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Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property

  Published:2015-10-29
 Printed: Feb 2016
  • David J. Fernández Bretón,
    Department of Mathematics and Statistics, York University , Toronto, Ontario, Canada
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Abstract

We answer two questions of Hindman, Steprāns and Strauss, namely we prove that every strongly summable ultrafilter on an abelian group is sparse and has the trivial sums property. Moreover we show that in most cases the sparseness of the given ultrafilter is a consequence of its being isomorphic to a union ultrafilter. However, this does not happen in all cases: we also construct (assuming Martin's Axiom for countable partial orders, i.e. $\operatorname{cov}(\mathcal{M})=\mathfrak c$), on the Boolean group, a strongly summable ultrafilter that is not additively isomorphic to any union ultrafilter.
Keywords: ultrafilter, Stone-Cech compactification, sparse ultrafilter, strongly summable ultrafilter, union ultrafilter, finite sum, additive isomorphism, trivial sums property, Boolean group, abelian group ultrafilter, Stone-Cech compactification, sparse ultrafilter, strongly summable ultrafilter, union ultrafilter, finite sum, additive isomorphism, trivial sums property, Boolean group, abelian group
MSC Classifications: 03E75, 54D35, 54D80, 05D10, 05A18, 20K99 show english descriptions Applications of set theory
Extensions of spaces (compactifications, supercompactifications, completions, etc.)
Special constructions of spaces (spaces of ultrafilters, etc.)
Ramsey theory [See also 05C55]
Partitions of sets
None of the above, but in this section
03E75 - Applications of set theory
54D35 - Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54D80 - Special constructions of spaces (spaces of ultrafilters, etc.)
05D10 - Ramsey theory [See also 05C55]
05A18 - Partitions of sets
20K99 - None of the above, but in this section
 

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