Definition:Inverse Semigroup

Let $$\left({S, \circ}\right)$$ be a semigroup such that:

$$\forall a \in S: \exists! b \in S: a = a \circ b \circ a, b = b \circ a \circ b$$

Such a structure is known as an inverse semigroup, and the element $$b$$ is called the inverse of $$a$$.