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 A G 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.

On the minimal number of generators of endomorphism monoids of full shifts

“…The question of finding the rank of a monoid is important in semigroup theory; it has been answered for several kinds of transformation monoids and Rees matrix semigroups (e.g., see [1,11]). For the case of monoids of endomorphisms of full shifts over finite groups, the question has been addressed in [6,7,8]; in particular, the rank of Aut(A G ) when G is a finite cyclic group has been examined in detail in [5].…”

“…This paper is an extended version of [8]. All sections have been restructured, the exposition has been improved, and the results of Section 4 have been greatly extended.…”