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.)