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'regular semigroup'
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| Title of object: |
regular semigroup |
| Canonical Name: |
RegularSemigroup |
| Type: |
Definition |
| Created on: |
2004-06-04 10:05:10 |
| Modified on: |
2006-10-04 05:43:39 |
| Classification: |
msc:20M17 |
| Defines: |
regular, $\pi$-regular, eventually regular, strongly $\pi$-regular, group-bound, inverse semigroup, Clifford semigroup, orthodox semigroup, completely regular, epigroup, regular element, inverse |
Revision comment (for changes between this and next version):
| add classification 20M18 (as it defines inverse semigroups) |
Preamble:
Content:
\PMlinkescapephrase{completely regular}
\PMlinkescapephrase{generated by}
\PMlinkescapeword{index}
\PMlinkescapeword{inverse}
\PMlinkescapeword{power}
Let $S$ be a semigroup.
$x\in S$ is \emph{regular} if there is a $y\in S$ such that $x=xyx$.\\
$y\in S$ is an \emph{inverse} for $x$ if $x=xyx$ and $y=yxy$.
\section{Regular semigroups}
$S$ is a \emph{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 $a$.
\section{Inverse semigroups}
$S$ is an \emph{inverse semigroup} if for all $x\in S$ there is a \emph{unique} $y\in S$ such that $x=xyx$ and $y=yxy$.
In an inverse semigroup every principal ideal is generated by a \emph{unique} idempotent.
In an inverse semigroup the set of idempotents is a subsemigroup, in particular a \PMlinkname{commutative band}{ASemilatticeIsACommutativeBand}.
An example of an inverse semigroup is the bicyclic semigroup.
\section{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 \emph{orthodox semigroups} where the set of idempotents is a subsemigroup, and \emph{Clifford semigroups} where the idempotents are central.
\section{Additional}
$S$ is called \emph{eventually regular} (or \emph{$\pi$-regular}) if a power of every element is regular.
$S$ is called \emph{group-bound} (or \emph{strongly $\pi$-regular}, or an \emph{epigroup}) if a power of every element is in a subgroup of $S$.
$S$ is called \emph{completely regular} if every element is in a subgroup of $S$. |
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