A generalization of cellular automata over groups

TL;DR

This paper extends cellular automata theory to a broader context by defining generalized automata over different groups, establishing foundational theorems, and analyzing automorphism groups within this generalized framework.

Contribution

It introduces the concept of generalized cellular automata over groups connected by homomorphisms, preserving core properties and expanding the theoretical landscape.

Findings

Proves a generalized Curtis-Hedlund Theorem for GCA

Establishes a composition theorem for GCA

Characterizes the invertible GCA group as a semidirect product

Abstract

Let $G$ be a group and let $A$ be a finite set with at least two elements. A cellular automaton (CA) over $A^G$ is a function $\tau : A^G \to A^G$ defined via a finite memory set $S \subseteq G$ and a local function $\mu :A^S \to A$. The goal of this paper is to introduce the definition of a generalized cellular automaton (GCA) $\tau : A^G \to A^H$, where $H$ is another arbitrary group, via a group homomorphism $\phi : H \to G$. Our definition preserves the essence of CA, as we prove analogous versions of three key results in the theory of CA: a generalized Curtis-Hedlund Theorem for GCA, a Theorem of Composition for GCA, and a Theorem of Invertibility for GCA. When $G=H$, we prove that the group of invertible GCA over $A^G$ is isomorphic to a semidirect product of $\text{Aut}(G)^{op}$ and the group of invertible CA. Finally, we apply our results to study automorphisms of the monoid $\text{CA}(G;A)$ consisting of all CA over $A^G$. In particular, we show that every $\phi \in \text{Aut}(G)$ defines an automorphism of $\text{CA}(G;A)$ via conjugation by the invertible GCA defined by $\phi$, and that, when $G$ is abelian, $\text{Aut}(G)$ is embedded in the outer automorphism group of $\text{CA}(G;A)$.