# Regular semigroup

In mathematics, a **regular semigroup** is a semigroup *S* in which every element is **regular**, i.e., for each element *a*, there exists an element *x* 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 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 α is any inverse for *a*, then *a* is -related to *αa* and -related to *aα*.[15]

**Theorem.** Let *S* be a regular semigroup, and let *a* and *b* be elements of *S*. Then

- if, and only if, there exist α in
*V*(*a*) and β in*V*(*b*) such that α*a*= β*b*; - if, and only if, there exist α in
*V*(*a*) and β in*V*(*b*) such that*a*α =*b*β.[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

## Notes

- Howie 1995 : 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 and Preston 1961 : Lemma 1.14.
- Howie 1995 : p. 52.
- Clifford and Preston 1961 : p. 26.
- Howie 1995 : Lemma 2.4.4.
- Howie 1995 : Theorem 5.1.1.
- Proof: http://planetmath.org/?op=getobj&from=objects&id=6391
- Howie 1995 : 55.
- Clifford and Preston 1961 : Lemma 1.13.
- Howie 1995 : Proposition 2.4.1.
- Howie 1995 : Section 2.4 & Chapter 6.
- Howie 1995 : 222.

## References

- A. H. Clifford and G. B. Preston,
*The Algebraic Theory of Semigroups*, Volume 1, Mathematical Surveys of the American Mathematical Society, No. 7, Providence, R.I., 1961. - J. M. Howie,
*Fundamentals of Semigroup Theory*, Clarendon Press, Oxford, 1995. - M. Kilp, U. Knauer, A.V. Mikhalev,
*Monoids, Acts and Categories with Applications to Wreath Products and Graphs*, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7. - J. A. Green (1951). "On the structure of semigroups".
*Annals of Mathematics*. Second Series.**54**(1): 163–172. doi:10.2307/1969317. hdl:10338.dmlcz/100067. JSTOR 1969317. - J. M. Howie, Semigroups, past, present and future,
*Proceedings of the International Conference on Algebra and Its Applications*, 2002, 6–20. - J. von Neumann (1936). "On regular rings".
*Proceedings of the National Academy of Sciences of the USA*.**22**(12): 707–713. doi:10.1073/pnas.22.12.707. PMC 1076849. PMID 16577757.