Remark

Let $M$ be the multiplicative monoid of strictly positive integers and let $\mathfrak{P}$ be the set of prime numbers (§ 4, no. 10, Definition 15). By Proposition 10 there exists a homomorphism $u$ of $\mathbf{N}^{(\mathfrak{P})}$ into $M$ characterized by $u(\delta_p) = p$ for every prime number $p$. Then $u(\alpha) = \prod_{p \in \mathfrak{P}} p^{\alpha(p)}$ for $\alpha$ in $\mathbf{N}^{(\mathfrak{P})}$ and Theorem 7 of § 4, no. 10 shows that $u$ is an isomorphism of $\mathbf{N}^{(\mathfrak{P})}$ onto $H$.