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arXiv · 2510.20167

A Linear Representation for Functions on Finite Sets

Roman Bacik
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Abstract

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 $\mathbb{Z}/m\mathbb{Z}$ and a constant $a \in \mathbb{Z}/m\mathbb{Z}$ such that $j(f(y)) = a \cdot j(y)$ in $\mathbb{Z}/m\mathbb{Z}$ 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 $a$.

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Roman Bacik. 2025-10-23. A Linear Representation for Functions on Finite Sets. https://arxiv.org/abs/2510.20167

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