Divisibility classes of ultrafilters and their patterns

TL;DR

This paper investigates the structure of ultrafilters on natural numbers through a divisibility relation, defining a topology on basic classes, analyzing patterns of ultrafilter limits, and characterizing possible patterns and divisibility classes.

Contribution

It introduces a topology on basic classes of ultrafilters, enabling the calculation of limit patterns and the characterization of which patterns can occur.

Findings

A topology on basic classes allows pattern calculation of ultrafilter limits.

Characterization of possible patterns of ultrafilters.

Identification of singleton classes and conditions for immediate predecessors.

Abstract

A divisibility relation on ultrafilters on the set $\mathbb{N}$ of natural numbers is defined as follows: ${\cal F}\hspace{1mm}\widetilde{\mid}\hspace{1mm}{\cal G}$ if and only if every set in $\cal F$ upward closed for divisibility also belongs to $\cal G$. Previously we isolated basic classes: powers of prime ultrafilters, and described the pattern of an ultrafilter, measuring the quantity of members of each basic class dividing a given ultrafilter. In this paper we define a topology on the set of basic classes which will allow us to calculate the pattern of the limit of a $\widetilde{\mid}$-increasing chain of ultrafilters. Using this we characterize which patterns can actually appear as patterns of an ultrafilter. Defining the $=_\sim$-divisibility classes by identifying mutually divisible ultrafilters, in the respective quotient order $(\beta\mathbb{N}/=_\sim,\widetilde{\mid})$ we identify singleton classes and consider their patterns. Finally, we give a sufficient condition for a $=_\sim$-divisibility class to have an immediate predecessor.