Skip to main content Accessibility help
×
Home

PARTIAL ORDERS ON SEMIGROUPS OF PARTIAL TRANSFORMATIONS WITH RESTRICTED RANGE

  • KRITSADA SANGKHANAN (a1) and JINTANA SANWONG (a2) (a3)

Abstract

Let X be any set and P(X) the set of all partial transformations defined on X, that is, all functions α:AB where A,B are subsets of X. Then P(X) is a semigroup under composition. Let Y be a subset of X. Recently, Fernandes and Sanwong defined PT(X,Y )={αP(X):Y } and defined I(X,Y ) to be the set of all injective transformations in PT(X,Y ) . Hence PT(X,Y ) and I(X,Y ) are subsemigroups of P(X) . In this paper, we study properties of the so-called natural partial order ≤ on PT(X,Y ) and I(X,Y ) in terms of domains, images and kernels, compare ≤ with the subset order, characterise the meet and join of these two orders, then find elements of PT(X,Y ) and I(X,Y ) which are compatible. Also, the minimal and maximal elements are described.

    • 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.

      PARTIAL ORDERS ON SEMIGROUPS OF PARTIAL TRANSFORMATIONS WITH RESTRICTED RANGE
      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.

      PARTIAL ORDERS ON SEMIGROUPS OF PARTIAL TRANSFORMATIONS WITH RESTRICTED RANGE
      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.

      PARTIAL ORDERS ON SEMIGROUPS OF PARTIAL TRANSFORMATIONS WITH RESTRICTED RANGE
      Available formats
      ×

Copyright

Corresponding author

For correspondence; e-mail: jintana.s@cmu.ac.th

Footnotes

Hide All

The first author thanks the Development and Promotion of Science and Technology talents project, Thailand, for its financial support. He also thanks the Graduate School, Chiang Mai University, Chiangmai, Thailand, for its financial support that he received during the preparation of this paper. The second author thanks the National Research University Project under the Office of the Higher Education Commission, Thailand, for its financial support.

Footnotes

References

Hide All
[1]Clifford, A. H. and Preston, G. B., The Algebraic Theory of Semigroups, Vol. 1, Mathematical Surveys, 7 (American Mathematical Society, Providence, RI, 1961).
[2]Fernandes, V. H. and Sanwong, J., ‘On the ranks of semigroups of transformations on a finite set with restricted range’, Algebra Colloq., to appear.
[3]Hartwig, R. E., ‘How to partially order regular elements’, Math. Jap. 25(1) (1980), 113.
[4]Howie, J. M., ‘‘Naturally ordered bands’’, Glasg. Math. J. 8 (1967), 5558.
[5]Howie, J. M., An Introduction to Semigroup Theory (Academic Press, London, 1976).
[6]Kowol, G. and Mitsch, H., ‘Naturally ordered transformation semigroups’, Monatsh. Math. 102 (1986), 115138.
[7]Marques-Smith, M. P. O. and Sullivan, R. P., ‘Partial orders on transformation semigroups’, Monatsh. Math. 140 (2003), 103118.
[8]Mitsch, H., ‘A natural partial order for semigroups’, Proc. Amer. Soc. 97(3) (1986), 384388.
[9]Nambooripad, K. S. S., ‘The natural partial order on a regular semigroup’, Proc. Edinb. Math. Soc. (2) 23 (1980), 249260.
[10]Sanwong, J. and Sommanee, W., ‘Regularity and Green’s relations on a semigroup of transformation with restricted range’, Int. J. Math. Math. Sci. 2008 (2008), Art. ID 794013, 11pp.
[11]Singha, B., Sanwong, J. and Sullivan, R. P., ‘Partial orders on partial Baer–Levi semigroups’, Bull. Aust. Math. Soc. 81 (2010), 195207.
[12]Symons, J. S. V., ‘Some results concerning a transformation semigroup’, J. Aust. Math. Soc. (Series A) 19 (1975), 413425.
[13]Vagner, V., ‘Generalized groups’, Dokl. Akad. Nauk SSSR 84 (1952), 11191122 (in Russian).
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

PARTIAL ORDERS ON SEMIGROUPS OF PARTIAL TRANSFORMATIONS WITH RESTRICTED RANGE

  • KRITSADA SANGKHANAN (a1) and JINTANA SANWONG (a2) (a3)

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