For all $f, g : \mathbb{N} \rightarrow \mathbb{N}$ let
$$ f \leq g \leftrightarrow \exists x {\in} \mathbb{N} \, \forall y {\in} \mathbb{N} \, (x \leq y \rightarrow f(y) \leq g(y)) $$
Let $S \subseteq \mathbb{N} \rightarrow \mathbb{N}$ be the set of natural-coefficient polynomial functions. Then the ordered semiring $(S, \leq, +, \cdot)$ is isomorphic to $(\omega^\omega, \leq, +_H, \cdot_H)$ where $+_H$ and $\cdot_H$ are the Hessenberg sum and product.