Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-17T13:57:00.916Z Has data issue: false hasContentIssue false
  • English
  • Français

LEFT MAXIMAL AND STRONGLY RIGHT MAXIMAL IDEMPOTENTS IN G*

Published online by Cambridge University Press:  21 March 2017

YEVHEN ZELENYUK
YEVHEN ZELENYUK*
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF THE WITWATERSRAND PRIVATE BAG 3, WITS 2050 SOUTH AFRICAE-mail: yevhen.zelenyuk@wits.ac.za
Get access Rights & Permissions [Opens in a new window]

Abstract

Let G be a countably infinite discrete group, let βG be the Stone–Čech compactification of G, and let ${G^{\rm{*}}} = \beta G \setminus G$. An idempotent $p \in {G^{\rm{*}}}$ is left (right) maximal if for every idempotent $q \in {G^{\rm{*}}}$, pq = p (qp = P) implies qp = q (qp = q). An idempotent $p \in {G^{\rm{*}}}$ is strongly right maximal if the equation xp = p has the unique solution x = p in G*. We show that there is an idempotent $p \in {G^{\rm{*}}}$ which is both left maximal and strongly right maximal.

Keywords

Primary 22A1554H20Secondary 54D8054H11Stone–Čechcompactificationultrafilterprincipal left idealorbit closureleft maximal idempotentstrongly right maximal idempotent
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 1 , March 2017 , pp. 26 - 34
Copyright
Copyright © The Association for Symbolic Logic 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Ellis, R., Lectures on Topological Dynamics, Benjamin, New York, 1969.Google Scholar
Hindman, N. and Strauss, D., Nearly prime subsemigroups of βℕ. Semigroup Forum, vol. 51 (1995), pp. 299–318.Google Scholar
Hindman, N. and Strauss, D., Algebra in the Stone-Čech Compactification, De Gruyter, Berlin, 1998.Google Scholar
Protasov, I., Maximal topologies on groups . Siberian Mathematical Journal, vol. 39 (1998), pp. 1184–1194.CrossRefGoogle Scholar
Ruppert, W., Compact Semitopological Semigroups: An Intrinsic Theory, Lecture Notes in Mathematics 1079, Springer-Verlag, Berlin, 1984.Google Scholar
Zelenyuk, Y., Ultrafilters and Topologies on Groups, De Gruyter, Berlin, 2011.CrossRefGoogle Scholar
Zelenyuk, Y., Principal left ideals of βG may be both minimal and maximal . Bulletin of the London Mathematical Society, vol. 45 (2013), pp. 613–617.CrossRefGoogle Scholar
Zelenyuk, Y., Left maximal idempotents in G * . Advances in Mathematics, vol. 262 (2014), pp. 593–603.Google Scholar