Let $(M, *)$ be a monoid. $(N, *)$ is a submonoid of $(M, *)$, written $N \leq M$, if $(N, *)$ itself is a monoid and $N \subseteq M$.
Our definition does not require a submonoid to inherit the neutral element.
Example. $(\N, \max)$ is a monoid with neutral element $0$; $(\N^+, \max)$ is a submonoid with neutral element $1$.
Let $(M, *)$ be a monoid. $N \subseteq M$ is a submonoid as soon as:
(Closure) For all $a$, $b \in N$,
\[ab \in N.\](Neutral element) There is $e \in N$ such that for all $a \in N$,
\[ae = ea = a.\]