# $\widetilde{\mid}\hspace{1mm}$-divisibility of ultrafilters

## TL;DR

This paper explores a divisibility relation among ultrafilters on natural numbers, identifying prime ultrafilters, establishing a hierarchy, and analyzing the structure and positions of ultrafilters within this framework.

## Contribution

It introduces a novel divisibility hierarchy for ultrafilters, characterizes prime ultrafilters, and analyzes their structural relationships and product positions.

## Key findings

- Prime ultrafilters are characterized within the hierarchy.
- A hierarchy of ultrafilters based on divisibility is established.
- Ultrafilters with many immediate successors are constructed.

## Abstract

We further investigate a divisibility relation on the set $\beta N$ of ultrafilters on the set of natural numbers. We single out prime ultrafilters (divisible only by 1 and themselves) and establish a hierarchy in which a position of every ultrafilter depends on the set of prime ultrafilters it is divisible by. We also construct ultrafilters with many immediate successors in this hierarchy and find positions of products of ultrafilters.