A generalization of cellular automata over groups. Issue 7 (3rd July 2023)

• Record Type: Journal Article
• Authors: Castillo-Ramirez, A.; Sanchez-Alvarez, M.; Vazquez-Aceves, A.; Zaldivar-Corichi, A.
• Is Part Of: Communications in algebra. Volume 51:Issue 7(2023)
• Page Start: 3114
• Page End: 3123
• Publication Date: 2023-07-03
• Subjects: Cellular automata -- Curtis-Hedlund theorem -- monoid of cellular automata -- outer automorphism group; 37B15 -- 68Q80; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2023.2177663
• Languages: English
• ISSNs: 0092-7872

Abstract

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