Skip to main content Accessibility help
×
Home

Semigroups of continuous selfmaps for which Green's and ℐ relations coincide

  • K. D. Magill (a1)

Extract

For algebraic terms which are not defined, one may consult [2]. The symbol S(X) denotes the semigroup, under composition, of all continuous selfmaps of the topological space X. When X is discrete, S(X) is simply the full transformation semigroup on the set X. It has long been known that Green's relations and ℐ coincide for [2, p. 52] and F. A. Cezus has shown in his doctoral dissertation [1, p. 34] that and ℐ also coincide for S(X) when X is the one-point compactification of the countably infinite discrete space. Our main purpose here is to point out the fact that among the 0-dimensional metric spaces, Cezus discovered the only nondiscrete space X with the property that and ℐ coincide on the semigroup S(X). Because of a result in a previous paper [6] by S. Subbiah and the author, this property (for 0-dimensional metric spaces) is in turn equivalent to the semigroup being regular. We gather all this together in the following

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Semigroups of continuous selfmaps for which Green's and ℐ relations coincide
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Semigroups of continuous selfmaps for which Green's and ℐ relations coincide
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Semigroups of continuous selfmaps for which Green's and ℐ relations coincide
      Available formats
      ×

Copyright

References

Hide All
1.Cezus, F. A., Green's relations in semigroups of functions, Ph.D. thesis at Australian National University, Canberra, Australia (1972).
2.Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Vol. 1, Math Surveys of the Amer. Math. Soc. 7 (Providence, R. I., 1961).
3.Dugundji, J., Topology (Allyn and Bacon, 1966).
4.de Groot, J., Groups represented by homeomorphism groups I, Math. Ann. 138 (1959), 80102.
5.Magill, K. D. Jr and Subbiah, S., Green's relations for regular elements of semigroups of endomorphisms, Canad. J. Math. 26 (1974), 14841497.
6.Magill, K. D. Jr and Subbiah, S., Green's relations for regular elements of sandwich semigroups II; semigroups of continuous functions, J. Austral. Math. Soc. 25 (1978), 4565.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed