regular semigroup


completely regularPlanetmathPlanetmathPlanetmath \PMlinkescapephrasegenerated by

Let S be a semigroup.

xS is regularPlanetmathPlanetmathPlanetmathPlanetmath if there is a yS such that x=xyx.
yS is an inverseMathworldPlanetmathPlanetmath (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 idealMathworldPlanetmathPlanetmath is generated by an idempotentPlanetmathPlanetmath.

Every regular element has at least one inverse. To show this, suppose aS is regular, so that a=aba for some bS. Put c=bab. Then




so c is an inverse of a.

2 Inverse semigroups

S is an inverse semigroup if for all xS there is a unique yS 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 commutativePlanetmathPlanetmathPlanetmath 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.

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 π-regular) if a power of every element is regular.

S is called group-bound (or strongly π-regular, or an epigroup) if a power of every element is in a subgroupMathworldPlanetmath 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 π-regular
Defines eventually regular
Defines strongly π-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