Definition 1. Let $(M, *)$ be a monoid. $(N, *)$ is a submonoid of $(M, *)$, written $N \leq M$, if $(N, *)$ itself is a monoid and $N \subseteq M$.
Definition 2. Let $M$ be a monoid with neutral element $e$ and $N$ be a submonoid. $N$ is neutral-preserving if the neutral element of $N$ is also $e$.
Example. $(\N, \max)$ is a monoid with neutral element $0$; $(\N^+, \max)$ is a submonoid, but with neutral element $1$, so it is not neutral preserving.
Proposition 1 (Submonoid Test). Let $(M, *)$ be a monoid. $N \subseteq M$ is a submonoid as soon as:
-
(Closure) For all $a$, $b \in N$,
\[ab \in N.\] -
(Unitality) There is $e \in N$ such that for all $a \in N$,
\[ae = ea = a.\]