A Linear Representation for Functions on Finite Sets

TL;DR

This paper shows that any function on a finite set can be represented linearly via an injective map into a modular ring, revealing algebraic structure and providing a constructive method for such representations.

Contribution

It introduces a novel linear representation of functions on finite sets using modular embeddings, with a constructive proof and method.

Findings

Existence of an injective map into a modular ring for any finite set function

Linear representation of functions via a modular ring and a constant multiplier

Constructive method to find the embedding, modulus, and multiplier

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 $a$.