More number theory in $\beta N$

TL;DR

This paper extends the divisibility relation to the Stone-cech compactification  N, proving properties of prime ultrafilters, exploring divisibility conditions, and constructing large well-ordered chains within this framework.

Contribution

It introduces new algebraic properties of prime ultrafilters, establishes equivalences for divisibility in  N, and constructs maximal chains, advancing the understanding of divisibility in this extended setting.

Findings

Prime ultrafilters are algebraically prime.

Multiple equivalent conditions for divisibility in  N are identified.

A well-ordered chain of maximal cardinality is constructed.

Abstract

We continue the research of an extension $\widetilde{\mid}$ of the divisibility relation to the Stone-\v Cech compactification $\beta N$. First we prove that ultrafilters we call prime actually possess the algebraic property of primality. Several questions concerning the connection between divisibilities in $\beta N$ and nonstandard extensions of $N$ are answered, providing a few more equivalent conditions for divisibility in $\beta N$. Results on uncountable chains in $(\beta N,\widetilde{\mid})$ are proved and used in a construction of a well-ordered chain of maximal cardinality. Finally, we consider ultrafilters without divisors in $N$ and among them find the maximal class.