A note on isomorphisms between submonoids of ${\mathbb{N}}^k$ and numerical semigroups

Abstract

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.

9 pages