# Bounding the minimal number of generators of groups and monoids of cellular automata

## TL;DR

This paper investigates the minimal number of generators needed for the group of invertible cellular automata over finite and infinite groups, providing bounds and showing certain cases are not finitely generated.

## Contribution

It offers new bounds on the rank of invertible cellular automata groups for specific finite groups and establishes non-finite generation results for infinite groups.

## Key findings

- Upper bounds for the rank when G is a finite group, especially dihedral or Dedekind groups.
- A basic lower bound for the rank of ICA(G;A).
- Infinite abelian groups have non-finitely generated CA monoids.

## Abstract

For a group $G$ and a finite set $A$, denote by $\text{CA}(G;A)$ the monoid of all cellular automata over $A^G$ and by $\text{ICA}(G;A)$ its group of units. We study the minimal cardinality of a generating set, known as the rank, of $\text{ICA}(G;A)$. In the first part, when $G$ is a finite group, we give upper bounds for the rank in terms of the number of conjugacy classes of subgroups of $G$. The case when $G$ is a finite cyclic group has been studied before, so here we focus on the cases when $G$ is a finite dihedral group or a finite Dedekind group. In the second part, we find a basic lower bound for the rank of $\text{ICA}(G;A)$ when $G$ is a finite group, and we apply this to show that, for any infinite abelian group $H$, the monoid $\text{CA}(H;A)$ is not finitely generated. The same is true for various kinds of infinite groups, so we ask if there exists an infinite group $H$ such that $\text{CA}(H;A)$ is finitely generated.