On Strongly Summable Ultrafilters and Union Ultrafilters

Abstract

Abstract.We prove that union ultrafilters are essentially the same as strongly summable ultrafilters but ordered-union ultrafilters are not. We also prove that the existence of ultrafilters of these sorts implies the existence of P-points and therefore cannot be established in ZFC.I. Introduction. The purpose of this paper is to clarify the connections between the strongly summable ultrafilters introduced in [9] and the union ultrafilters and ordered-union ultrafilters introduced in [3]. We show that strongly summable ultrafilters and union ultrafilters are essentially the same in that every ultrafilter of either sort is isomorphic to one of the other sort via an isomorphism that (almost everywhere) respects the operations of summation and union. We also show that the existence of such ultrafilters, deduced in [9, 3] from the continuum hypothesis (CH) or Martin's axiom (MA), cannot be deduced in ZFC alone, for it implies the existence of P-point ultrafilters. Finally, we show that union ultrafilters and ordered-union ultrafilters are not essentially equivalent; we construct, using CH or MA, a union ultrafilter that is not isomorphic to any ordered-union ultrafilter via an isomorphism that respects unions.For any set A of natural numbers, let FS(^4) be the set of all sums of nonempty finite subsets of A. An ultrafilter on the set w of natural numbers is strongly summable if it has a base consisting of sets of the form FS(^), with A infinite.We write F for the set of all finite nonempty subsets of to. For A QF, let FU(^) be the set of all unions of nonempty finite subsets of A. An ultrafilter on F is a

Elements of finite order in Stone-Čech compactifications

“…It has also been shown [2,8] that (3) is independent of ZFC. We might expect there to be analogues of these results for elements of order k. Write FS, (mod(k) <x n > for the subset of FS<x n > consisting of sums of the form x ni + •• • + *",.…”

On Union Ultrafilters

“…The one interesting non-implication missing is that a fortiori there consistently exist union ultrafilters that are not ordered union. However, the construction in [BH87] does not give any direct information on what it means to be an unordered union ultrafilter. In a manner of speaking it is a sledge hammer smashing orderedness so badly that is difficult to identify…”

“…[Hin72] and [HS98, notes to Chapter 5]; they were the first strongly right maximal idempotents known and, even stronger, they are the only known idempotent with a maximal group isomorphic to Z; their existence is independent of ZFC, since it implies the existence of (rapid) P-points, cf. [BH87]; since a strongly summable is strongly right maximal, its orbit is a van Douwen space, cf. [HS02].…”

The topological center of the semigroup of free ultrafilters

“…Let B be a countable group of exponent two, and let p be a strongly surnmable ultrafilter in B*. By Definition [4], for any subset P E p, there is a subgroup H such that H \ {e} C P and H E p, where e is the identity element of the group B. It is obvious that the mapping pp is continuous at the point p. Since p is an idempotent of the semigroup B*, it follows that p is not a P-point of B*.…”

“…In [4] it was proved that the existence of a strongly summable ultrafilter on the group B implies the existence of a P-point in the space N*, where N is the discrete space of positive integers. Problem 3.…”