
Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property
We answer two questions of Hindman, SteprÄns 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, StoneCech compactification, sparse ultrafilter, strongly summable ultrafilter, union ultrafilter, finite sum, additive isomorphism, trivial sums property, Boolean group, abelian group Categories:03E75, 54D35, 54D80, 05D10, 05A18, 20K99 