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.

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