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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]

Contents 1 Origins 2 The basics 3 Green's Relations 4 Special classes of regular semigroups 5 Notes 6 References 7 See also

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 J. von Neumann.^[3] It was his study of regular semigroups which led Green 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.

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 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) = x(axa)x = xax.^[4]

The set of inverses (in the above sense) of an element a in an arbitrary semigroup S is denoted by V(a).^[5] 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.^[6]

A regular semigroup in which idempotents commute is an inverse semigroup, that is, 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.

Then

b = bab = b(aca)b = bac(ac)(ab) = bac(ab)(ac) = (ca)(ba)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.^[7]

Theorem. The homomorphic image of a regular semigroup is regular.^[8]

Examples of regular semigroups:

• Every group is regular.
• Every inverse semigroup is regular.
• Every band is regular.
• The bicyclic semigroup is regular.
• Any full transformation semigroup is regular.
• A Rees matrix semigroup is regular.

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:

if, and only if, Sa = Sb;

if, and only if, aS = bS;

if, and only if, SaS = SbS.^[9]

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.^[10]

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 αa = βb;
• if, and only if, there exist α in V(a) and β in V(b) such that aα = bβ.^[11]

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

Some special classes of regular semigroups are:^[13]

• 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.^[14]

1. ^ Howie 1995 : 54.
2. ^ Howie 2002.
3. ^ von Neumann 1936.
4. ^ Clifford and Preston 1961 : Lemma 1.14.
5. ^ Howie 1995 : 52.
6. ^ Clifford and Preston 1961 : 26.
7. ^ Howie 1995 : Theorem 5.1.1.
8. ^ Howie 1995 : Lemma 2.4.4.
9. ^ Howie 1995 : 55.
10. ^ Clifford and Preston 1961 : Lemma 1.13.
11. ^ Howie 1995 : Proposition 2.4.1.
12. ^ Howie 1995 : Theorem 5.1.1.
13. ^ Howie 1995 : Section 2.4 & Chapter 6.
14. ^ Howie 1995 : 222.

• 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. A. Green (1951). "On the structure of semigroups". Annals of Mathematics (2) 54: 163-172. 
• J. M. Howie, Fundamentals of Semigroup Theory, Clarendon Press, Oxford, 1995.
• 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: 707-713. 

• Inverse semigroups
• Green's relations