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., cov(M) = c), a strongly summable ultrafilter on the Boolean group that is not additively isomorphic to any union ultrafilter.
Original language | English |
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Pages (from-to) | 44-66 |
Number of pages | 23 |
Journal | Canadian Journal of Mathematics |
Volume | 68 |
Issue number | 1 |
DOIs | |
State | Published - Feb 2016 |
Externally published | Yes |
Keywords
- Abelian group
- Additive isomorphism
- Boolean group
- Finite sum
- Sparse ultrafilter
- Stone-Čech compactification
- Strongly summable ultrafilter
- Trivial sums property
- Ultrafilter
- Union ultrafilter