\begin{definition}[Definition:Inverse Completion]
Let $\struct {S, \circ}$ be a semigroup.
Let $\struct {C, \circ} \subseteq \struct {S, \circ}$ be the subsemigroup of cancellable elements of $\struct {S, \circ}$.
Let $\struct {T, \circ'}$ be a semigroup defined such that:
:$(1): \quad \struct {S, \circ}$ is a subsemigroup of $\struct {T, \circ'}$
:$(2): \quad$ Every element of $C$ has an inverse in $T$ for $\circ'$
:$(3): \quad \gen {S \cup C^{-1} } = \struct {T, \circ'}$
where:
:$\gen {S \cup C^{-1} }$ denotes the subsemigroup of $\struct {T, \circ'}$ generated by $S \cup C^{-1}$
:$C^{-1}$ denotes the inverse of $C$.
Then $\struct {T, \circ'}$ is called an '''inverse completion''' of $\struct {S, \circ}$.
\end{definition}