For a group $G$ and a set $A$, let $\text{ End}(A^G)$ be the monoid of all cellular automata over $A^G$, and let $\text{ Aut}(A^G)$ be its group of units. By establishing a characterisation of surjunctuve groups in terms of the monoid $\text{ End}(A^G)$, we prove that the rank of $\text{ End}(A^G)$ (i.e. the smallest cardinality of a generating set) is equal to the rank of $\text{ Aut}(A^G)$ plus the relative rank of $\text{ Aut}(A^G)$ in $\text{ End}(A^G)$, and that the latter is infinite when $G$ has an infinite decreasing chain of normal subgroups of finite index, condition which is satisfied, for example, for any infinite residually finite group. Moreover, when $A=V$ is a vector space over a field $\mathbb{ F}$, we study the monoid $\text{ End}_{ \mathbb{ F}}(V^G)$ of all linear cellular automata over $V^G$ and its group of units $\text{ Aut}_{ \mathbb{ F}}(V^G)$. We show that if $G$ is an indicable group and $V$ is finite-dimensional, then $\text{ End}_{ \mathbb{ F}}(V^G)$ is not finitely generated; however, for any finitely generated indicable group $G$, the group $\text{ Aut}_{ \mathbb{ F}}(\mathbb{ F}^G)$ is finitely generated if and only if $\mathbb{ F}$ is finite.
Categories: math.GR math.DS