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

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

arXiv commentAccepted for publication in Canadian Journal of Mathematics. 23 pages, contains the answer to two questions from: Hindman, N., Steprāns, J. and Strauss, D., "Semigroups in which all strongly summable ultrafilters are sparse", New York J. Math. 18 (2012), 835-848

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• Finiteness classes arising from Ramsey-theoretic statements in set theory without choice
• Strongly productive ultrafilters on semigroups