# Definition:Inverse Semigroup

## Definition

An inverse semigroup is a semigroup $\struct {S, \circ}$ such that:

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

### Inverse

Let $a\in S$.

The inverse of $a$ is the unique element $b\in S$ such that:

$a = a \circ b \circ a$ and $b = b \circ a \circ b$