On Finite Monoids of Cellular Automata

TL;DR

This paper investigates the algebraic structure of finite monoids of cellular automata over finite groups and alphabets, providing descriptions of invertible automata groups and minimal generating sets.

Contribution

It offers a detailed algebraic analysis of finite cellular automata monoids, including structure of invertible automata groups and minimal generating sets for the monoid.

Findings

Describes the structure of invertible cellular automata groups using group theory.

Shows that cellular automata with small memory sets cannot generate the entire monoid.

Determines minimal generating sets for the monoid when the group is finite abelian.

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 $\text{CA}(G;A) = \langle \text{ICA}(G;A) \cup V \rangle$.