\begin{definition}[Definition:Cancellable Monoid]
Let $\struct {S, \circ}$ be a monoid.
$\struct {S, \circ}$ is defined as being '''cancellable''' {{iff}}:
:$\forall a, b, c \in S: a \circ c = b \circ c \implies a \circ b$
:$\forall a, b, c \in S: a \circ b = a \circ c \implies b \circ c$
That is, {{iff}} $\circ$ is a cancellable operation.
\end{definition}