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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-10-04926-3
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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  • 1. A. Clifford and G. Preston, The algebraic theory of semigroups, Vol. 1, Amer. Math. Soc., Providence, RI, 1961. MR 0132791 (24:A2627)
  • 2. W. Comfort and S. Negrepontis, The theory of ultrafilters, Springer-Verlag, Berlin, 1974. MR 0396267 (53:135)
  • 3. E. van Douwen, Simultaneous extensions of continuous functions, Dissertation (1975), Vrije Universiteit.
  • 4. E. van Douwen, Applications of maximal topologies, Topology Appl. 51 (1993), 125-139. MR 1229708 (94h:54012)
  • 5. A. El'kin, Regular maximal spaces, Mat. Zametki 27 (1980), 301-305, 320 (Russian). MR 568408 (82a:54052)
  • 6. R. Ellis, Lectures on topological dynamics, Benjamin, New York, 1969. MR 0267561 (42:2463)
  • 7. S. Ferri, N. Hindman, and D. Strauss, Digital representation of semigroups and groups, Semigroup Forum 77 (2008), 36-63. MR 2413260
  • 8. E. Hewitt, A problem of set-theoretic topology, Duke Math. J. 10 (1943), 309-333. MR 0008692 (5:46e)
  • 9. N. Hindman and D. Strauss, Algebra in the Stone-Čech compactification, De Gruyter, Berlin, 1998. MR 1642231 (99j:54001)
  • 10. N. Hindman, D. Strauss, and I. Protasov, Topologies on $ S$ determined by idempotents in $ \beta S$, Topology Proc. 23 (1998), 155-190. MR 1803247 (2001j:54048)
  • 11. M. Katětov, On nearly discrete spaces, Časopis Pěst. Mat. Fys. 75 (1950), 69-78. MR 0036984 (12:195b)
  • 12. V. Malykhin, Extremally disconnected and nearby groups, Dokl. Akad. Nauk SSSR 220 (1975), 27-30 (Russian). MR 0382536 (52:3419)
  • 13. V. Malykhin, On extremally disconnected topological groups, Uspekhi Mat. Nauk 34 (1979), 59-66 (Russian). MR 562819 (81c:54007)
  • 14. T. Papazyan, Extremal topologies on a semigroup, Topology Appl. 39 (1991), 229-243. MR 1110567 (92f:22003)
  • 15. I. Protasov, Filters and topologies on semigroups, Matematychni Studii 3 (1994), 15-28 (Russian). MR 1692845 (2000k:22005)
  • 16. I. Protasov, Maximal topologies on groups, Sib. Math. J. 39 (1998), 1184-1194. MR 1672661 (99m:22001)
  • 17. W. Ruppert, Compact semitopological semigroups: An intrinsic theory, Lecture Notes in Math. 1079, Springer-Verlag, Berlin, 1984. MR 762985 (86e:22001)
  • 18. Y. Zelenyuk, Finite groups in $ \beta\mathbb{N}$ are trivial, Semigroup Forum 55 (1997), 131-132. MR 1446665 (99b:22004)
  • 19. Y. Zelenyuk, On the ultrafilter semigroup of a topological group, Semigroup Forum 73 (2006), 301-307. MR 2280826 (2007i:22004)
  • 20. Y. Zelenyuk, Almost maximal spaces, Topology Appl. 154 (2007), 339-357. MR 2278682 (2007i:54072)
  • 21. Y. Zelenyuk, Finite groups in Stone-Čech compactifications, Bull. London Math. Soc. 40 (2008), 337-346. MR 2414792
  • 22. Y. Zelenyuk, Topologies on groups determined by discrete subsets, Topology Appl. 155 (2008), 1332-1339. MR 2423971

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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: https://doi.org/10.1090/S0002-9947-10-04926-3
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.

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