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 = ØfØ for any transformation f. The inverse of Ø is unique however, because only one f satisfies the additional constraint that f = fØf, 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.

Green's relations

Recall that the principal ideals of a semigroup S are defined in terms of S¹, 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:

${\displaystyle a\,{\mathcal {L))\,b}$ if, and only if, Sa = Sb;

${\displaystyle a\,{\mathcal {R))\,b}$ if, and only if, aS = bS;

${\displaystyle a\,{\mathcal {J))\,b}$ if, and only if, SaS = SbS.^[13]

In a regular semigroup S, every ${\displaystyle {\mathcal {L))}$- and ${\displaystyle {\mathcal {R))}$-class contains at least one idempotent. If a is any element of S and a' is any inverse for a, then a is ${\displaystyle {\mathcal {L))}$-related to a'a and ${\displaystyle {\mathcal {R))}$-related to aa'.^[14]

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

• ${\displaystyle a\,{\mathcal {L))\,b}$ iff there exist a' in V(a) and b' in V(b) such that a'a = b'b;
• ${\displaystyle a\,{\mathcal {R))\,b}$ iff there exist a' in V(a) and b' in V(b) such that aa' = bb',
• ${\displaystyle a\,{\mathcal {H))\,b}$ iff there exist a' in V(a) and b' in V(b) such that a'a = b'b and aa' = bb'.^[15]

If S is an inverse semigroup, then the idempotent in each ${\displaystyle {\mathcal {L))}$- and ${\displaystyle {\mathcal {R))}$-class is unique.^[12]

Special classes of regular semigroups

Some special classes of regular semigroups are:^[16]

• 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.^[17]

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

Generalizations

• eventually regular semigroup
• E-dense (aka E-inversive) semigroup