regular semigroup

completely regular generated by

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 $y=yxy$.

1 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 $a$.

2 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 (http://planetmath.org/ASemilatticeIsACommutativeBand).

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.

3 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.

4 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 $S$.

Title	regular semigroup
Canonical name	RegularSemigroup
Date of creation	2013-03-22 14:23:17
Last modified on	2013-03-22 14:23:17
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	25
Author	yark (2760)
Entry type	Definition
Classification	msc 20M17
Classification	msc 20M18
Related topic	ACharacterizationOfGroups
Defines	regular
Defines	$\pi$-regular
Defines	eventually regular
Defines	strongly $\pi$-regular
Defines	group-bound
Defines	inverse semigroup
Defines	Clifford semigroup
Defines	orthodox semigroup
Defines	completely regular
Defines	epigroup
Defines	regular element
Defines	inverse
Defines	relative inverse