$\mathbb{N}$-polyregular functions arise from well-quasi-orderings

A fundamental construction in formal language theory is the Myhill-Nerode congruence on words, whose finitedness characterizes regular language. This construction was generalized to functions from ${\mathrm{\Sigma }}^{\ast }$ to $\mathbb{Z}$ by Colcombet, Douéneau-Tabot, and Lopez to characterize the class of so-called $\mathbb{Z}$-polyregular functions. In this paper, we relax the notion of equivalence relation to quasi-ordering in order to study the class of $\mathbb{N}$-polyregular functions, that plays the role of $\mathbb{Z}$-polyregular functions among functions from ${\mathrm{\Sigma }}^{\ast }$ to $\mathbb{N}$. The analogue of having a finite index is then being a well-quasi-ordering. This provides a canonical object to describe $\mathbb{N}$-polyregular functions, together with a powerful new characterization of this class.