Divisibility in the Stone-Čech compactification

Abstract

After defining continuous extensions of binary relations on the set N of natural numbers to its Stone-Cech compactification βN, we establish some results about one of such extensions. This provides us with one possible divisibility relation on βN 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 (βN, \cdot). The motivation behind all this is to try to translate problems in number theory into βN.