Generation of CA(G,A) by Lazy and Invertible Cellular Automata

Prove or disprove that the monoid of all cellular automata over A^G equals the submonoid generated by invertible cellular automata together with lazy cellular automata; specifically, establish whether CA(G,A) = ⟨ICA(G,A) ∪ L(G,A)⟩. If this equality does not hold, determine properties and structure of the submonoids ⟨ICA(G,A) ∪ L(G,A)⟩ and ⟨L(G,A)⟩.

Background

Cellular automata over A^G form a monoid under composition, with invertible CAs forming the group ICA(G,A). In classical semigroup theory, the full transformation monoid on a finite set is generated by idempotents of defect 1 together with the symmetric group. Lazy CAs with minimal neighborhood {e} correspond to such idempotents, suggesting a possible generation result for CA(G,A) using lazy and invertible CAs.

Motivated by this analogy, the authors ask whether every CA can be expressed as a composition of lazy and invertible CAs; if not, they seek structural insights into the submonoids generated by these classes.