Self-divisible ultrafilters and congruences in $\beta \mathbb{Z}$

Abstract

We introduce self-divisible ultrafilters, which we prove to be precisely those $w$ such that the weak congruence relation ${\equiv }_w$ introduced by Šobot is an equivalence relation on $\beta \mathbb{Z}$. We provide several examples and additional characterisations; notably we show that $w$ is self-divisible if and only if ${\equiv }_w$ coincides with the strong congruence relation ${\equiv }_w^{\mathrm{s}}$, if and only if the quotient $(\beta \mathbb{Z},\oplus )/{\equiv }_w^{\mathrm{s}}$ is a profinite group. We also construct an ultrafilter $w$ such that ${\equiv }_w$ fails to be symmetric, and describe the interaction between the aforementioned quotient and the profinite completion $\hat{\mathbb{Z}}$ of the integers.

17 pages, 1 figure