On Finite Monoids of Cellular Automata

Abstract: For any group $G$ and set $A$, a cellular automaton over $G$ and $A$ is a transformation $\tau : A^G \to A^G$ defined via a finite neighborhood $S \subseteq G$ (called a memory set of $\tau$) and a local function $\mu : A^S \to A$. In this paper, we assume that $G$ and $A$ are both finite and study various algebraic properties of the finite monoid $\text{CA}(G,A)$ consisting of all cellular automata over $G$ and $A$. Let $\text{ICA}(G;A)$ be the group of invertible cellular automata over $G$ and $A$. In the first part, using information on the conjugacy classes of subgroups of $G$, we give a detailed description of the structure of $\text{ICA}(G;A)$ in terms of direct and wreath products. In the second part, we study generating sets of $\text{CA}(G;A)$. In particular, we prove that $\text{CA}(G,A)$ cannot be generated by cellular automata with small memory set, and, when $G$ is finite abelian, we determine the minimal size of a set $V \subseteq \text{CA}(G;A)$ such that $\text{CA}(G;A) = \langle \text{ICA}(G;A) \cup V \rangle$.