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# cancellative semigroup

Let $S$ be a semigroup.

$S$ is *left cancellative* if, for all $a,b,c\in S$, $ab=ac\Rightarrow b=c$

$S$ is *right cancellative* if, for all $a,b,c\in S$, $ba=ca\Rightarrow b=c$

$S$ is *cancellative* if it is both left and right cancellative.

# 1 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 *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.

# 2 Individual elements

An element $x\in S$ is called *left cancellative* if, for all $b,c\in S$, $xb=xc\Rightarrow b=c$

An element $x\in S$ is called *right cancellative* if, for all $b,c\in S$, $bx=cx\Rightarrow b=c$

## Mathematics Subject Classification

20M10*no label found*

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