Definition:Inverse Completion

Definition
Let $\left({S, \circ}\right)$ be a monoid whose identity is $e$.

Let $\left ({C, \circ}\right) \subseteq \left({S, \circ}\right)$ be the subsemigroup of cancellable elements of $\left({S, \circ}\right)$.

Let $\left({T, \circ'}\right)$ be defined such that:


 * $\left({S, \circ}\right)$ is a subsemigroup of $\left({T, \circ'}\right)$
 * $\forall x \in C: \exists y \in T: x \circ' y = e, y \circ' x = e$, that is, $C^{-1} \subseteq T$ for $\circ'$
 * $\left\langle{S \cup C^{-1}}\right\rangle = \left({T, \circ'}\right)$.

Then $\left({T, \circ'}\right)$ is called an inverse completion of $\left({S, \circ}\right)$.