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[parent] regular semigroup (Definition)

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 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

\begin{displaymath} a=aba=(aba)ba=a(bab)a=aca \end{displaymath}

and
\begin{displaymath} c=bab=b(aba)b=(bab)ab=cab=c(aba)b=ca(bab)=cac, \end{displaymath}

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 $S$.



"regular semigroup" is owned by yark. [ full author list (2) | owner history (1) ]
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See Also: a characterization of groups

Also defines:  regular, $\pi$-regular, eventually regular, strongly $\pi$-regular, group-bound, inverse semigroup, Clifford semigroup, orthodox semigroup, completely regular, epigroup, regular element, inverse

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McAlister covering theorem (Theorem) by mathcam
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Cross-references: subgroup, subclasses, group, symmetric inverse semigroup, bicyclic semigroup, subsemigroup, idempotent, principal ideal, rings, von Neumann regular, semigroup
There are 15 references to this entry.

This is version 20 of regular semigroup, born on 2004-06-04, modified 2008-07-15.
Object id is 5883, canonical name is RegularSemigroup.
Accessed 11715 times total.

Classification:
AMS MSC: 20M17 (Group theory and generalizations :: Semigroups :: Regular semigroups)
 20M18 (Group theory and generalizations :: Semigroups :: Inverse semigroups)
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Discussion
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last remark by HkBst on 2004-10-16 12:13:23
seems like a completely regular semigroup is a group
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