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

Abstract

For a group G and a finite set A, denote by CA(G;A) the monoid of all cellular automata over AG and by ICA(G;A) its group of units. We study the minimal cardinality of a generating set, known as the rank, of 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 ICA(G;A) when G is a finite group, and we apply this to show that, for any infinite abelian group H, the monoid 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 CA(H;A) is finitely generated.