Strongly summable ultrafilters on 
abelian groups

N. Hindman; I. Protasov; D. Strauss

Strongly summable ultrafilters on a commutative semigroup are those that are generated by sets of finite sums. We establish several facts about strongly summable ultrafilters on a countable abelian group $G$ that were previously known to hold only for the group $(\Bbb Z,+)$ and for Boolean groups. It is shown that Martin's Axiom implies the existence of nonprincipal strongly summable ultrafilters, that their existence cannot be established in ZFC, and that, if $G$ is embeddable in the circle group, they satisfy strong algebraic properties regarding uniqueness of solutions to certain equations.

1. A. Blass and N. Hindman On strongly summable and union ultrafilters// Trans. Amer. Math. Soc.V.304--1987 p.83--99

2. N. Hindman Summable ultrafilters and finite sums Logic and CombinatoricsS. Simpson, Contemporary Math.V.65–1987p. 263–274

3. N. Hindman Strongly summable ultrafilters on $\Bbb N$ and small maximal subgroups of $\beta\Bbb N$// Semigroup ForumV. 42--1991p.63--75

4. N. Hindman and D. Strauss Nearly prime subsemigroups of $\beta \Bbb N$// Semigroup ForumV. 51--1995 p.379--397

5. N. Hindman and D. Strauss Algebra in the Stone-Čech compactification – theory and applications , de Gruyter,addr Berlin– 1998

6. V. Malykhin Extremally disconnected and similar groups// Soviet Math. Dokl.V. 16–1975p. 21–25

7. P. Matet Some filters of partitions// J. Symbolic LogicV. 53–1988p.540–553

8. I. Protasov Ultrafilters on abelian groups close to being Ramsey ultrafilters// Matem. StudiiV. 7–1997p.133–138

9. I. Protasov Finite groups in $\beta G$ // Algebra and Discrete Math\toappear

10. S. Shelah Proper forcing, Springer-Verlag ,addr Berlin– 1982

	Strongly summable ultrafilters on abelian groups
Author	N. Hindman, I. Protasov, D. Strauss Neil Hindman, Department of Mathematics, Howard University,Washington, DC 20059, USAI. Protasov, Department of Mathematics, Kyiv State University,Kyiv, UkraineDona Strauss, Department of Pure Mathematics,University of Hull,Hull HU6 7RX, UK
Abstract	Strongly summable ultrafilters on a commutative semigroup are those that are generated by sets of finite sums. We establish several facts about strongly summable ultrafilters on a countable abelian group $G$ that were previously known to hold only for the group $(\Bbb Z,+)$ and for Boolean groups. It is shown that Martin's Axiom implies the existence of nonprincipal strongly summable ultrafilters, that their existence cannot be established in ZFC, and that, if $G$ is embeddable in the circle group, they satisfy strong algebraic properties regarding uniqueness of solutions to certain equations.
Keywords	strongly summable ultrafilters, commutative semigroup, finite sums, countable abelian group, Martin's Axiom, nonprincipal ultrafilters, algebraic properties, uniqueness of solutions
DOI	doi:10.30970/ms.10.2.121-132
Reference	1. A. Blass and N. Hindman On strongly summable and union ultrafilters// Trans. Amer. Math. Soc.V.304--1987 p.83--99 2. N. Hindman Summable ultrafilters and finite sums Logic and CombinatoricsS. Simpson, Contemporary Math.V.65–1987p. 263–274 3. N. Hindman Strongly summable ultrafilters on $\Bbb N$ and small maximal subgroups of $\beta\Bbb N$// Semigroup ForumV. 42--1991p.63--75 4. N. Hindman and D. Strauss Nearly prime subsemigroups of $\beta \Bbb N$// Semigroup ForumV. 51--1995 p.379--397 5. N. Hindman and D. Strauss Algebra in the Stone-Čech compactification – theory and applications , de Gruyter,addr Berlin– 1998 6. V. Malykhin Extremally disconnected and similar groups// Soviet Math. Dokl.V. 16–1975p. 21–25 7. P. Matet Some filters of partitions// J. Symbolic LogicV. 53–1988p.540–553 8. I. Protasov Ultrafilters on abelian groups close to being Ramsey ultrafilters// Matem. StudiiV. 7–1997p.133–138 9. I. Protasov Finite groups in $\beta G$ // Algebra and Discrete Math\toappear 10. S. Shelah Proper forcing, Springer-Verlag ,addr Berlin– 1982
Pages	121-132
Volume	10
Issue	2
Year	1998
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Strongly summable ultrafilters on abelian groups