Cellular automata over algebraic structures

Abstract

Let $G$ be a group and $A$ a set equipped with a collection of finitary operations. We study cellular automata $τ: A^G \to A^G$ that preserve the operations of $A^G$ induced componentwise from the operations of $A$. We show that $τ$ 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 $A$.