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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Regular idempotents in $ \beta S$


Author: Yevhen Zelenyuk
Journal: Trans. Amer. Math. Soc. 362 (2010), 3183-3201
MSC (2000): Primary 22A05, 54G05; Secondary 22A30, 54H11
Published electronically: January 20, 2010
MathSciNet review: 2592952
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Abstract: Let $ S$ be a discrete semigroup and let $ \beta S$ be the Stone-Čech compactification of $ S$. We take the points of $ \beta S$ to be the ultrafilters on $ S$. Being a compact Hausdorff right topological semigroup, $ \beta S$ has idempotents. Every idempotent $ p\in\beta S$ determines a left invariant topology $ \mathcal{T}_p$ on $ S$ with a neighborhood base at $ a\in S$ consisting of subsets $ aB\cup\{a\}$, where $ B\in p$. If $ S$ is a group and $ p$ is an idempotent in $ S^*=\beta S\setminus S$, $ (S,\mathcal{T}_p)$ is a homogeneous Hausdorff maximal space. An idempotent $ p\in\beta S$ is regular if $ p$ is uniform and the topology $ \mathcal{T}_p$ is regular. We show that for every infinite cancellative semigroup $ S$, there exists a regular idempotent in $ \beta S$. As a consequence, we obtain that for every infinite cardinal $ \kappa$, there exists a homogeneous regular maximal space of dispersion character $ \kappa$. Another consequence says that there exists a translation invariant regular maximal topology on the real line of dispersion character $ \mathfrak{c}$ stronger than the natural topology.


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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9947-10-04926-3
PII: S 0002-9947(10)04926-3
Keywords: Stone-\v Cech compactification, ultrafilter, idempotent, left invariant topology, homogeneous regular maximal space.
Received by editor(s): June 23, 2008
Published electronically: January 20, 2010
Additional Notes: This work was supported by NRF grant FA2007041200005 and The John Knopfmacher Centre for Applicable Analysis and Number Theory.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.