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

## TL;DR

This paper characterizes when submonoids of $\\mathbb{N}^k$ are isomorphic to submonoids of lower dimensions and proves the uniqueness of isomorphisms for numerical semigroups within $\\mathbb{N}$.

## Contribution

It provides new characterizations of the minimal dimension for isomorphic embeddings of submonoids and establishes the uniqueness of isomorphisms for numerical semigroups.

## Key findings

- Characterization of minimal $r$ such that $H$ is isomorphic to a submonoid of $\\mathbb{N}^r$
- Proof that isomorphic numerical semigroups in $\\mathbb{N}$ are identical
- Uniqueness of the isomorphism as the identity map for 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.