regular semigroup RegularSemigroup 2004-06-04 10:05:10 2008-12-17 18:05:05 Definition semigroup regular $\pi$-regular eventually regular strongly $\pi$-regular group-bound inverse semigroup Clifford semigroup orthodox semigroup completely regular epigroup regular element inverse relative inverse \PMlinkescapephrase{completely regular} \PMlinkescapephrase{generated by} \PMlinkescapeword{index} \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} (or a \emph{relative inverse}) % Bruck, Survey of Binary Systems 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}. 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. \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 $S$.