  Given a submonoid $H$ of $\mathbb N^k$, we give some characterizations of the
minimum $r\in \mathbb N^+$ such that $H$ is isomorphic to a submonoid of
$\mathbb N^r$. In the context of submonoids of $\mathbb N$, we prove that if
two numerical semigroups are isomorphic submonoids of $\mathbb N$, then they
are equal and the identity map is the unique isomorphism between them.
