The Natural Numbers with Zero and Their Embedding into the Integers

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $\mathbb{N}_{0}$ and $0$ be as in The Class Omega of Natural Numbers with Zero §omega and The Class Omega of Natural Numbers with Zero §zero, with the addition, the multiplication and the order of $\mathbb{N}_{0}$, which on $\mathbb{N}$ are those of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion §numbers. Let $\mathbb{Z}$, its operations and order, $0_{\mathbb{Z}}$ and $\iota$ be as in The Integers §integers, The Integers §operations, The Integers §constants and The Integers §embedding.

¶ $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$ and $0\notin\mathbb{N}$.

¶ There is exactly one map $\iota_{0}:\mathbb{N}_{0}\to\mathbb{Z}$ with $\iota_{0}(0)=0_{\mathbb{Z}}$ and $\iota_{0}(n)=\iota(n)$ for every $n\in\mathbb{N}$. It is injective; for all $m,n\in\mathbb{N}_{0}$, $\iota_{0}(m+n)=\iota_{0}(m)+\iota_{0}(n)$, $\iota_{0}(mn)=\iota_{0}(m)\,\iota_{0}(n)$, and $m\le n$ if and only if $\iota_{0}(m)\le\iota_{0}(n)$; and its image is $\{x\in\mathbb{Z}:0_{\mathbb{Z}}\le x\}$.