Submonoid

PROP-SMO-A.

Let $(M,*)$ be a monoid. $N\subseteq M$ is a submonoid as soon as

1. $N$ is closed under $*$;
2. $N$ is unital.

REM-SMO-NEU.

A submonoid does not always inherit the neutral element.

Example.$(\N,\max)$ is a monoid with neutral element $0$, but $(\N^+,\max)$ is a submonoid with neutral element $1$.