# 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 $b,c\in S$, $bx=cx\Rightarrow b=c$

 Title cancellative semigroup Canonical name CancellativeSemigroup Date of creation 2013-03-22 14:25:09 Last modified on 2013-03-22 14:25:09 Owner yark (2760) Last modified by yark (2760) Numerical id 9 Author yark (2760) Entry type Definition Classification msc 20M10 Synonym cancellation semigroup Related topic CancellationIdeal Defines cancellative Defines weakly cancellative Defines left cancellative Defines right cancellative Defines weakly cancellative semigroup Defines left cancellative semigroup Defines right cancellative semigroup