Levels of Ultrafilters with Extension Divisibilities

TL;DR

This paper investigates the structure of ultrafilters in the Stone Cech compactification of natural numbers under multiplication, classifying them into finite level ultrafilters and others, and analyzing their divisibility properties.

Contribution

It introduces a classification of ultrafilters into finite levels and non-finite levels, providing new insights into their divisibility and irreducibility properties.

Findings

Ultrafilters on finite levels are either irreducible or products of irreducibles.

Higher level ultrafilters are extension divided by lower level ultrafilters.

Characterization of ultrafilters not on finite levels.

Abstract

To work more accurately with elements of the semigroup of the Stone Cech compactification of the discrete semigroup of natural numbers N under multiplication. We divided these elements into ultrafilters which are on finite levels and ultrafilters which are not on finite levels. For the ultrafilters that are on finite levels we prove that any element is irreducible or product of irreducible elements and all elements on higher levels are extension divided by some elements on lower levels. We characterize ultrafilters that are not on finite levels and the effect of extension divisibility on the ultrafilters which are not on finite levels.