Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Large rectangular semigroups in Stone-Cech compactifications


Authors: Neil Hindman, Dona Strauss and Yevhen Zelenyuk
Journal: Trans. Amer. Math. Soc. 355 (2003), 2795-2812
MSC (2000): Primary 20M10; Secondary 22A15, 54H13
DOI: https://doi.org/10.1090/S0002-9947-03-03276-8
Published electronically: March 12, 2003
MathSciNet review: 1975400
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that large rectangular semigroups can be found in certain Stone-Cech compactifications. In particular, there are copies of the $2^{\mathfrak{c}}\times 2^{\mathfrak{c}}$rectangular semigroup in the smallest ideal of $(\beta\mathbb{N},+)$, and so, a semigroup consisting of idempotents can be embedded in the smallest ideal of $(\beta\mathbb{N},+)$ if and only if it is a subsemigroup of the $2^{\mathfrak{c}}\times 2^{\mathfrak{c}}$ rectangular semigroup. In fact, we show that for any ordinal $\lambda$ with cardinality at most $\mathfrak{c}$, $\beta{\mathbb{N}}$ contains a semigroup of idempotents whose rectangular components are all copies of the $2^{\mathfrak{c}}\times 2^{\mathfrak{c}}$ rectangular semigroup and form a decreasing chain indexed by $\lambda+1$, with the minimum component contained in the smallest ideal of $\beta\mathbb{N}$.

As a fortuitous corollary we obtain the fact that there are $\leq_{L}$-chains of idempotents of length $\mathfrak{c}$ in $\beta \mathbb{N}$. We show also that there are copies of the direct product of the $2^{\mathfrak{c}}\times 2^{\mathfrak{c}}$ rectangular semigroup with the free group on $2^{\mathfrak{c}}$ generators contained in the smallest ideal of $\beta\mathbb{N}$.


References [Enhancements On Off] (What's this?)

  • 1. J. Berglund, H. Junghenn, and P. Milnes, Analysis on semigroups, Wiley, N.Y., 1989. MR 91b:43001
  • 2. W. Comfort and S. Negrepontis, The theory of ultrafilters, Springer-Verlag, Berlin, 1974. MR 53:135
  • 3. L. Gillman and M. Jerison, Rings of continuous functions, van Nostrand, Princeton, 1960. MR 22:6994
  • 4. N. Hindman and J. Pym, Free groups and semigroups in $\beta{\mathbb N}$, Semigroup Forum 30 (1984), 177-193. MR 86c:22002
  • 5. N. Hindman and D. Strauss, Chains of idempotents in $\beta{\mathbb N}$, Proc. Amer. Math. Soc. 123 (1995), 3881-3888. MR 96b:54037
  • 6. N. Hindman and D. Strauss, Algebra in the Stone-Cech compactification: Theory and applications, de Gruyter, Berlin, 1998. MR 99j:54001
  • 7. D. McLean, Idempotent semigroups, Amer. Math. Monthly 61 (1954), 110-113. MR 15:681a
  • 8. J. Pym, Semigroup structure in Stone-Cech compactifications, J. London Math. Soc. 36 (1987), 421-428. MR 89b:54043
  • 9. W. Ruppert, Rechstopologische Halbgruppen, J. Reine Angew. Math. 261 (1973), 123-133. MR 47:6933
  • 10. Y. Zelenyuk, On subsemigroups of $\beta{\mathbb N}$ and absolute coretracts, Semigroup Forum 63 (2001), 457-465. MR 2002f:22005

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20M10, 22A15, 54H13

Retrieve articles in all journals with MSC (2000): 20M10, 22A15, 54H13


Additional Information

Neil Hindman
Affiliation: Department of Mathematics, Howard University, Washington, DC 20059
Email: nhindman@aol.com

Dona Strauss
Affiliation: Department of Pure Mathematics, University of Hull, Hull HU6 7RX, United Kingdom
Email: d.strauss@maths.hull.ac.uk

Yevhen Zelenyuk
Affiliation: Faculty of Cybernetics, Kyiv Taras Shevchenko University, Volodymyrska Street 64, 01033 Kyiv, Ukraine
Email: grishko@i.com.ua

DOI: https://doi.org/10.1090/S0002-9947-03-03276-8
Received by editor(s): April 12, 2002
Received by editor(s) in revised form: November 14, 2002
Published electronically: March 12, 2003
Additional Notes: The first author acknowledges support received from the National Science Foundation (USA) via grant DMS-0070593
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society