More about divisibility in βN

Abstract

We continue the research of an extension ∣∼ of the divisibility relation to the Stone‐Čech compactification βN. First we prove that ultrafilters we call prime actually possess the algebraic property of primality. Several questions concerning the connection between divisibilities in βN and nonstandard extensions of double-struckN are answered, providing a few more equivalent conditions for divisibility in βN. Results on uncountable chains in (βN,true∣∼) are proved and used in a construction of a well‐ordered chain of maximal cardinality. Probably the most interesting result is the existence of a chain of type (R,<) in (βN,true∣∼). Finally, we consider ultrafilters without divisors in double-struckN and among them find the greatest class.

Multiplicative finite embeddability vs divisibility of ultrafilters

“…Theorem 2. 13 In βN/ = f e there is a strictly ≤ f e -increasing chain without a smallest upper bound.…”

“…, i k )) and F is selective, there is B (i1,i2,...,i k ) ∈ F such that |B (i1,i2,...,i k ) ∩ A Recall that U N is the family of all N-free sets from U, and V N = {A c : A ∈ U N }. In [13] it was shown that U N ∪ {(nN) c : n ∈ N \ {1}} has the finite intersection property, so there is a | -greatest N-free class that we denote N M AX. A ⊆ N is a strong antichain if every two elements m, n ∈ A are mutually prime.…”

Congruence of ultrafilters

“…In the previous two papers, [15] and [16], we employed nonstandard methods (more precisely, the superstructure approach) to get more information on the relation | . We will continue that practice here.…”

“…Let N be the set of natural numbers. The relation | , an extension of the divisibility relation | on N to the set βN of ultrafilters on N, was introduced in [12] and further investigated in [13][14][15][16]. The main idea was to understand the impact of various properties of | to | and possibly, learning about the | -hierarchy, to acquire better understanding of |.…”

“…It contains the maximal = ∼ -class, M AX, consisting of ultrafilters divisible by all n ∈ N, and consequently by all F ∈ βN ( [15], Lemma 4.6). Another interesting class is N M AX, maximal among N-free ultrafiters (those that are not divisible by any n ∈ N), see [16], Theorem 5.4. A set belonging to an N-free ultrafilter is called an N-free set.…”