Self-divisible ultrafilters and congruences in $β\mathbb{Z}$
Mauro Di Nasso, Lorenzo Luperi Baglini, Rosario Mennuni, Moreno Pierobon, Mariaclara Ragosta
TL;DR
The paper identifies self-divisible ultrafilters as precisely those for which the weak congruence ≡_w on βℤ is an equivalence and coincides with the strong congruence ≡^s_w. It proves that self-divisible ultrafilters are exactly those for which the quotient (βℤ,⊕)/≡^s_w is a profinite group, with an explicit description of the p-component factors G_{p,w} determined by φ_w(p) = max{k: p^kℤ ∈ w}. The work connects βℤ-congruences to the profinite completion ĉℤ via a natural map π and shows a tight decomposition into products of p-adic or finite p-power components, thereby enriching the interaction between nonstandard methods and topological-algebraic structure. It also provides concrete examples and topological observations about self-divisible and division-linear ultrafilters, and raises questions about symmetry versus transitivity of ≡_w, with potential implications for additive combinatorics and Ramsey theory.
Abstract
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 $β\mathbb{Z}$. 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 $(β\mathbb{Z},\oplus)/\mathord{\equiv^{\mathrm{s}}_w}$ 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.