A survey on divisibility of ultrafilters

Boris Sobot

Published 2024-01-04Version 1

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 a few simple results connecting the themes of different stages of the research. The main highlights include: separation 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; estimating cardinalities of divisibility-equivalence classes; extending the congruence relation (in two ways) and checking properties of the obtained relations.