In mathematics, a **regular semigroup** is a semigroup *S* in which every element is **regular**, i.e., for each element *a* in *S* there exists an element *x* in *S* such that *axa* = *a*.^{[1]} Regular semigroups are one of the most-studied classes of semigroups, and their structure is particularly amenable to study via Green's relations.^{[2]}

## History

Regular semigroups were introduced by J. A. Green in his influential 1951 paper "On the structure of semigroups"; this was also the paper in which Green's relations were introduced. The concept of *regularity* in a semigroup was adapted from an analogous condition for rings, already considered by John von Neumann.^{[3]} It was Green's study of regular semigroups which led him to define his celebrated relations. According to a footnote in Green 1951, the suggestion that the notion of regularity be applied to semigroups was first made by David Rees.

The term **inversive semigroup** (French: demi-groupe inversif) was historically used as synonym in the papers of Gabriel Thierrin (a student of Paul Dubreil) in the 1950s,^{[4]}^{[5]} and it is still used occasionally.^{[6]}

## The basics

There are two equivalent ways in which to define a regular semigroup *S*:

- (1) for each
*a*in*S*, there is an*x*in*S*, which is called a**pseudoinverse**,^{[7]}with*axa*=*a*; - (2) every element
*a*has at least one**inverse***b*, in the sense that*aba*=*a*and*bab*=*b*.

To see the equivalence of these definitions, first suppose that *S* is defined by (2). Then *b* serves as the required *x* in (1). Conversely, if *S* is defined by (1), then *xax* is an inverse for *a*, since *a*(*xax*)*a* = *axa*(*xa*) = *axa* = *a* and (*xax*)*a*(*xax*) = *x*(*axa*)(*xax*) = *xa*(*xax*) = *x*(*axa*)*x* = *xax*.^{[8]}

The set of inverses (in the above sense) of an element *a* in an arbitrary semigroup *S* is denoted by *V*(*a*).^{[9]} Thus, another way of expressing definition (2) above is to say that in a regular semigroup, *V*(*a*) is nonempty, for every *a* in *S*. The product of any element *a* with any *b* in *V*(*a*) is always idempotent: *abab* = *ab*, since *aba* = *a*.^{[10]}

### Examples of regular semigroups

- Every group is a regular semigroup.
- Every band (idempotent semigroup) is regular in the sense of this article, though this is not what is meant by a regular band.
- The bicyclic semigroup is regular.
- Any full transformation semigroup is regular.
- A Rees matrix semigroup is regular.
- The homomorphic image of a regular semigroup is regular.
^{[11]}

### Unique inverses and unique pseudoinverses

A regular semigroup in which idempotents commute (with idempotents) is an inverse semigroup, or equivalently, every element has a *unique* inverse. To see this, let *S* be a regular semigroup in which idempotents commute. Then every element of *S* has at least one inverse. Suppose that *a* in *S* has two inverses *b* and *c*, i.e.,

*aba*=*a*,*bab*=*b*,*aca*=*a*and*cac*=*c*. Also*ab*,*ba*,*ac*and*ca*are idempotents as above.

Then

*b*=*bab*=*b*(*aca*)*b*=*bac*(*a*)*b*=*bac*(*aca*)*b*=*bac*(*ac*)(*ab*) =*bac*(*ab*)(*ac*) =*ba*(*ca*)*bac*=*ca*(*ba*)*bac*=*c*(*aba*)*bac*=*cabac*=*cac*=*c*.

So, by commuting the pairs of idempotents *ab* & *ac* and *ba* & *ca*, the inverse of *a* is shown to be unique. Conversely, it can be shown that any inverse semigroup is a regular semigroup in which idempotents commute.^{[12]}

The existence of a unique pseudoinverse implies the existence of a unique inverse, but the opposite is not true. For example, in the symmetric inverse semigroup, the empty transformation Ø does not have a unique pseudoinverse, because Ø = Ø*f*Ø for any transformation *f*. The inverse of Ø is unique however, because only one *f* satisfies the additional constraint that *f* = *f*Ø*f*, namely *f* = Ø. This remark holds more generally in any semigroup with zero. Furthermore, if every element has a unique pseudoinverse, then the semigroup is a group, and the unique pseudoinverse of an element coincides with the group inverse.^{[13]}

## Green's relations

Recall that the principal ideals of a semigroup *S* are defined in terms of *S*^{1}, the *semigroup with identity adjoined*; this is to ensure that an element *a* belongs to the principal right, left and two-sided ideals which it generates. In a regular semigroup *S*, however, an element *a* = *axa* automatically belongs to these ideals, without recourse to adjoining an identity. Green's relations can therefore be redefined for regular semigroups as follows:

- if, and only if,
*Sa*=*Sb*; - if, and only if,
*aS*=*bS*; - if, and only if,
*SaS*=*SbS*.^{[14]}

In a regular semigroup *S*, every - and -class contains at least one idempotent. If *a* is any element of *S* and *a'* is any inverse for *a*, then *a* is -related to *a'a* and -related to *aa'*.^{[15]}

**Theorem.** Let *S* be a regular semigroup; let *a* and *b* be elements of *S*, and let *V(x)* denote the set of inverses of *x* in *S*. Then

- iff there exist
*a'*in*V*(*a*) and*b'*in*V*(*b*) such that*a'a*=*b'b*; - iff there exist
*a'*in*V*(*a*) and*b'*in*V*(*b*) such that*aa'*=*bb'*, - iff there exist
*a'*in*V*(*a*) and*b'*in*V*(*b*) such that*a'a*=*b'b*and*aa'*=*bb'*.^{[16]}

If *S* is an inverse semigroup, then the idempotent in each - and -class is unique.^{[12]}

## Special classes of regular semigroups

Some special classes of regular semigroups are:^{[17]}

*Locally inverse semigroups*: a regular semigroup*S*is**locally inverse**if*eSe*is an inverse semigroup, for each idempotent*e*.*Orthodox semigroups*: a regular semigroup*S*is**orthodox**if its subset of idempotents forms a subsemigroup.*Generalised inverse semigroups*: a regular semigroup*S*is called a**generalised inverse semigroup**if its idempotents form a normal band, i.e.,*xyzx*=*xzyx*for all idempotents*x*,*y*,*z*.

The class of generalised inverse semigroups is the intersection of the class of locally inverse semigroups and the class of orthodox semigroups.^{[18]}

All inverse semigroups are orthodox and locally inverse. The converse statements do not hold.

## Generalizations

## See also

## References

**^**Howie 1995 p. 54**^**Howie 2002.**^**von Neumann 1936.**^**Christopher Hollings (16 July 2014).*Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups*. American Mathematical Society. p. 181. ISBN 978-1-4704-1493-1.**^**http://www.csd.uwo.ca/~gab/pubr.html**^**Jonathan S. Golan (1999).*Power Algebras over Semirings: With Applications in Mathematics and Computer Science*. Springer Science & Business Media. p. 104. ISBN 978-0-7923-5834-3.**^**Klip, Knauer and Mikhalev : p. 33**^**Clifford & Preston 2010 Lemma 1.14.**^**Howie 1995 p. 52**^**Clifford & Preston 2010 p. 26**^**Howie 1995 Lemma 2.4.4- ^
^{a}^{b}Howie 1995 Theorem 5.1.1 **^**Proof: https://planetmath.org/acharacterizationofgroups**^**Howie 1995 p. 55**^**Clifford & Preston 2010 Lemma 1.13**^**Howie 1995 Proposition 2.4.1**^**Howie 1995 ch. 6, § 2.4**^**Howie 1995 p. 222

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