Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-28T07:12:53.391Z Has data issue: false hasContentIssue false
  • English
  • Français

Compact subsemigroups of (βℕ, +) containing the idempotents

Published online by Cambridge University Press:  20 January 2009

Neil Hindman  and
Dona Strauss
Neil Hindman
Affiliation:
Department of Mathematics Howard University Washington D.C. 20059, U.S.A.
Dona Strauss
Affiliation:
Department of Pure Mathematics University of Hull Cottingham Road Hull HU6 7RX
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The space βℕ is the Stone-Čech compactification of the discrete space of positive integers. The set of elements of βℕ which are in the kernel of every continuous homomorpnism from βℕ to a topological group is a compact semigroup containing the idempotents. At first glance it would seem a good candidate for the smallest such semigroup. We produce an infinite nested sequence of smaller such semigroups all defined naturally in terms of addition on ℕ.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 39 , Issue 2 , June 1996 , pp. 291 - 307
Copyright
Copyright © Edinburgh Mathematical Society 1996

References

REFERENCES

1. Baker, J., Hindman, N. and Pym, J., Elements of finite order in Stone-Čech compactifications, Proc. Edinburgh Math. Soc. 36 (1992), 498–54.Google Scholar
2. Berg, K., Gove, D. and Haddad, K., On the connection between beta compactifications and enveloping semigroups of subshifts, Abstracts Amer. Math. Soc. 15 (1994), 320.Google Scholar
3. Bergelson, V. and Hindman, N., A combinatorially large cell of a partition of ℕ, J. Combin. Theory Ser. A 48 (1988), 39–52.CrossRefGoogle Scholar
4. Berglund, J., Junghenn, H. and Milnes, P., Analysis on Semigroups (Wiley, New York, 1989).Google Scholar
5. Comfort, W., Ultrafilters; some old and some new results, Bull. Amer. Math. Soc. 83 (1977), 417–455.CrossRefGoogle Scholar
6. Ellis, R., Lectures on topological dynamics (Benjamin, New York, 1969).Google Scholar
7. Frankl, P., Graham, R. and Rödl, V., Induced restricted Ramsey theorems for spaces, J. Combin. Theory Ser. A 44 (1987), 120–128.CrossRefGoogle Scholar
8. Graham, R., Rothschild, B. and Spencer, J., Ramsey Theory (Wiley, New York, 1990).Google Scholar
9. Hindman, N., Partitions and sums and products of integers, Trans. Amer. Math. Soc. 247 (1979), 227–245.CrossRefGoogle Scholar
10. Hindman, N., Summable ultrafilters and finite sums, in Logic and Combinatorics, (S. Simpson ed.), Contemp. Math. 65 (1987), 263–274.CrossRefGoogle Scholar
11. Hindman, N., The semigroup βℕ and its applications to number theory, in The Analytical and Topological Theory of Semigroups—Trends and Developments (Hoffmann, K., Lawson, J. and Pym, J. eds., de Gruyter Expositions in Math. 1, 1990), 347–360.CrossRefGoogle Scholar
12. Hindman, N., Ultrafilters and Ramsey Theory—an update, in Set Theory and its Applications (Stephrāns, J. and Watson, S. eds., Lecture Notes in Math. 1401, 1989), 97–118.Google Scholar
13. Hindman, N. and Pym, J., Free groups and semigroups in βℕ, Semigroup Forum 30 (1984), 177–193.Google Scholar
14. Hindman, N. and Strauss, D., Cancellation in the Stone-Čech compactification of a discrete semigroup, Proc. Edinburgh Math. Soc. 37 (1994), 379–397.CrossRefGoogle Scholar
15. Hindman, N. and Woan, W., Central sets in semigroups and partition regularity of systems of linear equations, Mathematika 40 (1993), 169–186.CrossRefGoogle Scholar
16. El-Mabhouh, A., Pym, J. and Strauss, D., Subsemigroups of βℕ, Topology Appl., to appear.Google Scholar
17. Milnes, P., Compactifications of topological semigroups, J. Australian Math. Soc. 15 (1973), 488–503.CrossRefGoogle Scholar
18. Neřil, J. and Rödl, V., Finite union theorem with restrictions, Graphs Combin. 2 (1986), 357–361.Google Scholar