A generalization of cellular automata over groups
Alonso Castillo-Ramirez, Miguel Sánchez‐Álvarez, Alejandro Vazquez-Aceves, Ángel Zaldívar · Communications in Algebra · 2023
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 τ:AG→AG defined via a finite memory set S⊆G and a local function μ:AS→A. The goal of this paper is to introduce the definition of a generalized cellular automaton (GCA) τ:AG→AH, where H is another arbitrary group, via a group homomorphism ϕ:H→G. 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 Aut(G)op and the group of invertible CA. Finally, we apply our results to study automorphisms of the monoid CA(G;A) consisting of all CA over AG . In particular, we show that every ϕ∈Aut(G) defines an automorphism of CA(G;A) via conjugation by the invertible GCA defined by ϕ, and that, when G is abelian, Aut(G) is embedded in the outer automorphism group of CA(G;A).Communicated by Pedro Garcia-Sanchez