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Divisibility in β N and *N

Šobot Boris

Reports on Mathematical Logic

2019

Vol. 54

65--82

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-Čech 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.

Divisibility in the Stone-Čech compactification

Šobot B.

Reports on Mathematical Logic

2015

Vol. 50

53--66

After defining continuous extensions of binary relations on the set N of natural numbers to its Stone-Čech 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, ·). The motivation behind all this is to try to translate problems in elementary number theory into βN.