A generalization of cellular automata over groups. Issue 7 (3rd July 2023) - Castillo-Ramirez, A.; Sanchez-Alvarez, M.; Vazquez-Aceves, A.; Zaldivar-Corichi, A.
Abstract: Let G be a group and let A be a finite set with at least two elements. A cellular automaton (CA) over A G is a function τ : A G → A G defined via a finite memory set S ⊆ G and a local function μ : A S → A . The goal of this paper is to introduce the definition of a generalized cellular automaton (GCA) τ : A G → A H, 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 A G 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 A G . 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