Abstract
Given a finite set A and a group homomorphism , a -cellular automaton is a function that is continuous with respect to the prodiscrete topologies and -equivariant in the sense that , for all , where denotes the shift actions of G and H on A G and AH, respectively. When G = H and , the definition of -cellular automata coincides with the classical definition of cellular automata. The purpose of this paper is to expand the theory of -cellular automata by focusing on the differences and similarities with their classical counterparts. After discussing some basic results, we introduce the following definition: a -cellular automaton has the unique homomorphism property (UHP) if is not ψ-equivariant for any group homomorphism . We show that if the difference set is infinite, then is not ψ-equivariant; it follows that when G is torsion-free abelian, every non-constant has the UHP. Furthermore, inspired by the theory of classical cellular automata, we study -cellular automata over quotient groups, as well as their restriction and induction to subgroups and supergroups, respectively.
Acknowledgments
We sincerely thank the anonymous reviewer of this paper for their careful reading and insightful comments; in particular, they greatly simplified the proof of Lemma 8.