Congruence of ultrafilters

Abstract

We continue the research of the relation $\tilde{\mid }$ 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 $\mathcal{F}{=}_{\sim }\mathcal{G}$ means that $\mathcal{F}$ and $\mathcal{G}$ are mutually $\tilde{\mid }$-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 $\tilde{\mid }$ and show that it also behaves well in relation to the congruence relation.

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References in corpus (1)

• Hyperintegers and Nonstandard Techniques in Combinatorics of Numbers

Cited by in corpus (1)

• Self-divisible ultrafilters and congruences in $\beta \mathbb{Z}$