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.
SELF-DIVISIBLE ULTRAFILTERS AND CONGRUENCES IN βZ / M. Di Nasso, L. Luperi Baglini, R. Mennuni, M. Pierobon, M. Ragosta. - In: THE JOURNAL OF SYMBOLIC LOGIC. - ISSN 0022-4812. - (2023), pp. 1-18. [Epub ahead of print] [10.1017/jsl.2023.51]
SELF-DIVISIBLE ULTRAFILTERS AND CONGRUENCES IN βZ
L. Luperi BagliniSecondo
;
2023
Abstract
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.File | Dimensione | Formato | |
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self-divisible-ultrafilters-and-congruences-in-dollarbeta-mathbb-zdollar.pdf
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