Order on the Natural Numbers

Let $\mathbb{N}$ be the set of natural numbers, with addition $+$ as in that definition, and let $m,n\in\mathbb{N}$.

We write

if there exists $k\in\mathbb{N}$ with $n=m+k$, and we write

if $m<n$ or $m=n$. We also write $n>m$ for $m<n$, and $n\ge m$ for $m\le n$.