Divisibility in beta N and *N

Abstract

The paper first covers several properties of the extension of the divisibility relation to a set *N of nonstandard integers, including an analogue of the basic theorem of arithmetic. After that, a connection is established with the divisibility in the Stone–Cech compactification βN, proving that the divisibility of ultrafilters introduced by the author is equivalent to divisibility of some elements belonging to their respective monads in an enlargement. Some earlier results on ultrafilters on lower levels on the divisibility hierarchy are illuminated by nonstandard methods. Using limits by ultrafilters we obtain results on ultrafilters above these finite levels, showing that for them a distribution by levels is not possible. Received 16 July 2018 AMS subject classification: Primary 54D80; Secondary 11U10, 03H15, 54D35 Keywords: divisibility, nonstandard integer, Stone-Cech compactification, ultrafilter