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PARTIAL ORDERS ON SEMIGROUPS OF PARTIAL TRANSFORMATIONS WITH RESTRICTED RANGE

Part of: Semigroups

Published online by Cambridge University Press:  16 February 2012

KRITSADA SANGKHANAN  and
JINTANA SANWONG
KRITSADA SANGKHANAN
Affiliation:
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiangmai 50200, Thailand (email: kritsada_kst@hotmail.com)
JINTANA SANWONG*
Affiliation:
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiangmai 50200, Thailand Material Science Research Center, Faculty of Science, Chiang Mai University, Thailand (email: jintana.s@cmu.ac.th)
*
✉For correspondence; e-mail: jintana.s@cmu.ac.th
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Abstract

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Let X be any set and P(X) the set of all partial transformations defined on X, that is, all functions α:A→B 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 )={α∈P(X):Xα⊆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.

Keywords

partial transformation semigroupsnatural ordercompatibilitymaximal and minimal elements

MSC classification

Secondary: 20M20: Semigroups of transformations, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 1 , August 2012 , pp. 100 - 118
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

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.

References

[1]Clifford, A. H. and Preston, G. B., The Algebraic Theory of Semigroups, Vol. 1, Mathematical Surveys, 7 (American Mathematical Society, Providence, RI, 1961).Google Scholar
[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.Google Scholar
[3]Hartwig, R. E., ‘How to partially order regular elements’, Math. Jap. 25(1) (1980), 1–13.Google Scholar
[4]Howie, J. M., ‘‘Naturally ordered bands’’, Glasg. Math. J. 8 (1967), 55–58.CrossRefGoogle Scholar
[5]Howie, J. M., An Introduction to Semigroup Theory (Academic Press, London, 1976).Google Scholar
[6]Kowol, G. and Mitsch, H., ‘Naturally ordered transformation semigroups’, Monatsh. Math. 102 (1986), 115–138.CrossRefGoogle Scholar
[7]Marques-Smith, M. P. O. and Sullivan, R. P., ‘Partial orders on transformation semigroups’, Monatsh. Math. 140 (2003), 103–118.CrossRefGoogle Scholar
[8]Mitsch, H., ‘A natural partial order for semigroups’, Proc. Amer. Soc. 97(3) (1986), 384–388.CrossRefGoogle Scholar
[9]Nambooripad, K. S. S., ‘The natural partial order on a regular semigroup’, Proc. Edinb. Math. Soc. (2) 23 (1980), 249–260.CrossRefGoogle Scholar
[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.CrossRefGoogle Scholar
[11]Singha, B., Sanwong, J. and Sullivan, R. P., ‘Partial orders on partial Baer–Levi semigroups’, Bull. Aust. Math. Soc. 81 (2010), 195–207.CrossRefGoogle Scholar
[12]Symons, J. S. V., ‘Some results concerning a transformation semigroup’, J. Aust. Math. Soc. (Series A) 19 (1975), 413–425.CrossRefGoogle Scholar
[13]Vagner, V., ‘Generalized groups’, Dokl. Akad. Nauk SSSR 84 (1952), 1119–1122 (in Russian).Google Scholar