2025/10/23 by Roman Bacik, Bacik, Roman
Computer Science · Mathematics · #Polynomial and algebraic computation #Rings, Modules, and Algebras #Advanced Algebra and Logic
paper · pdf · doi:10.48550/arxiv.2510.20167
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 ℤ/mℤ and a constant a ∈ ℤ/mℤ such that j(f(y)) = a ⋅ j(y) in ℤ/mℤ holds for all y ∈ 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.