Inverse semigroup
Encyclopedia
In mathematics, 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 x = xyx and y = yxy. Inverse semigroups appear in a range of contexts; for
example, they can be employed in the study of partial symmetries.
(The convention followed in this article will be that of writing a function on the right of its argument, and
composing functions from left to right — a convention often observed in semigroup theory.)
in 1952, and by Gordon Preston
in Great Britain
in 1954. Both authors arrived at inverse semigroups via the study of partial one-one transformations 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 (from left to right) 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 pseudogroup
s. It was Wagner, however, who
was the first to observe that the composition of partial transformations is a special case of the multiplication 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 (or monoid) on X. 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 (see below).
inverse semigroup have many of the same properties as inverses in a group
, for example,
(ab)−1 = b−1a−1. 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 xx−1 = 1 =
x−1x, for all x in S (a unipotent inverse monoid), is, of course, a group
.
There are a number of equivalent characterisations of an inverse semigroup S:
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:
Examples of inverse semigroups:
Unless stated otherwise, E(S) will denote the semilattice of idempotents of an inverse semigroup S.
which is defined by the following:
for some idempotent e in S. Equivalently,
for some (in general, different) 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., α ≤ β if, and only if, the domain of α is contained in the domain of β and
xα = xβ, for all x in the domain of α.
The natural partial order on an inverse semigroup interacts with Green's relations
as follows: if s ≤
t and s
t, then s = t. Similarly, if
s
t.
On E(S), the natural partial order becomes:
so the product of any two idempotents in S is equal to the lesser of the two, with respect to
≤. If E(S) forms a chain (i.e., E(S) is totally ordered by ≤), then
S is a union
of groups
.
(or morphism) 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 (sθ)(tθ) = (st)θ, for all s,t in S. The definition of a
morphism of inverse semigroups could be augmented by including the condition (sθ)−1 =
s−1θ, 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.
congruence ρ is an equivalence relation
which is compatible with semigroup multiplication, i.e.,
Of particular interest is the relation
, defined on an inverse semigroup S by
there exists a 
with 
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 (for the partial order defined by inclusion of sets) 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 (see below).
A congruence ρ on an inverse semigroup S is called idempotent pure if
inverse semigroups: an inverse semigroup S (with semilattice
E of idempotents) 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:
where
is the compatibility relation on S, defined by
are idempotent.
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
Now let G be a group
which acts
on
(on the left), such that
The triple
is also assumed to have the following properties:
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
(A,g)−1 = (g−1A, g−1). 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.
Theorem. Every inverse semigroup has an F-inverse cover.
McAlister's P-theorem has been used to characterize F-inverse semigroups as well. A McAlister triple
is an F-inverse semigroups if and only if 
is a principal ideal of 
and 
is a semilattice.
is possible for inverse semigroups. A presentation
of the free inverse semigroup on a set X may be obtained by considering the free semigroup with involution, where involution is the taking of the inverse, and then taking the quotient by the Vagner congruence
The word problem
for free inverse semigroups is much more intricate than that of free groups. A celebrated result in this area due to W. D. Munn who showed that elements of the free inverse semigroup can be naturally regarded as trees, known as Munn trees. Multiplication in the free inverse semigroup has a correspondent on Munn trees, which essentially consists of overlapping common portions of the trees. (see Lawson 1998 for further details)
Any free inverse semigroup is F-inverse.
another way of composing partial transformations, which is more restrictive than that used above: two partial
transformations α and β are composed if, and only if, the image of α is equal to the domain of
β; otherwise, the composition αβ is undefined. Under this alternative composition, the collection
of all partial one-one transformations of a set forms not an inverse semigroup but an inductive groupoid, in the
sense of category theory
. This close connection between inverse semigroups and inductive groupoids is
embodied in the Ehresmann-Schein-Nambooripad Theorem, which states that an inductive groupoid can always be
constructed from an inverse semigroup, and conversely.
,
and (2) the idempotents in S commute; this has led to two distinct classes of generalisations of
an inverse semigroup: semigroups in which (1) holds, but (2) does not, and vice versa.
Examples of regular generalisations of an inverse semigroup are:
The class
of generalised inverse semigroups is the intersection
of the class of
locally inverse semigroups and the class of orthodox semigroups.
Amongst the non-regular generalisations of an inverse semigroup are:http://www-users.york.ac.uk/~varg1/finitela.ps
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...
in which every element x in S has a unique inverse
y in S in the sense that x = xyx and y = yxy. Inverse semigroups appear in a range of contexts; for
example, they can be employed in the study of partial symmetries.
(The convention followed in this article will be that of writing a function on the right of its argument, and
composing functions from left to right — a convention often observed in semigroup theory.)
Origins
Inverse semigroups were introduced independently by Viktor Vladimirovich Wagner in the Soviet UnionSoviet Union
The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....
in 1952, and by Gordon Preston
Gordon Preston
Gordon Bamford Preston is an English mathematician who is known for his work on semigroups. He received his D.Phil. in mathematics in 1954 from the University of Oxford.He was born in Workington and brought up in Carlisle...
in Great Britain
Great Britain
Great Britain or Britain is an island situated to the northwest of Continental Europe. It is the ninth largest island in the world, and the largest European island, as well as the largest of the British Isles...
in 1954. Both authors arrived at inverse semigroups via the study of partial one-one transformations of a set: a partial transformation
Partial function
In mathematics, a partial function from X to Y is a function ƒ: X' → Y, where X' is a subset of X. It generalizes the concept of a function by not forcing f to map every element of X to an element of Y . If X' = X, then ƒ is called a total function and is equivalent to a function...
α of a set X is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
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 (from left to right) on the largest domain
Domain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
upon
which it "makes sense" to compose them:
- dom αβ = [im α
dom β]α−1
where α−1 denotes the preimage under α. Partial transformations had already been studied
in the context of pseudogroup
Pseudogroup
In mathematics, a pseudogroup is an extension of the group concept, but one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra...
s. It was Wagner, however, who
was the first to observe that the composition of partial transformations is a special case of the multiplication of
binary relations. He recognised also that the domain of composition of two partial
transformations may be the empty set
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
, 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
Binary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
. Under this composition, the collection
of all partial one-one transformations of a set X forms an inverse semigroup, calledthe symmetric inverse semigroup (or monoid) on X. This is the "archetypal"
inverse semigroup, in the same way that a symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
is the archetypal 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...
. For
example, just as every 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...
can be embedded in a symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
, every inverse
semigroup can be embedded in a symmetric inverse semigroup (see below).
The basics
The inverse of an element x of an inverse semigroup S is usually written x−1. Inverses in aninverse semigroup have many of the same properties as inverses in 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...
, for example,
(ab)−1 = b−1a−1. In an inverse monoid
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
, xx−1 and
x−1x are not (necessarily) equal to the identity, but they are both idempotent. An inverse monoid S in which xx−1 = 1 =
x−1x, for all x in S (a unipotent inverse monoid), is, of course, 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...
.
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 (S is a regular semigroupRegular semigroupA 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.- Origins...
) and idempotents commute (that is, the idempotents of S form a semilatticeSemilatticeIn mathematics, a join-semilattice is a partially ordered set which has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for any nonempty finite subset...
). - Every
-class and every 
-class contains precisely one idempotent, where 
and 
are two of Green's relationsGreen's relationsIn 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...
.
The idempotent in the
-class of s is s−1s, whilst theidempotent in the
-class of s is ss−1. There is therefore a simplecharacterisation of 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...
in an inverse semigroup:
Examples of inverse 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 an inverse semigroup. - 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 inverse, with (a,b)−1 = (b,a). - Every semilatticeSemilatticeIn mathematics, a join-semilattice is a partially ordered set which has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for any nonempty finite subset...
is inverse. - The Brandt semigroupBrandt semigroupIn mathematics, Brandt semigroups are completely 0-simple inverse semigroups. In other words, they are semigroups without proper ideal and which are also inverse semigroups. They are build in the same way that completely 0-simple semigroups:...
is inverse. - The Munn semigroup is inverse.
Unless stated otherwise, E(S) will denote the semilattice of idempotents of an inverse semigroup S.
The natural partial order
An inverse semigroup S possesses a natural partial order relation ≤ (sometimes denoted by ω)which is defined by the following:
for some idempotent e in S. Equivalently,
for some (in general, different) 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
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...
, 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., α ≤ β if, and only if, the domain of α is contained in the domain of β and
xα = xβ, for all x in the domain of α.
The natural partial order on an inverse semigroup interacts with 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...
as follows: if s ≤
t and s
t, then s = t. Similarly, ifs
t.On E(S), the natural partial order becomes:
so the product of any two idempotents in S is equal to the lesser of the two, with respect to
≤. If E(S) forms a chain (i.e., E(S) is totally ordered by ≤), then
S is a union
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
of groups
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...
.
Homomorphisms and representations of inverse semigroups
A homomorphismHomomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
(or morphism) of inverse semigroups is defined in exactly the same way as for any other
semigroup: for inverse semigroups S and T, a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
θ from S to T
is a morphism if (sθ)(tθ) = (st)θ, for all s,t in S. The definition of a
morphism of inverse semigroups could be augmented by including the condition (sθ)−1 =
s−1θ, however, there is no need to do so, since this property follows from the above
definition, via the following theorem:
Theorem. The homomorphic image
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
of an inverse semigroup is an inverse semigroup; the
inverse of an element is always mapped to the inverse of the image
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
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
Cayley's theorem
In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting on G...
for groups
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...
:
Wagner-Preston Theorem. If S is an inverse semigroup, then the function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
φ
from S to
, given by
- dom (aφ) = Sa−1 and x(aφ) = xa
is a faithful representation
Faithful representation
In mathematics, especially in the area of abstract algebra known as representation theory, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings ρ.In more abstract language, this means...
of S.
Thus, any inverse semigroup can be embedded in a symmetric inverse semigroup.
Congruences on inverse semigroups
Congruences are defined on inverse semigroups in exactly the same way as for any other semigroup: acongruence ρ is an equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
which is compatible with semigroup multiplication, i.e.,
Of particular interest is the relation
, defined on an inverse semigroup S by there exists a with 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 (for the partial order defined by inclusion of sets) 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 (see below).
A congruence ρ on an inverse semigroup S is called idempotent pure if
E-unitary inverse semigroups
One class of inverse semigroups which has been studied extensively over the years is the class of E-unitaryinverse semigroups: an inverse semigroup S (with semilattice
Semilattice
In mathematics, a join-semilattice is a partially ordered set which has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for any nonempty finite subset...
E of idempotents) 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
Semilattice
In mathematics, a join-semilattice is a partially ordered set which has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for any nonempty finite subset...
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 are idempotent.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 setPartially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
, with ordering ≤, and let
be a subsetSubset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of
with the properties that
is a lower semilatticeSemilatticeIn mathematics, a join-semilattice is a partially ordered set which has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for any nonempty finite subset...
, that is, every pair of elements A, B in
has a greatest lower bound A 
B in 
(with respect to ≤);
is an order ideal of 
, that is, for A, B in 
, if A is in 
and B ≤ A, then B is in 
.
Now let G be 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...
which acts
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
on
(on the left), such that
- for all g in G and all A, B in
, gA = gB if, and only if, A = B; - for each g in G and each B in
, there exists an A in 
such that gA = B; - for all A, B in
, A ≤ B if, and only if, gA ≤ gB; - for all g, h in G and all A in
, g(hA) = (gh)A.
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 gA = X; - for all g in G, g
and 
have nonempty intersection.
Such a triple
is called a McAlister triple. A McAlister triple isused to define the following:
together with multiplication
.Then
is an inverse semigroup under this multiplication, with(A,g)−1 = (g−1A, g−1). 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 inversesemigroup is isomorphic to one of this type.
F-inverse semigroups
An inverse semigroup is said to be F-inverse if every element has a unique maximal element above it in the natural partial order, i.e. ever σ-class has a maximal element. Every F-inverse semigroup is an E-unitary monoid. McAlister's covering theorem has been refined by M.V. Lawson to:Theorem. Every inverse semigroup has an F-inverse cover.
McAlister's P-theorem has been used to characterize F-inverse semigroups as well. A McAlister triple
is an F-inverse semigroups if and only if is a principal ideal of and is a semilattice.Free inverse semigroups
A construction similar to a free groupFree group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
is possible for inverse semigroups. A presentation
Presentation
Presentation is the practice of showing and explaining the content of a topic to an audience or learner. Presentations come in nearly as many forms as there are life situations...
of the free inverse semigroup on a set X may be obtained by considering the free semigroup with involution, where involution is the taking of the inverse, and then taking the quotient by the Vagner congruence
The word problem
Word problem
The term word problem has several meanings:* word problem is a type of textbook problem designed to help students apply abstract mathematical concepts to "real-world" situations...
for free inverse semigroups is much more intricate than that of free groups. A celebrated result in this area due to W. D. Munn who showed that elements of the free inverse semigroup can be naturally regarded as trees, known as Munn trees. Multiplication in the free inverse semigroup has a correspondent on Munn trees, which essentially consists of overlapping common portions of the trees. (see Lawson 1998 for further details)
Any free inverse semigroup is F-inverse.
Connections with category theory
The above composition of partial transformations of a set gives rise to a symmetric inverse semigroup. There isanother way of composing partial transformations, which is more restrictive than that used above: two partial
transformations α and β are composed if, and only if, the image of α is equal to the domain of
β; otherwise, the composition αβ is undefined. Under this alternative composition, the collection
of all partial one-one transformations of a set forms not an inverse semigroup but an inductive groupoid, in the
sense of category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
. This close connection between inverse semigroups and inductive groupoids is
embodied in the Ehresmann-Schein-Nambooripad Theorem, which states that an inductive groupoid can always be
constructed from an inverse semigroup, and conversely.
Generalisations of inverse semigroups
As noted above, an inverse semigroup S can be defined by the conditions (1) S is a regular semigroupRegular semigroup
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.- Origins...
,
and (2) the idempotents in S commute; this has led to two distinct classes of generalisations of
an inverse semigroup: semigroups in which (1) holds, but (2) does not, and vice versa.
Examples of regular generalisations of an inverse semigroup are:
- Regular semigroupRegular semigroupA 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.- Origins...
s: a semigroupSemigroupIn 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 regular if every element has at least one inverse; equivalently, for each a in S, there is an x in S such that axa = a. - Locally inverse semigroups: a regular semigroupRegular semigroupA 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.- Origins...
S is locally inverse if eSe is an inverse semigroup, for each idempotent e. - Orthodox semigroups: a regular semigroupRegular semigroupA 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.- Origins...
S is orthodox if its subset of idempotents forms a subsemigroup. - Generalised inverse semigroups: a regular semigroupRegular semigroupA 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.- Origins...
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.
Amongst the non-regular generalisations of an inverse semigroup are:http://www-users.york.ac.uk/~varg1/finitela.ps
- (Left, right, two-sided) adequate semigroups.
- (Left, right, two-sided) ample semigroups.
- (Left, right, two-sided) semiadequate semigroups.
- Weakly (left, right, two-sided) ample semigroups.
See also
- Biordered setBiordered setA biordered set is a mathematical object that occurs in the description of the structure of the set of idempotents in a semigroup. The concept and the terminology were developed by K S S Nambooripad of Kerala, India, in the early 1970s....
- PseudogroupPseudogroupIn mathematics, a pseudogroup is an extension of the group concept, but one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra...
- Partial symmetries
- Regular semigroupRegular semigroupA 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.- Origins...
- SemilatticeSemilatticeIn mathematics, a join-semilattice is a partially ordered set which has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for any nonempty finite subset...
- Green's relationsGreen's relationsIn 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...
- Category theoryCategory theoryCategory theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
- Special classes of semigroupsSpecial classes of semigroupsIn mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions...
- Weak inverseWeak inverse- Theory of semigroups :In the theory of semigroups, a weak inverse of an element x in a semigroup is an element y such that .An element x of S for which there is an element y of S such that is called regular...
- Nambooripad orderNambooripad orderIn mathematics, Nambooripad order is a certain natural partial order on a regular semigroup discovered by K S S Nambooripad in late seventies...
Further reading
- For a brief introduction to inverse semigroups, see either Clifford & Preston 1967 : Chapter 7 or Howie 1995 : Chapter 5.
- More comprehensive introductions can be found in Petrich 1984 and Lawson 1998.