SimpleSemigroup 2002-10-17 18:04:26 2007-11-24 12:57:20 Definition simple zero simple right simple left 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 \PMlinkescapeword{structure} \PMlinkescapeword{theory} Let $S$ be a semigroup. If $S$ has no ideals other than itself, then $S$ is said to be \emph{simple}. If $S$ has no left ideals [resp. right ideals] other than itself, then $S$ is said to be \emph{left simple} [resp. \emph{right simple}]. Right simple and left simple are stronger conditions than simple. A semigroup $S$ is left simple if and only if $Sa = S$ for all $a \in S$. A semigroup is both left and right simple if and only if it is a group. If $S$ has a zero element $\theta$, then $0 = \{ \theta \}$ is always an ideal of $S$, so $S$ is not simple (unless it has only one element). So in studying semigroups with a zero, a slightly weaker definition is required. Let $S$ be a semigroup with a zero. Then $S$ is \emph{zero simple}, or $0$-simple, if the following conditions hold: \begin{itemize} \item $S^2 \neq 0$ \item $S$ has no ideals except $0$ and $S$ itself \end{itemize} The condition $S^2 = 0$ really only eliminates one semigroup: the 2-element null semigroup. Excluding this semigroup makes parts of the structure theory of semigroups cleaner.