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THE NATURAL PARTIAL ORDER ON SOME TRANSFORMATION SEMIGROUPS

Published online by Cambridge University Press:  28 June 2013

SUREEPORN CHAOPRAKNOI*
Affiliation:
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand email [email protected][email protected]
TEERAPHONG PHONGPATTANACHAROEN
Affiliation:
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand email [email protected][email protected]
PATTANACHAI RAWIWAN
Affiliation:
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand email [email protected][email protected]
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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.

Type
Research Article
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

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