New York Journal of Mathematics
Volume 18 (2012) 835-848


Neil Hindman, Juris Steprans, and Dona Strauss

Semigroups in which all strongly summable ultrafilters are sparse

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Published: October 16, 2012
Keywords: ultrafilters, sparse, strongly summable, Stone-Čech compactification, uniqueness of finite sums
Subject: 54D35, 54D80, 03E75, 05D10

We show that if (S,+) is a commutative semigroup which can be embedded in the circle group T, in particular if S=(N,+), then all nonprincipal, strongly summable ultrafilters on S are sparse and can be written as sums in βS only trivially. We develop a simple condition on a strongly summable ultrafilter which guarantees that it is sparse and show that this holds for many ultrafilters on semigroups which are embeddable in the direct sum of countably many copies of T.


The first author acknowledges support received from the National Science Foundation via Grants DMS-0852512 and DMS-1160566.
The second author was partially supported by an NSERC grant.

Author information

Neil Hindman:
Department of Mathematics, Howard University, Washington, DC 20059, USA

Juris Steprans:
Department of Mathematics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3

Dona Strauss:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9J2, UK