Further results on generalized cellular automata

Abstract

Given a finite set A and a group homomorphism ϕ : H → G , a ϕ -cellular automaton is a function T : A G → A H that is continuous with respect to the prodiscrete topologies and ϕ -equivariant in the sense that h · T ( x ) = T ( ϕ ( h ) · x ) , for all x ∈ A G , h ∈ H , where · denotes the shift actions of G and H on A G and AH , respectively. When G = H and ϕ = id , the definition of id -cellular automata coincides with the classical definition of cellular automata. The purpose of this paper is to expand the theory of ϕ -cellular automata by focusing on the differences and similarities with their classical counterparts. After discussing some basic results, we introduce the following definition: a ϕ -cellular automaton T : A G → A H has the unique homomorphism property (UHP) if T is not ψ-equivariant for any group homomorphism ψ : H → G , ψ ≠ ϕ . We show that if the difference set Δ ( ϕ , ψ ) is infinite, then T is not ψ-equivariant; it follows that when G is torsion-free abelian, every non-constant T has the UHP. Furthermore, inspired by the theory of classical cellular automata, we study ϕ -cellular automata over quotient groups, as well as their restriction and induction to subgroups and supergroups, respectively.