Divisibility in *N and $\beta N$

TL;DR

This paper explores divisibility properties in nonstandard integers and the Stone-ch compactification, linking these concepts and providing new insights into ultrafilters and their divisibility hierarchy.

Contribution

It introduces a novel connection between nonstandard divisibility and ultrafilter divisibility in ch compactification, offering new results and clarifying earlier findings.

Findings

Established an equivalent condition for ultrafilter divisibility

Connected nonstandard and ultrafilter divisibility properties

Derived new results on ultrafilters at higher levels of the hierarchy

Abstract

The paper first covers several properties of the extension of the divisibility relation to a set ${}^*\hspace{-0.5mm}N$ of nonstandard integers. After that, a connection is established with the divisibility in the Stone-\v{C}ech compactification $\beta N$, obtaining an equivalent condition for divisibility of ultrafilters introduced by the author. Some earlier results are illuminated by nonstandard methods, and new results on ultrafilters on the higher levels of the divisibility hierarchy are obtained by means of limits.