Congruence of ultrafilters

TL;DR

This paper explores the structure of ultrafilters on natural numbers through a divisibility relation extension, introducing a congruence relation modulo ultrafilters and analyzing their algebraic properties.

Contribution

It introduces a new congruence relation modulo ultrafilters and studies its properties, extending the divisibility relation on ultrafilters with nonstandard analysis tools.

Findings

Ultrafilters equivalent under the divisibility relation can have different residues modulo an integer.

A generalized congruence relation modulo ultrafilters is introduced and analyzed.

A stronger relation with better algebraic properties is developed using iterated nonstandard extensions.

Abstract

We continue the research of the relation $\hspace{1mm}\widetilde{\mid}\hspace{1mm}$ on the set $\beta {\mathbb{N}}$ of ultrafilters on ${\mathbb{N}}$, defined as an extension of the divisibility relation. It is a quasiorder, so we see it as an order on the set of $=_\sim$-equivalence classes, where ${\cal F}=_\sim{\cal G}$ means that ${\cal F}$ and ${\cal G}$ are mutually $\hspace{1mm}\widetilde{\mid}\hspace{1mm}$-divisible. Here we introduce a new tool: a relation of congruence modulo an ultrafilter. We first recall the congruence of ultrafilters modulo an integer and show that $=_\sim$-equivalent ultrafilters do not necessarily have the same residue modulo $m\in {\mathbb{N}}$. Then we generalize this relation to congruence modulo an ultrafilter in a natural way. After that, using iterated nonstandard extensions, we introduce a stronger relation, which has nicer properties with respect to addition and multiplication of ultrafilters. Finally, we also introduce a strengthening of $\hspace{1mm}\widetilde{\mid}\hspace{1mm}$ and show that it also behaves well in relation to the congruence relation.