Divisibility in the Stone-\v{C}ech compactification

TL;DR

This paper explores how divisibility relations can be extended to the Stone-Cech compactification of natural numbers, aiming to translate number theory problems into this topological setting.

Contribution

It introduces new divisibility relations on  N, establishes conditions for divisibility, and discusses prime and irreducible elements in this context.

Findings

Defined continuous extensions of binary relations on N to  N

Established equivalent conditions for some divisibility relations

Discussed properties of prime and irreducible elements in ( N,  

Abstract

After defining continuous extensions of binary relations on the set N of natural numbers to its Stone-Cech compactification \beta N, we establish some results about one of such extensions. This provides us with one possible divisibility relation on \beta N and we introduce a few more, defined in a natural way. For some of them we find equivalent conditions for divisibility. Finally, we mention a few facts about prime and irreducible elements of (\beta N, \cdot). The motivation behind all this is to try to translate problems in number theory into \beta N.