Self-divisible ultrafilters and congruences in βZ

We introduce self-divisible ultrafilters, which we prove to be precisely those w such that the weak congruence relation w introduced by Sobot is an equivalence relation on βZ. We provide several examples and additional characterisations; notably we show that w is self-divisible if and only if ≡w coincides with the strong congruence relation s w, if and only if the quotient (βZ,⊕)/≡s w is a profinite group. We also construct an ultrafilter w such that ≡w fails to be symmetric, and describe the interaction between the aforementioned quotient and the profinite completion Z of the integers.