Open Collections

Description

For any group $G$ and set $A$, let $\text{CA}(G;A)$ be the monoid of all cellular automata over the configuration space $A^G$. In this talk, we present some algebraic results on $\text{CA}(G;A)$ when $G$ and $A$ are both finite. First, we show that any generating set of $\text{CA}(G;A)$ must have a cellular automaton with minimal memory set equal to $G$ itself. Second, we describe the structure of the group of units of $\text{CA}(G;A)$ in terms of a set of representatives of the conjugacy classes of subgroups of $G$. Third, we discuss the minimal cardinality of a generating set of $\text{CA}(G;A)$: in some cases we give it precisely, while in others we give some bounds. We apply this to provide a simple proof that $\text{CA}(G;A)$ is not finitely generated for various kinds of infinite groups $G$.