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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Semiprincipal closed ideals of $ \beta S$


Authors: Wilson Toko and Yuliya Zelenyuk
Journal: Proc. Amer. Math. Soc. 138 (2010), 2217-2220
MSC (2000): Primary 22A30, 05E99; Secondary 22A15, 20M12
Published electronically: January 20, 2010
MathSciNet review: 2596062
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Abstract: Let $ S$ be an infinite discrete semigroup and let $ \beta S$ be the Stone-Čech compactification of $ S$. For every $ p\in\beta S$, $ \operatorname{cl}((\beta S)p(\beta S))$ is a closed two-sided ideal of $ \beta S$ called the semiprincipal closed ideal generated by $ p$. We show that if $ S$ can be embedded into a group, then $ \beta S$ contains $ 2^{2^{\vert S\vert}}$ pairwise incomparable semiprincipal closed ideals.


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Additional Information

Wilson Toko
Affiliation: School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa
Email: wilson.toko@students.wits.ac.za

Yuliya Zelenyuk
Affiliation: School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa
Email: yuliya.zelenyuk@wits.ac.za

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10209-3
PII: S 0002-9939(10)10209-3
Keywords: Stone-\v {C}ech compactification, ultrafilter, semigroup, semiprincipal closed ideal
Received by editor(s): July 5, 2009
Received by editor(s) in revised form: August 28, 2009
Published electronically: January 20, 2010
Additional Notes: The second author was supported by NRF grant IFR2008041600015 and the John Knopfmacher Centre for Applicable Analysis and Number Theory.
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2010 American Mathematical Society