A generalization of cellular automata over groups

Abstract

Let G be a group and let A be a finite set with at least two elements. A cellular automaton (CA) over AG is a function defined via a finite memory set and a local function . The goal of this paper is to introduce the definition of a generalized cellular automaton (GCA) , where H is another arbitrary group, via a group homomorphism . Our definition preserves the essence of CA, as we prove analogous versions of three key results in the theory of CA: a generalized Curtis-Hedlund Theorem for GCA, a Theorem of Composition for GCA, and a Theorem of Invertibility for GCA. When G = H, we prove that the group of invertible GCA over AG is isomorphic to a semidirect product of and the group of invertible CA. Finally, we apply our results to study automorphisms of the monoid consisting of all CA over AG . In particular, we show that every defines an automorphism of via conjugation by the invertible GCA defined by , and that, when G is abelian, is embedded in the outer automorphism group of . Communicated by Pedro Garcia-Sanchez