A Linear Representation for Functions on Finite Sets
Roman Bacik
TL;DR
The paper shows that any function on a finite set can be represented linearly after embedding into a modular ring. It constructs an explicit embedding using the function's adjacency matrix $A_f$, the adjugate $M = \operatorname{adj}(\chi_A(x))$ of the characteristic matrix $\chi_A(x) = xI - A_f$, and the modulus $m = \det(xI - A_f)$ with a chosen vector $v = (1,2,...,n)^T$ to obtain $y = Mv$ and $j(i) = y_i$; taking $a = x$ yields $j(f(i)) \equiv a\, j(i) \pmod{m}$ for large enough $x$, hence a linear representation $j(f(i)) = a\, j(i)$ in $\mathbb{Z}/m\mathbb{Z}$. The method is constructive and guarantees an injective $j$ with $0 < y_0 < \cdots < y_{n-1} < m$, and thus $m$ as a modulus with $a \in \{0,1,...,m-1\}$. An explicit example shows $f(x) = x^2$ on $\mathbb{Z}/3\mathbb{Z}$ has a linear representation with $m = 36$ and $a = 4$. The result reveals a hidden linear structure in all finite functions, linking discrete maps to linear systems and offering a theoretical bridge to concepts like polynomial interpolation and LFSR-based design.
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 $a$.