Inverse semigroup

In group 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.) 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: where α−1 denotes the 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 (or monoid) on X, with inverse the functional inverse defined from image to domain (equivalently, the converse relation).