Inverse semigroup

In group theory, an inverse semigroup S is a semigroup in which every element x in S has a unique inverse ''y in S'' in the sense that and, i.e. a regular semigroup in which every element has a unique inverse. Inverse semigroups appear in a range of contexts; for example, they can be employed in the study of partial symmetries.

Origins

Inverse semigroups were introduced independently by Viktor Vladimirovich Wagner in the Soviet Union in 1952, and by Gordon Preston in the United Kingdom in 1954. Both authors arrived at inverse semigroups via the study of partial bijections of a set: a partial transformation α of a set X is a function from A to B, where A and B are subsets of X. Let α and β be partial transformations of a set X; α and β can be composed on the largest domain upon which it "makes sense" to compose them:
where α⁻¹ denotes the preimage under α. Partial transformations had already been studied in the context of pseudogroups. It was Wagner, however, who was the first to observe that the composition of partial transformations is a special case of the composition of binary relations. He recognised also that the domain of composition of two partial transformations may be the empty set, so he introduced an empty transformation to take account of this. With the addition of this empty transformation, the composition of partial transformations of a set becomes an everywhere-defined associative binary operation. Under this composition, the collection of all partial one-one transformations of a set X forms an inverse semigroup, called the symmetric inverse semigroup on X, with inverse the functional inverse defined from image to domain. This is the "archetypal" inverse semigroup, in the same way that a symmetric group is the archetypal group. For example, just as every group can be embedded in a symmetric group, every inverse semigroup can be embedded in a symmetric inverse semigroup.

The basics

The inverse of an element x of an inverse semigroup S is usually written x⁻¹. Inverses in an inverse semigroup have many of the same properties as inverses in a group, for example,. In an inverse monoid, xx⁻¹ and x⁻¹x are not necessarily equal to the identity, but they are both idempotent. An inverse monoid S in which, for all x in S, is, of course, a group.
There are a number of equivalent characterisations of an inverse semigroup S:

Every element of S has a unique inverse, in the above sense.
Every element of S has at least one inverse and idempotents commute.
Every -class and every -class contains precisely one idempotent, where and are two of Green's relations.

The idempotent in the -class of s is s⁻¹s, whilst the
idempotent in the -class of s is ss⁻¹. There is therefore a simple
characterisation of Green's relations in an inverse semigroup:
Unless stated otherwise, E will denote the semilattice of idempotents of an inverse semigroup S.

Examples of inverse semigroups

Partial bijections on a set X form an inverse semigroup under composition.
Every group is an inverse semigroup.
The bicyclic semigroup is inverse, with.
Every semilattice is inverse.
The Brandt semigroup is inverse.
The Munn semigroup is inverse.

Multiplication table example. It is associative and every element has its own inverse according to,. It has no identity and is not commutative.

	a	b	c	d	e
a	a	a	a	a	a
b	a	b	c	a	a
c	a	a	a	b	c
d	a	d	e	a	a
e	a	a	a	d	e

The natural partial order

An inverse semigroup S possesses a natural partial order relation ≤,
which is defined by the following:
for some idempotent e in S. Equivalently,
for some idempotent f in S. In fact, e can be taken to be
aa⁻¹ and f to be a⁻¹a.
The natural partial order is compatible with both multiplication and inversion, that is,
and
In a group, this partial order simply reduces to equality, since the identity is the only idempotent. In a symmetric inverse semigroup, the partial order reduces to restriction of mappings, i.e., if, and only if, the domain of α is contained in the domain of β and, for all x in the domain of α.
The natural partial order on an inverse semigroup interacts with Green's relations as follows: if and s''t'', then. Similarly, if.
On E, the natural partial order becomes:
so, since the idempotents form a semilattice under the product operation, products on E give least upper bounds with respect to ≤.
If E is finite and forms a chain, then S is a union of groups. If E is an infinite chain it is possible to obtain an analogous result under additional hypotheses on S and E.

Homomorphisms and representations of inverse semigroups

A homomorphism of inverse semigroups is defined in exactly the same way as for any other
semigroup: for inverse semigroups S and T, a function θ from S to T
is a morphism if, for all s,''t in S''. The definition of a
morphism of inverse semigroups could be augmented by including the condition, however, there is no need to do so, since this property follows from the above definition, via the following theorem:
Theorem. The homomorphic image of an inverse semigroup is an inverse semigroup; the inverse of an element is always mapped to the inverse of the image of that element.
One of the earliest results proved about inverse semigroups was the Wagner–Preston Theorem, which is an analogue of Cayley's theorem for groups:
Wagner–Preston Theorem. If S is an inverse semigroup, then the function φ
from S to, given by
is a faithful representation of S.
Thus, any inverse semigroup can be embedded in a symmetric inverse semigroup, and with image closed under the inverse operation on partial bijections. Conversely, any subsemigroup of the symmetric inverse semigroup closed under the inverse operation is an inverse semigroup. Hence a semigroup S is isomorphic to a subsemigroup of the symmetric inverse semigroup closed under inverses if and only if S is an inverse semigroup.

Congruences on inverse semigroups

are defined on inverse semigroups in exactly the same way as for any other semigroup: a
congruence ''ρ is an equivalence relation that is compatible with semigroup multiplication, i.e.,
Of particular interest is the relation, defined on an inverse semigroup S'' by
It can be shown that σ is a congruence and, in fact, it is a group congruence, meaning that the factor semigroup S/''σ is a group. In the set of all group congruences on a semigroup S'', the minimal element need not be the smallest element. In the specific case in which S is an inverse semigroup σ is the smallest congruence on S such that S/''σ is a group, that is, if τ'' is any other congruence on S with S/''τ a group, then σ'' is contained in τ. The congruence σ is called the minimum group congruence on S. The minimum group congruence can be used to give a characterisation of E-unitary inverse semigroups.
A congruence ρ on an inverse semigroup S is called idempotent pure if

''E''-unitary inverse semigroups

One class of inverse semigroups that has been studied extensively over the years is the class of E-unitary inverse semigroups: an inverse semigroup S is E-''unitary if, for all e'' in E and all s in S,
Equivalently,
One further characterisation of an E-unitary inverse semigroup S is the following: if e is in E and
e ≤ s, for some s in S, then s is in E.
Theorem. Let S be an inverse semigroup with semilattice E of idempotents, and minimum group congruence σ. Then the following are equivalent:

S is E-unitary;
σ is idempotent pure;
= σ,

where is the compatibility relation on S, defined by
McAlister's Covering Theorem. Every inverse semigroup S has a E-unitary cover; that is there exists an idempotent separating surjective homomorphism from some E-unitary semigroup T onto S.
Central to the study of E-unitary inverse semigroups is the following construction. Let be a partially ordered set, with ordering ≤, and let be a subset of with the properties that

is a lower semilattice, that is, every pair of elements A, B in has a greatest lower bound A ''B in ;
is an order ideal of, that is, for A'', B in, if A is in and, then B is in.

Now let G be a group that acts on , such that

for all g in G and all A, B in, if, and only if, ;
for each g in G and each B in, there exists an A in such that ;
for all A, B in, if, and only if, ;
for all g, h in G and all A in,.

The triple is also assumed to have the following properties:

for every X in, there exists a g in G and an A in such that ;
for all g in G, g and have nonempty intersection.

Such a triple is called a McAlister triple. A McAlister triple is used to define the following:
together with multiplication
Then is an inverse semigroup under this multiplication, with. One of the main results in the study of E-unitary inverse semigroups is McAlister's P-Theorem:
McAlister's P-Theorem. Let be a McAlister triple. Then is an E-unitary inverse semigroup. Conversely, every E-unitary inverse semigroup is isomorphic to one of this type.