Further results on generalized cellular automata

TL;DR

This work generalizes cellular automata from a single group action to φ-cellular automata between configuration spaces $A^G$ and $A^H$, with emphasis on continuity and $\phi$-equivariance. It develops a robust algebraic framework: a canonical decomposition $\mathcal{T}=\phi^*\circ\tau$, a monoid structure under composition, and a criterion based on the infinite difference set $\Delta(\phi,\psi)$ for the unique homomorphism property (UHP). The paper then extends GCAs to quotient groups, establishing a commuting diagram and a monoid-homomorphism when $N$ is fully invariant, and finally studies induction and restriction, characterizing injectivity/bijectivity transfer to restricted and induced GCAs while highlighting open questions about surjectivity transfer. Overall, it broadens the toolkit for analyzing group-symmetric discrete dynamics across group extensions, quotients, and subgroup relations, via precise algebraic and topological constructions. $GCA$ theory thus connects symbolic dynamics with deeper group-theoretic properties such as Hopfianity and surjunctivity.$

Abstract

Given a finite set $A$ and a group homomorphism $φ: H \to G$, a $φ$-cellular automaton is a function $\mathcal{T} : A^G \to A^H$ that is continuous with respect to the prodiscrete topologies and $φ$-equivariant in the sense that $h \cdot \mathcal{T}(x) = \mathcal{T}( φ(h) \cdot x)$, for all $x \in A^G, h \in H$, where $\cdot$ denotes the shift actions of $G$ and $H$ on $A^G$ and $A^H$, respectively. When $G=H$ and $φ= \text{id}$, the definition of $\text{id}$-cellular automata coincides with the classical definition of cellular automata. The purpose of this paper is to expand the theory of $φ$-cellular automata by focusing on the differences and similarities with their classical counterparts. After discussing some basic results, we introduce the following definition: a $φ$-cellular automaton $\mathcal{T} : A^G \to A^H$ has the unique homomorphism property (UHP) if $\mathcal{T}$ is not $ψ$-equivariant for any group homomorphism $ψ: H \to G$, $ψ\neq φ$. We show that if the difference set $Δ(φ, ψ)$ is infinite, then $\mathcal{T}$ is not $ψ$-equivariant; it follows that when $G$ is torsion-free abelian, every non-constant $\mathcal{T}$ has the UHP. Furthermore, inspired by the theory of classical cellular automata, we study $φ$-cellular automata over quotient groups, as well as their restriction and induction to subgroups and supergroups, respectively.

Theorems & Definitions (61)

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