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THE NATURAL PARTIAL ORDER ON SOME TRANSFORMATION SEMIGROUPS

Part of: Semigroups Ordered sets

Published online by Cambridge University Press:  28 June 2013

SUREEPORN CHAOPRAKNOI ,
TEERAPHONG PHONGPATTANACHAROEN  and
PATTANACHAI RAWIWAN
SUREEPORN CHAOPRAKNOI*
Affiliation:
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand email teeraphong.p@chula.ac.thball_ll@hotmail.com
TEERAPHONG PHONGPATTANACHAROEN
Affiliation:
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand email teeraphong.p@chula.ac.thball_ll@hotmail.com
PATTANACHAI RAWIWAN
Affiliation:
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand email teeraphong.p@chula.ac.thball_ll@hotmail.com
*
✉sureeporn.c@chula.ac.th
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Abstract

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For a semigroup $S$, let ${S}^{1} $ be the semigroup obtained from $S$ by adding a new symbol 1 as its identity if $S$ has no identity; otherwise let ${S}^{1} = S$. Mitsch defined the natural partial order $\leqslant $ on a semigroup $S$ as follows: for $a, b\in S$, $a\leqslant b$ if and only if $a= xb= by$ and $a= ay$ for some $x, y\in {S}^{1} $. In this paper, we characterise the natural partial order on some transformation semigroups. In these partially ordered sets, we determine the compatibility of their elements, and find all minimal and maximal elements.

Keywords

transformation semigroupsnatural partial orderleft (right) compatibleminimal (maximal) elements

MSC classification

Secondary: 20M20: Semigroups of transformations, etc. 06A06: Partial order, general

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 2 , April 2014 , pp. 279 - 292
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

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