# regular semigroup

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 |