome 22 (2025) no. 30, p. 491 . doi: 10.64191/zr24040410115
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     title = {A survey on divisibility of ultrafilters},
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