Automaticity of spacetime diagrams generated by cellular automata on commutative monoids

Abstract: It is well-known that the spacetime diagrams of some cellular automata have a fractal structure: for instance Pascal's triangle modulo 2 generates a Sierpi\'nski triangle. It has been shown that such patterns can occur when the alphabet is endowed with the structure of an Abelian group, provided the cellular automaton is a morphism with respect to this structure. The spacetime diagram then has a property related to $k$-automaticity. We show that this condition can be relaxed from an Abelian group to a commutative monoid, and that in this case the spacetime diagrams still exhibit the same regularity.