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 |
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:
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.[5]
The set of inverses (in the above sense) of an element a in an arbitrary semigroup S is denoted by V(a).[6] 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.[7]
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.,
Then
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.[8]
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 = ØfØ, 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.[9]
Theorem. The homomorphic image of a regular semigroup is regular.[10]
Examples of regular semigroups:
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:
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.[12]
Theorem. Let S be a regular semigroup, and let a and b be elements of S. Then
If S is an inverse semigroup, then the idempotent in each - and -class is unique.[8]
Some special classes of regular semigroups are:[14]
The class of generalised inverse semigroups is the intersection of the class of locally inverse semigroups and the class of orthodox semigroups.[15]