Characterize idempotency for cellular automata with a unique active transition over arbitrary groups

Establish necessary and sufficient conditions, in terms of the pattern p ∈ A^S, under which a cellular automaton τ: A^G → A^G defined over an arbitrary group G with finite neighborhood S ⊂ G containing the identity element and local function μ: A^S → A satisfying μ(z) = z(identity) if and only if z ≠ p (i.e., τ has a unique active transition p), is idempotent (τ ∘ τ = τ).

Background

The paper studies one-dimensional cellular automata with a unique active transition and provides a complete characterization of idempotency when the neighborhood in Z is an interval containing 0. Prior work for general groups G showed idempotency in special cases, such as constant or symmetric patterns, and gave a full characterization when the pattern is quasi-constant.

Despite these partial results, a comprehensive, general criterion for idempotency in the arbitrary group setting—covering all patterns p ∈ A^S and all finite neighborhoods S containing the identity—has not been obtained. Resolving this would unify known special cases and extend the algebraic understanding of idempotents in the broader monoid of cellular automata over groups.