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

TL;DR

This paper characterizes self-divisible ultrafilters in the Stone-Čech compactification of integers, showing their relation to congruence relations and profinite groups, with examples and structural insights.

Contribution

It introduces and characterizes self-divisible ultrafilters, linking weak and strong congruences, and explores their algebraic and topological properties in $eta\mathbb{Z}$.

Findings

Self-divisible ultrafilters are precisely those for which $ eg ext{equiv}_w$ is an equivalence.

The quotient by the strong congruence is a profinite group.

An ultrafilter exists where $ ext{equiv}_w$ is not symmetric.

Abstract

We introduce self-divisible ultrafilters, which we prove to be precisely those $w$ such that the weak congruence relation $\equiv_w$ introduced by \v{S}obot 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 $\equiv^{\mathrm{s}}_{w}$, if and only if the quotient $(\beta\mathbb{Z},\oplus)/\mathord{\equiv^{\mathrm{s}}_w}$ 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.