\begin{definition}[Definition:Normal Subgroup of Monoid]
Let $\struct {S, \circ}$ be a monoid whose identity element is $e$.
Let $\struct {H, \circ}$ be a subgroup of $\struct {S, \circ}$.
$H$ is a '''normal subgroup of $S$''' {{iff}}:
:$e$ is the identity element of $H$
:$\forall s \in S: s \circ H = H \circ s$
where $s \circ H$ denotes the subset product of $s$ with $H$.
\end{definition}