Inverse semigroups appear in a range of contexts; for example, they can be employed in the study of partial symmetries.
Inverse semigroups were introduced independently by Viktor Vladimirovich Wagner[3] in the Soviet Union in 1952,[4] and by Gordon Preston in the United Kingdom in 1954.
[5] 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.
In an inverse monoid, xx−1 and x−1x are not necessarily equal to the identity, but they are both idempotent.
There are a number of equivalent characterisations of an inverse semigroup S:[10] The idempotent in the
There is therefore a simple characterisation of Green's relations in an inverse semigroup:[11] Unless stated otherwise, E(S) will denote the semilattice of idempotents of an inverse semigroup S. Multiplication table example.
An inverse semigroup S possesses a natural partial order relation ≤ (sometimes denoted by ω), which is defined by the following:[12] 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.
[13] The natural partial order is compatible with both multiplication and inversion, that is,[14] and In a group, this partial order simply reduces to equality, since the identity is the only idempotent.
[15] The natural partial order on an inverse semigroup interacts with Green's relations as follows: if s ≤ t and s
If E(S) is finite and forms a chain (i.e., E(S) is totally ordered by ≤), then S is a union of groups.
[17] If E(S) is an infinite chain it is possible to obtain an analogous result under additional hypotheses on S and E(S).
, 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 (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.
Let S be an inverse semigroup with semilattice E of idempotents, and minimum group congruence σ.
is the compatibility relation on S, defined by McAlister's Covering Theorem.
Conversely, every E-unitary inverse semigroup is isomorphic to one of this type.
A construction similar to a free group is possible for inverse semigroups.
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.
(see Lawson 1998 for further details) Any free inverse semigroup is F-inverse.
[31] 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.
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.
[32] 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 and whose poset of objects is a meet-semilattice.
The category of sets and partial bijections is the prime example.
[36] Inverse categories have found various applications in theoretical computer science.