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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
Published electronically: March 12, 2003
MathSciNet review: 1975400
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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 $\beta\mathbb{N}$.


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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