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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
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Abstract

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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. 

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