regular semigroup

Let $S$ be a semigroup.

$x\in S$ is regular if there is a $y\in S$ such that $x=xyx$ .
$y\in S$ is an inverse (or a relative inverse) for $x$ if $x=xyx$ and $y=yxy$ .

Regular semigroups

$S$ is a regular semigroup if all its elements are regular. The phrase 'von Neumann regular' is sometimes used, after the definition for rings.

In a regular semigroup, every principal ideal is generated by an idempotent.

Every regular element has at least one inverse. To show this, suppose $a\in S$ is regular, so that $a = aba$ for some $b\in S$ . Put $c=bab$ . Then$$ a=aba=(aba)ba=a(bab)a=aca$$ and$$ c=bab=b(aba)b=(bab)ab=cab=c(aba)b=ca(bab)=cac,$$ so $c$ is an inverse of $a$ .

Inverse semigroups

$S$ is an inverse semigroup if for all $x\in S$ there is a unique $y\in S$ such that $x=xyx$ and $y=yxy$ .

In an inverse semigroup every principal ideal is generated by a unique idempotent.

In an inverse semigroup the set of idempotents is a subsemigroup, in particular a commutative band.

The bicyclic semigroup is an example of an inverse semigroup. The symmetric inverse semigroup (on some set $X$ ) is another example. Of course, every group is also an inverse semigroup.

Motivation

Both of these notions generalise the definition of a group. In particular, a regular semigroup with one idempotent is a group: as such, many interesting subclasses of regular semigroups arise from putting conditions on the idempotents. Apart from inverse semigroups, there are orthodox semigroups where the set of idempotents is a subsemigroup, and Clifford semigroups where the idempotents are central.

Additional

$S$ is called eventually regular (or $\pi$ -regular) if a power of every element is regular.

$S$ is called group-bound (or strongly $\pi$ -regular, or an epigroup) if a power of every element is in a subgroup of $S$ .

$S$ is called completely regular if every element is in a subgroup of $S$ .