In semigroup theory, an inverse semigroup (occasionally called an inversion 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, 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. (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 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 (from left to right) on the largest domain upon which it "makes sense" to compose them:
dom α β = [ im α ∩ dom β ] α − 1 {\displaystyle \operatorname {dom} \alpha \beta =[\operatorname {im} \alpha \cap \operatorname {dom} \beta ]\alpha ^{-1}\,}
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 I X {\displaystyle {\mathcal {I}}_{X}} of all partial one-one transformations of a set X forms an inverse semigroup, called the symmetric inverse semigroup (or monoid) on X, with inverse the functional inverse defined from image to domain (equivalently, the converse relation). 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 § Homomorphisms and representations of inverse semigroups 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, 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:
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) and idempotents commute (that is, the idempotents of S form a semilattice). Every L {\displaystyle {\mathcal {L}}} -class and every R {\displaystyle {\mathcal {R}}} -class contains precisely one idempotent, where L {\displaystyle {\mathcal {L}}} and R {\displaystyle {\mathcal {R}}} are two of Green's relations. The idempotent in the L {\displaystyle {\mathcal {L}}} -class of s is s−1s, whilst the idempotent in the R {\displaystyle {\mathcal {R}}} -class of s is ss−1. There is therefore a simple characterisation of Green's relations in an inverse semigroup:
a L b ⟺ a − 1 a = b − 1 b , a R b ⟺ a a − 1 = b b − 1 {\displaystyle a\,{\mathcal {L}}\,b\Longleftrightarrow a^{-1}a=b^{-1}b,\quad a\,{\mathcal {R}}\,b\Longleftrightarrow aa^{-1}=bb^{-1}}
Unless stated otherwise, E(S) 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 (a, b)−1 = (b, a). 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 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 relation ≤ (sometimes denoted by ω), which is defined by the following:
a ≤ b ⟺ a = e b , {\displaystyle a\leq b\Longleftrightarrow a=eb,}
for some idempotent e in S. Equivalently,
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