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 $$\left({S, \circ}\right)$$.