| Version current |
Version 5 |
| Let $S$ be a semigroup. |
Let $S$ be a semigroup. |
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| $S$ is \emph{left cancellative} if, for all $a,b,c\in S$, $ab=ac\Rightarrow b=c$\\ |
$S$ 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{right cancellative} if, for all $a,b,c\in S$, $ba=ca\Rightarrow b=c$ |
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| $S$ is \emph{cancellative} if it is both left and right cancellative. |
$S$ is \emph{cancellative} if it is both left and right cancellative. |
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| \section{Relationship to some other types of semigroup} |
\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. |
This is a generalisation of groups, and in fact being cancellative is a necessary condition for a semigroup to be embeddable in a group. |
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| Note that a non-empty semigroup is a group if and only if it is cancellative and regular. |
Note that a non-empty semigroup is a group if and only if it is cancellative and regular. |
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| $S$ is \emph{weakly cancellative} if, for all $a,b,c\in S$, $(ab=ac~\&~ba=ca)\Rightarrow b=c$ |
$S$ is \emph{weakly cancellative} if, for all $a,b,c\in S$, $(ab=ac~\&~ba=ca)\Rightarrow b=c$ |
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| A semigroup is completely simple if and only if it is weakly cancellative and regular. |
A semigroup is completely simple if and only if it is weakly cancellative and regular. |
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| \section{Individual elements} |
\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{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 $b,c\in S$, $bx=cx\Rightarrow b=c$ |
An element $x\in S$ is called \emph{right cancellative} if, for all $b,c\in S$, $bx=cx\Rightarrow b=c$ |
