A Linear Representation for Functions on Finite Sets

We demonstrate that any function $f$ from a finite set $Y$ to itself can be represented linearly. Specifically, we prove the existence of an injective map $j$ from $Y$ into a modular ring $\mathbb{Z}\mathrm{/}m\mathbb{Z}$ and a constant $a\in \mathbb{Z}\mathrm{/}m\mathbb{Z}$ such that $j(f(y))=a\cdot j(y)$ in $\mathbb{Z}\mathrm{/}m\mathbb{Z}$ holds for all $y\in Y$. This result is established by analyzing the algebraic properties of the adjugate of the characteristic matrix associated with the function’s digraph. The proof is constructive, providing a method for finding the embedding $j$, the modulus $m$, and the linear multiplier $a$.