Congruence of ultrafilters

Boris Šobot

Published 2020-08-06Version 1

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 ${\cal F}=_\sim{\cal G}$ means that ${\cal F}$ and ${\cal G}$ are mutually $\hspace{1mm}\widetilde{\mid}\hspace{1mm}$-divisible. Here we introduce a new tool: a relation of congruence modulo an ultrafilter. We first recall the congruence of ultrafilters modulo an integer and show that $=_\sim$-equivalent ultrafilters do not necessarily have the same residue modulo $m\in {\mathbb{N}}$. Then we generalize this relation to congruence modulo an ultrafilter in a natural way. After that, using iterated nonstandard extensions, we introduce a stronger relation, which has nicer properties with respect to addition and multiplication of ultrafilters. Finally, we also introduce a strengthening of $\hspace{1mm}\widetilde{\mid}\hspace{1mm}$ and show that it also behaves well in relation to the congruence relation.