Regular semigroup
Encyclopedia
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. Regular semigroups are one of the most-studied classes of semigroups, and their structure is particularly amenable to study via 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. 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 semigroup
s was first made by David Rees
.
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.
The set of inverses (in the above sense) of an element a in an arbitrary semigroup
S is denoted by V(a). 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.
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.
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.
Theorem. The homomorphic image of a regular semigroup is regular.
Examples of regular semigroups:
s 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 ideal
s 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.
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.
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.
The class
of generalised inverse semigroups is the intersection
of the class of locally inverse semigroups and the class of orthodox semigroups.
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
S in which every element is regular, i.e., for each element a, there exists an element x such that axa = a. Regular semigroups are one of the most-studied classes of semigroups, and their structure is particularly amenable to study via Green's relations
Green's relations
In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. The relations are named for James Alexander Green, who introduced them in a paper of 1951...
.
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 relationsGreen's relations
In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. The relations are named for James Alexander Green, who introduced them in a paper of 1951...
were introduced. The concept of regularity in a semigroup was adapted from an analogous condition for rings
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
, already considered by J. von Neumann. It was his study of regular semigroups which led Green to define his celebrated relations
Green's relations
In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. The relations are named for James Alexander Green, who introduced them in a paper of 1951...
. According to a footnote in Green 1951, the suggestion that the notion of regularity be applied to semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
s was first made by David Rees
David Rees (mathematician)
David Rees ScD Cantab, FIMA, FRS is an emeritus professor of pure mathematics at the University of Exeter, having been head of the Mathematics / Mathematical Sciences Department at Exeter for many years....
.
The basics
There are two equivalent ways in which to define a regular semigroup S: for each a in S, there is an x in S, which is called a pseudoinverse, with axa = a; 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.
The set of inverses (in the above sense) of an element a in an arbitrary semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
S is denoted by V(a). 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.
A regular semigroup in which idempotents commute is an inverse semigroup
Inverse semigroup
In mathematics, an inverse semigroup S is a semigroup in which every element x in S has a unique inversey in S in the sense that x = xyx and y = yxy...
, 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. 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
Inverse semigroup
In mathematics, an inverse semigroup S is a semigroup in which every element x in S has a unique inversey in S in the sense that x = xyx and y = yxy...
is a regular semigroup in which idempotents commute.
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
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
, and the unique pseudoinverse of an element coincides with the group inverse.
Theorem. The homomorphic image of a regular semigroup is regular.
Examples of regular semigroups:
- Every groupGroup (mathematics)In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
is regular. - Every inverse semigroupInverse semigroupIn mathematics, an inverse semigroup S is a semigroup in which every element x in S has a unique inversey in S in the sense that x = xyx and y = yxy...
is regular. - 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 semigroupBicyclic semigroupIn mathematics, the bicyclic semigroup is an algebraic object important for the structure theory of semigroups. Although it is in fact a monoid, it is usually referred to as simply a semigroup. The first published description of this object was given by Evgenii Lyapin in 1953. Alfred H...
is regular. - Any full transformation semigroupTransformation semigroupIn algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup is associated with the composite of the two corresponding...
is regular. - A Rees matrix semigroup is regular.
Green's relations
Recall that the principal idealPrincipal ideal
In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R.More specifically:...
s of a semigroup S are defined in terms of S1, the semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
with identity adjoined; this is to ensure that an element a belongs to the principal right, left and two-sided ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
s 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
Green's relations
In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. The relations are named for James Alexander Green, who introduced them in a paper of 1951...
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.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.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.
If S is an inverse semigroup
Inverse semigroup
In mathematics, an inverse semigroup S is a semigroup in which every element x in S has a unique inversey in S in the sense that x = xyx and y = yxy...
, then the idempotent in each
- and -class is unique.Special classes of regular semigroups
Some special classes of regular semigroups are:- 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
Class (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...
of generalised inverse semigroups is the intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
of the class of locally inverse semigroups and the class of orthodox semigroups.