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 ss Z. 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 (ss Z,circle plus)/(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) over cap of the integers.