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

Published online by Cambridge University Press:  20 November 2018

David J. Fernández Bretón*
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, ON e-mail: difernan@umich.edu
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Abstract

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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., $\operatorname{cov}\left( \text{M} \right)=\mathfrak{c})$, a strongly summable ultrafilter on the Boolean group that is not additively isomorphic to any union ultrafilter.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2016

References

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