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SELF-DIVISIBLE ULTRAFILTERS AND CONGRUENCES IN
$\beta {\mathbb {Z}}$
- Part of
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 17 July 2023, pp. 1-18
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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 
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$, if and only if the quotient 
$(\beta {\mathbb Z},\oplus )/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$ 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.
