Regular semigroup

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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]

1 Origins
2 The basics
3 Green's Relations
4 Special classes of regular semigroups
5 References

[edit] Origins

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.

[edit] 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 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.

[edit] 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:

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

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

$a\,\mathcal{J}\,b$ if, and only if, SaS = SbS.^[9]

In a regular semigroup S, every $\mathcal{L}$ - and $\mathcal{R}$ -class contains at least one idempotent. If a is any element of S and α is any inverse for a, then a is $\mathcal{L}$ -related to αa and $\mathcal{R}$ -related to aα.^[10]

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

$a\,\mathcal{L}\,b$ if, and only if, there exist α in V(a) and β in V(b) such that αa = βb;
$a\,\mathcal{R}\,b$ 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 $\mathcal{L}$ - and $\mathcal{R}$ -class is unique.^[12]

[edit] Special classes of regular semigroups

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]

[edit] 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. A. Green (1951). "On the structure of semigroups". Annals of Mathematics (2) 54: 163–172. doi:10.2307/1969317.
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. doi:10.1073/pnas.22.12.707.