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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.

An example of an inverse semigroup is the bicyclic 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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Attachments:
McAlister covering theorem (Theorem) by mathcam
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Cross-references: subgroup, subclasses, group, bicyclic semigroup, subsemigroup, idempotent, principal ideal, rings, von Neumann regular, semigroup
There are 14 references to this entry.

This is version 19 of regular semigroup, born on 2004-06-04, modified 2006-10-04.
Object id is 5883, canonical name is RegularSemigroup.
Accessed 9814 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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