Inverse semigroup
In mathematics, 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, 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 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:
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