CompletelySimpleSemigroup 2004-09-09 13:02:25 2009-01-01 14:46:59 Definition simple semigroup primitive completely $0$-simple completely simple % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $S$ be a semigroup. An idempotent $e\in S$ is \emph{primitive} if for every other idempotent $f\in S$, $ef=fe=f\not= 0\Rightarrow e=f$ A semigroup $S$ (without zero) is \emph{completely \PMlinkescapetext{simple}} if it is simple and contains a primitive idempotent. A semigroup $S$ is \emph{completely $0$-simple} if it is \PMlinkname{$0$-simple}{SimpleSemigroup} and contains a primitive idempotent. Completely simple and completely $0$-simple semigroups maybe characterised by the Rees Theorem (\cite{ReesRef}, Theorem 3.2.3). Note: A semigroup (without zero) is completely simple if and only if it is regular and weakly cancellative. A simple semigroup (without zero) is completely simple if and only if it is completely regular. A $0$-simple semigroup is completely $0$-simple if and only if it is group-bound. \begin{thebibliography}{2} \bibitem[Ho95]{ReesRef} Howie, John M. \emph{Fundamentals of Semigroup Theory}. Oxford University Press, 1995. \end{thebibliography}