Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T21:30:00.195Z Has data issue: false hasContentIssue false
  • English
  • Français

THE NATURAL PARTIAL ORDER ON LINEAR SEMIGROUPS WITH NULLITY AND CO-RANK BOUNDED BELOW

Part of: Semigroups Ordered sets

Published online by Cambridge University Press:  14 October 2014

SUREEPORN CHAOPRAKNOI ,
TEERAPHONG PHONGPATTANACHAROEN  and
PONGSAN PRAKITSRI
SUREEPORN CHAOPRAKNOI
Affiliation:
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand email sureeporn.c@chula.ac.th
TEERAPHONG PHONGPATTANACHAROEN*
Affiliation:
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand email teeraphong.p@chula.ac.th
PONGSAN PRAKITSRI
Affiliation:
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand email pongsan.p@gmail.com
*
✉teeraphong.p@chula.ac.th
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Higgins [‘The Mitsch order on a semigroup’, Semigroup Forum 49 (1994), 261–266] showed that the natural partial orders on a semigroup and its regular subsemigroups coincide. This is why we are interested in the study of the natural partial order on nonregular semigroups. Of particular interest are the nonregular semigroups of linear transformations with lower bounds on the nullity or the co-rank. In this paper, we determine when they exist, characterise the natural partial order on these nonregular semigroups and consider questions of compatibility, minimality and maximality. In addition, we provide many examples associated with our results.

Keywords

linear transformation semigroupnatural partial orderleft (right) compatible elementminimal (maximal) element

MSC classification

Primary: 20M20: Semigroups of transformations, etc.
Secondary: 06A06: Partial order, general
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 1 , February 2015 , pp. 104 - 115
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Chaopraknoi, S., Phongpattanacharoen, T. and Rawiwan, P., ‘The natural partial order on some transformation semigroups’, Bull. Aust. Math. Soc. 89 (2014), 279–292.CrossRefGoogle Scholar
Chaopraknoi, S. and Kemprasit, Y., ‘Linear transformation semigroups admitting the structure of a semihyperring with zero’, Ital. J. Pure Appl. Math. 17 (2005), 213–222.Google Scholar
Hartwig, R., ‘How to partially order regular elements’, Math. Japonica 25 (1980), 1–13.Google Scholar
Higgins, P. M., ‘The Mitsch order on a semigroup’, Semigroup Forum 49 (1994), 261–266.CrossRefGoogle Scholar
Huisheng, P. and Weina, D., ‘Naturally ordered semigroups of partial transformations preserving an equivalence relation’, Comm. Algebra 41 (2013), 3308–3324.CrossRefGoogle Scholar
Kemprasit, Y., Algebraic Semigroup Theory (Pitak Press, Bangkok, 2002), (in Thai).Google Scholar
Marques-Smith, M. P. O. and Sullivan, R. P., ‘Partial orders on transformation semigroups’, Monatsh. Math. 140 (2003), 103–118.CrossRefGoogle Scholar
Mendez-Gonçalves, S. and Sullivan, R. P., ‘The ideal structure of semigroups of linear transformation with lower bounds on their nullity or defect’, Algebra Colloq. 17(1) (2010), 109–120.CrossRefGoogle Scholar
Mitsch, H., ‘A natural partial order for semigroups’, Proc. Amer. Math. Soc. 97 (1986), 384–388.Google Scholar
Nambooripad, K., ‘The natural partial order on a regular semigroup’, Proc. Edinb. Math. Soc. (2) 23 (1980), 249–260.CrossRefGoogle Scholar
Prakitsri, P., ‘Nearring Structure on Variants of Some Transformation Semigroups’, Master’s Thesis, Chulalongkorn University, 2010.Google Scholar
Prakitsri, P., Chaopraknoi, S. and Phongpattanacharoen, T., ‘The natural partial order on almost monomorphism semigroups and almost epimorphism semigroups’, Proc. 19th Annual Meeting Mathematics (2014), 303–308.Google Scholar
Sullivan, R. P., ‘Partial orders on linear transformation semigroups’, Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), 413–437.Google Scholar
Wagner, V., ‘Generalized groups’, Dokl. Akad. Nauk SSSR 84 (1952), 1119–1122 (in Russian).Google Scholar