cancellative semigroupCancellativeSemigroup2004-06-17 05:50:402006-10-04 05:25:41Definitionsemigroupcancellativeweakly cancellativeleft cancellativeright cancellativeweakly cancellative semigroupleft cancellative semigroupright cancellative semigroup\usepackage{amssymb}
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Let $S$ be a semigroup.
$S$ is \emph{left cancellative} if, for all $a,b,c\in S$, $ab=ac\Rightarrow b=c$\\
$S$ is \emph{right cancellative} if, for all $a,b,c\in S$, $ba=ca\Rightarrow b=c$
$S$ is \emph{cancellative} if it is both left and right cancellative.
\section{Relationship to some other types of semigroup}
This is a generalisation of groups, and in fact being cancellative is a necessary condition for a semigroup to be embeddable in a group.
Note that a non-empty semigroup is a group if and only if it is cancellative and regular.
$S$ is \emph{weakly cancellative} if, for all $a,b,c\in S$, $(ab=ac~\&~ba=ca)\Rightarrow b=c$
A semigroup is completely simple if and only if it is weakly cancellative and regular.
\section{Individual elements}
An element $x\in S$ is called \emph{left cancellative} if, for all $b,c\in S$, $xb=xc\Rightarrow b=c$\\
An element $x\in S$ is called \emph{right cancellative} if, for all $b,c\in S$, $bx=cx\Rightarrow b=c$