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'simple semigroup'
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| Title of object: |
simple semigroup |
| Canonical Name: |
SimpleSemigroup |
| Type: |
Definition |
| Created on: |
2002-10-17 18:04:26 |
| Modified on: |
2004-03-17 15:57:50 |
| Classification: |
msc:20M10 |
| Defines: |
simple, zero simple, right simple, left simple |
Revision comment (for changes between this and next version):
| Changes for correction #11025 ('linking policy'). |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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%\usepackage{amsthm}
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%\usepackage{xypic}
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Content:
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. |
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