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PARTIAL ORDERS ON PARTIAL BAER–LEVI SEMIGROUPS

Part of: Ordered sets Semigroups

Published online by Cambridge University Press:  26 January 2010

BOORAPA SINGHA ,
JINTANA SANWONG  and
R. P. SULLIVAN
BOORAPA SINGHA
Affiliation:
Department of Mathematics, Chiang Mai University, Chiangmai 50200, Thailand (email: boorapas@yahoo.com)
JINTANA SANWONG
Affiliation:
Department of Mathematics, Chiang Mai University, Chiangmai 50200, Thailand (email: scmti004@chiangmai.ac.th)
R. P. SULLIVAN*
Affiliation:
School of Mathematics and Statistics, University of Western Australia, Nedlands 6009, Australia (email: bob@maths.uwa.edu.au)
*
✉For correspondence; e-mail: bob@maths.uwa.edu.au
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Abstract

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Marques-Smith and Sullivan [‘Partial orders on transformation semigroups’, Monatsh. Math.140 (2003), 103–118] studied various properties of two partial orders on P(X), the semigroup (under composition) consisting of all partial transformations of an arbitrary set X. One partial order was the ‘containment order’: namely, if α,β∈P(X) then α⊆β means xα=xβ for all x∈dom α, the domain of α. The other order was the so-called ‘natural order’ defined by Mitsch [‘A natural partial order for semigroups’, Proc. Amer. Math. Soc.97(3) (1986), 384–388] for any semigroup. In this paper, we consider these and other orders defined on the symmetric inverse semigroup I(X) and the partial Baer–Levi semigroup PS(q). We show that there are surprising differences between the orders on these semigroups, concerned with their compatibility with respect to composition and the existence of maximal and minimal elements.

Keywords

partial transformation semigroupBaer–Levi semigroupnatural partial ordermaximal (minimal) elementcompatible

MSC classification

Secondary: 20M20: Semigroups of transformations, etc. 06A06: Partial order, general
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 2 , April 2010 , pp. 195 - 207
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

The first author thanks the Office of the Higher Education Commission, Thailand, for its support by a ‘Strategic Scholarships for Frontier Research Network’ grant that enabled him to join a Thai PhD program and complete this research for his doctoral degree. The third author thanks the Faculty of Science, Chiangmai University, Thailand, for its financial assistance during his visit in May 2009.

References

[1]Clifford, A. H. and Preston, G. B., The Algebraic Theory of Semigroups, Vols 1 and 2, Mathematical Surveys, No. 7 (American Mathematical Society, Providence, RI, 1961 and 1967).Google Scholar
[2]Howie, J. M., An Introduction to Semigroup Theory (Academic Press, London, 1976).Google Scholar
[3]Kowol, G. and Mitsch, H., ‘Naturally ordered transformation semigroups’, Monatsh. Math. 102 (1986), 115–138.CrossRefGoogle Scholar
[4]Marques-Smith, M. P. O. and Sullivan, R. P., ‘Partial orders on transformation semigroups’, Monatsh. Math. 140 (2003), 103–118.CrossRefGoogle Scholar
[5]Mitsch, H., ‘A natural partial order for semigroups’, Proc. Amer. Math. Soc. 97(3) (1986), 384–388.CrossRefGoogle Scholar
[6]Pinto, F. A. and Sullivan, R. P., ‘Baer–Levi semigroups of partial transformations’, Bull. Aust. Math. Soc. 69(1) (2004), 87–106.CrossRefGoogle Scholar