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NATURAL PARTIAL ORDER IN SEMIGROUPS OF TRANSFORMATIONS WITH INVARIANT SET

Part of: Semigroups

Published online by Cambridge University Press:  22 May 2012

LEI SUN  and
LIMIN WANG
LEI SUN*
Affiliation:
Department of Mathematics, South China Normal University, Guangzhou, Guangdong, 510631, PR China School of Mathematics and Information Science, Henan Polytechnic University, Henan, Jiaozuo, 454003, PR China (email: sunlei97@163.com)
LIMIN WANG
Affiliation:
Department of Mathematics, South China Normal University, Guangzhou, Guangdong, 510631, PR China
*
✉For correspondence; e-mail: sunlei97@163.com
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Abstract

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Let 𝒯X be the full transformation semigroup on the nonempty set X. We fix a nonempty subset Y of X and consider the semigroup of transformations that leave Y invariant, and endow it with the so-called natural partial order. Under this partial order, we determine when two elements of S(X,Y ) are related, find the elements which are compatible and describe the maximal elements, the minimal elements and the greatest lower bound of two elements. Also, we show that the semigroup S(X,Y ) is abundant.

Keywords

transformation semigroupsnatural orderabundance

MSC classification

Secondary: 20M20: Semigroups of transformations, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 1 , February 2013 , pp. 94 - 107
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

This paper is partially supported by National Natural Science Foundation of China (No. 10971086) and Natural Science Foundation of Henan Province (Nos 112300410120, 122300410276).

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