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单词 RegularSemigroup
释义

regular semigroup


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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

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 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 (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 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.

Titleregular semigroup
Canonical nameRegularSemigroup
Date of creation2013-03-22 14:23:17
Last modified on2013-03-22 14:23:17
Owneryark (2760)
Last modified byyark (2760)
Numerical id25
Authoryark (2760)
Entry typeDefinition
Classificationmsc 20M17
Classificationmsc 20M18
Related topicACharacterizationOfGroups
Definesregular
Definesπ-regular
Defineseventually regular
Definesstrongly π-regular
Definesgroup-bound
Definesinverse semigroup
DefinesClifford semigroup
Definesorthodox semigroup
Definescompletely regular
Definesepigroup
Definesregular element
Definesinverse
Definesrelative inverse
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