Definition:Inverse Semigroup

Definition
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 $a$.