Cellular automata over algebraic structures

TL;DR

This paper characterizes cellular automata over algebraic structures, showing their endomorphisms correspond to homomorphisms, and explores their structure in cases like modules and Boolean algebras.

Contribution

It provides a comprehensive algebraic framework for cellular automata over various algebraic structures, including entropic sets, modules, and Boolean algebras.

Findings

Endomorphisms correspond to local homomorphisms.

EndCA(G;A) is isomorphic to a direct limit of homomorphism sets.

Number of endomorphic CA over finite Boolean algebras is explicitly counted.

Abstract

Let $G$ be a group and $A$ a set equipped with a collection of finitary operations. We study cellular automata $\tau : A^G \to A^G$ that preserve the operations of $A^G$ induced componentwise from the operations of $A$. We show that $\tau$ is an endomorphism of $A^G$ if and only if its local function is a homomorphism. When $A$ is entropic (i.e. all finitary operations are homomorphisms), we establish that the set $\text{EndCA}(G;A)$, consisting of all such cellular automata, is isomorphic to the direct limit of $\text{Hom}(A^S, A)$, where $S$ runs among all finite subsets of $G$. In particular, when $A$ is an $R$-module, we show that $\text{EndCA}(G;A)$ is isomorphic to the group algebra $\text{End}(A)[G]$. Moreover, when $A$ is a finite Boolean algebra, we establish that the number of endomorphic cellular automata over $A^G$ admitting a memory set $S$ is precisely $(k \vert S \vert)^k$, where $k$ is the number of atoms of $A$.