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In
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
theory, an inverse semigroup (occasionally called an inversion semigroup) ''S'' is a
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'' ...
in which every element ''x'' in ''S'' has a unique ''inverse'' ''y'' in ''S'' in the sense that ''x = xyx'' and ''y = yxy'', i.e. a
regular semigroup In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of 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 __NOTOC__ In abstract algebra, the set of all partial bijections on a set ''X'' ( one-to-one partial transformations) forms an inverse semigroup, called the symmetric inverse semigroup (actually a monoid) on ''X''. The conventional notation for the ...
. (The convention followed in this article will be that of writing a function on the right of its argument, e.g. ''x f'' rather than ''f(x)'', 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 Union The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, ...
in 1952, and by
Gordon Preston Gordon Bamford Preston (28 April 1925 – 14 April 2015) was an English mathematician best known for his work on semigroups. He received his D.Phil. in mathematics in 1954 from Magdalen College, Oxford. He was born in Workington and broug ...
in the
United Kingdom The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain, is a country in Europe, off the north-western coast of the continental mainland. It comprises England, Scotland, Wales and No ...
in 1954. Both authors arrived at inverse semigroups via the study of
partial bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
s 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 Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function ** Domain of holomorphy of a function * ...
upon which it "makes sense" to compose them: : \operatorname\alpha\beta = operatorname \alpha \cap \operatorname \beta\alpha^ \, where ''α''−1 denotes the
preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through ...
under ''α''. Partial transformations had already been studied in the context of
pseudogroup In mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a group, originating however from the geometric approach of Sophus Li ...
s. 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 In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
, 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 In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...
. Under this composition, the collection \mathcal_X of all partial one-one transformations of a set ''X'' forms an inverse semigroup, called the ''
symmetric inverse semigroup __NOTOC__ In abstract algebra, the set of all partial bijections on a set ''X'' ( one-to-one partial transformations) forms an inverse semigroup, called the symmetric inverse semigroup (actually a monoid) on ''X''. The conventional notation for t ...
'' (or monoid) on ''X'', with inverse the functional inverse defined from image to domain (equivalently, the
converse relation In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent&nb ...
). This is the "archetypal" inverse semigroup, in the same way that a
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
is the archetypal
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
. For example, just as every
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
can be embedded in a
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
, 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 an inverse semigroup have many of the same properties as inverses in a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
, for example, (''ab'')−1 = ''b''−1''a''−1. In an inverse
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids a ...
, ''xx''−1 and ''x''−1''x'' are not necessarily equal to the identity, but they are both
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of p ...
. An inverse monoid ''S'' in which ''xx''−1 = 1 = ''x''−1''x'', for all ''x'' in ''S'' (a ''unipotent'' inverse monoid), is, of course, a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
. 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 semigroup In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of semigroup ...
) and
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of p ...
s commute (that is, the
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of p ...
s of ''S'' form a
semilattice In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a ...
). * Every \mathcal-class and every \mathcal-class contains precisely one
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of p ...
, where \mathcal and \mathcal are two of
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 195 ...
. The
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of p ...
in the \mathcal-class of ''s'' is ''s''−1''s'', whilst the
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of p ...
in the \mathcal-class of ''s'' is ''ss''−1. There is therefore a simple characterisation of
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 195 ...
in an inverse semigroup: :a\,\mathcal\,b\Longleftrightarrow a^a=b^b,\quad a\,\mathcal\,b\Longleftrightarrow aa^=bb^ Unless stated otherwise, ''E(S)'' will denote the semilattice of idempotents of an inverse semigroup ''S''.


Examples of inverse semigroups

* Partial
bijections In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
on a set ''X'' form an inverse semigroup under composition. * Every
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
is an inverse semigroup. * The
bicyclic semigroup In 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. It is perhaps most easily understood as the syntactic m ...
is inverse, with (''a'',''b'')−1 = (''b'',''a''). * Every
semilattice In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a ...
is inverse. * The
Brandt semigroup In mathematics, Brandt semigroups are completely 0-simple inverse semigroups. In other words, they are semigroups without proper ideals and which are also inverse semigroups. They are built in the same way as completely 0-simple semigroups: Let '' ...
is inverse. * The
Munn semigroup In mathematics, the Munn semigroup is the inverse semigroup of isomorphisms between principal ideals of a semilattice (a commutative semigroup of idempotents). Munn semigroups are named for the Scottish mathematician Walter Douglas Munn (1929–200 ...
is inverse. Multiplication table example. It is associative and every element has its own inverse according to aba = a, bab = b. It has no identity and is not commutative.


The natural partial order

An inverse semigroup ''S'' possesses a ''natural
partial order In mathematics, especially order theory, a partially ordered set (also poset) 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 ≤ (sometimes denoted by ω), which is defined by the following: :a \leq b \Longleftrightarrow a=eb, for some
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of p ...
''e'' in ''S''. Equivalently, :a \leq b \Longleftrightarrow a=bf, for some (in general, different)
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of p ...
''f'' in ''S''. In fact, ''e'' can be taken to be ''aa''−1 and ''f'' to be ''a''−1''a''. The natural
partial order In mathematics, especially order theory, a partially ordered set (also poset) 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 ...
is compatible with both multiplication and inversion, that is, :a \leq b, c \leq d \Longrightarrow ac \leq bd and :a \leq b \Longrightarrow a^ \leq b^. In a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
, this
partial order In mathematics, especially order theory, a partially ordered set (also poset) 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 ...
simply reduces to equality, since the identity is the only
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of p ...
. In a symmetric inverse semigroup, the
partial order In mathematics, especially order theory, a partially ordered set (also poset) 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 ...
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 ''x''α = ''x''β, for all ''x'' in the domain of α. The natural partial order on an inverse semigroup interacts with
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 195 ...
as follows: if ''s'' ≤ ''t'' and ''s''\,\mathcal\,''t'', then ''s'' = ''t''. Similarly, if ''s''\,\mathcal\,''t''. On ''E(S)'', the natural
partial order In mathematics, especially order theory, a partially ordered set (also poset) 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 ...
becomes: :e \leq f \Longleftrightarrow e = ef, so, since the
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of p ...
s form a semilattice under the product operation, products on ''E(S)'' give least upper bounds with respect to ≤. If ''E(S)'' is finite and forms a
chain A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A ...
(i.e., ''E(S)'' is
totally ordered In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexiv ...
by ≤), then ''S'' is a
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''U ...
of groups. If ''E(S)'' is an infinite
chain A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A ...
it is possible to obtain an analogous result under additional hypotheses on ''S'' and ''E(S).''


Homomorphisms and representations of inverse semigroups

A
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...
(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''−1''θ'', however, there is no need to do so, since this property follows from the above definition, via the following theorem: Theorem. The homomorphic
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of an inverse semigroup is an inverse semigroup; the inverse of an element is always mapped to the inverse of the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
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 In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group is isomorphic to a subgroup of a symmetric group. More specifically, is isomorphic to a subgroup of the symmetric group \operatorname(G) whose ele ...
for groups: Wagner–Preston Theorem. If ''S'' is an inverse semigroup, then the function φ from ''S'' to \mathcal_S, given by :dom (''a''φ) = ''Sa''−1 and ''x''(''a''φ) = ''xa'' is a faithful
representation Representation may refer to: Law and politics * Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a ...
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

Congruences In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done w ...
are defined on inverse semigroups in exactly the same way as for any other semigroup: a ''congruence'' ''ρ'' is an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
that is compatible with semigroup multiplication, i.e., :a\,\rho\,b,\quad c\,\rho\,d\Longrightarrow ac\,\rho\,bd. Of particular interest is the relation \sigma, defined on an inverse semigroup ''S'' by :a\,\sigma\,b\Longleftrightarrow there exists a c\in S with c\leq a,b. 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 :a\in S, e\in E(S), a\,\rho\,e\Longrightarrow a\in E(S).


''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'' (with
semilattice In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a ...
''E'' of
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of p ...
s) is ''E''-''unitary'' if, for all ''e'' in ''E'' and all ''s'' in ''S'', :es \in E \Longrightarrow s \in E. Equivalently, :se \in E \Rightarrow s \in E. 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 In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a ...
''E'' of idempotents, and minimum group congruence ''σ''. Then the following are equivalent: *''S'' is ''E''-unitary; *''σ'' is idempotent pure; *\sim = ''σ'', where \sim is the ''compatibility relation'' on ''S'', defined by :a\sim b\Longleftrightarrow ab^,a^b 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 \mathcal be a
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) 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 ...
, with ordering ≤, and let \mathcal be a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of \mathcal with the properties that *\mathcal is a lower semilattice, that is, every pair of elements ''A'', ''B'' in \mathcal has a
greatest lower bound In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
''A'' \wedge ''B'' in \mathcal (with respect to ≤); *\mathcal is an
order ideal In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different no ...
of \mathcal, that is, for ''A'', ''B'' in \mathcal, if ''A'' is in \mathcal and ''B'' ≤ ''A'', then ''B'' is in \mathcal. Now let ''G'' be a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
that
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
on \mathcal (on the left), such that *for all ''g'' in ''G'' and all ''A'', ''B'' in \mathcal, ''gA'' = ''gB'' if, and only if, ''A'' = ''B''; *for each ''g'' in ''G'' and each ''B'' in \mathcal, there exists an ''A'' in \mathcal such that ''gA'' = ''B''; *for all ''A'', ''B'' in \mathcal, ''A'' ≤ ''B'' if, and only if, ''gA'' ≤ ''gB''; *for all ''g'', ''h'' in ''G'' and all ''A'' in \mathcal, ''g''(''hA'') = (''gh'')''A''. The triple (G, \mathcal, \mathcal) is also assumed to have the following properties: *for every ''X'' in \mathcal, there exists a ''g'' in ''G'' and an ''A'' in \mathcal such that ''gA'' = ''X''; *for all ''g'' in ''G'', ''g''\mathcal and \mathcal have nonempty intersection. Such a triple (G, \mathcal, \mathcal) is called a ''McAlister triple''. A McAlister triple is used to define the following: :P(G, \mathcal, \mathcal) = \ together with multiplication :(A,g)(B,h)=(A \wedge gB, gh). Then P(G, \mathcal, \mathcal) is an inverse semigroup under this multiplication, with (''A'',''g'')−1 = (''g''−1''A'', ''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 (G, \mathcal, \mathcal) be a McAlister triple. Then P(G, \mathcal, \mathcal) is an ''E''-unitary inverse semigroup. Conversely, every ''E''-unitary inverse semigroup is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
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. every ''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 (G, \mathcal, \mathcal) is an ''F''-inverse semigroup if and only if \mathcal is a principal ideal of \mathcal and \mathcal is a semilattice.


Free inverse semigroups

A construction similar to a
free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1 ...
is possible for inverse semigroups. A
presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
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.


Connections with category theory

The above composition of partial transformations of a set gives rise to a symmetric inverse semigroup. There is 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 Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cat ...
. 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. More precisely, an inverse semigroup is precisely a groupoid in the category of posets that is an étale groupoid with respect to its (dual)
Alexandrov topology In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any ''finite'' family of open sets is open; in Alexandrov topologies the finite restr ...
and whose poset of objects is a meet-semilattice.


Generalisations of inverse semigroups

As noted above, an inverse semigroup ''S'' can be defined by the conditions (1) ''S'' is a
regular semigroup In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of semigroup ...
, and (2) the
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of p ...
s 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 semigroup In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of semigroup ...
s'': a
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'' ...
''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 semigroup In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of semigroup ...
''S'' is ''locally inverse'' if ''eSe'' is an inverse semigroup, for each
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of p ...
''e''. *''Orthodox semigroups'': a
regular semigroup In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of semigroup ...
''S'' is ''orthodox'' if its subset of
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of p ...
s forms a subsemigroup. *''Generalised inverse semigroups'': a
regular semigroup In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of semigroup ...
''S'' is called a ''generalised inverse semigroup'' if its
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of p ...
s form a normal band, i.e., ''xyzx'' = ''xzyx'', for all
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of p ...
s ''x'', ''y'', ''z''. The class of generalised inverse semigroups is the
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
of the class of locally inverse semigroups and the class of orthodox semigroups. Amongst the non-regular generalisations of an inverse semigroup are:, *(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.


Inverse category

This notion of inverse also readily generalizes to
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) * Category (Kant) *Categories (Peirce) * ...
. An inverse category is simply a category in which every
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
has a generalized inverse such that and . An inverse category is selfdual. The category of sets and
partial bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
s is the prime example. Inverse categories have found various applications in
theoretical computer science computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the ...
.


See also


Notes


References

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English translation
PDF) *


Further reading

*For a brief introduction to inverse semigroups, see either or . *More comprehensive introductions can be found in and . *{{Cite journal , doi = 10.1017/S0013091512000211, title = On inverse categories and transfer in cohomology, journal = Proceedings of the Edinburgh Mathematical Society, volume = 56, pages = 187, year = 2012, last1 = Linckelmann , first1 = M. , url = http://openaccess.city.ac.uk/7351/1/invcat.pdf}
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Algebraic structures Semigroup theory