A generalization of cellular automata over groups

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 $\text{CA}(G;A)$.