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.

翻译：众所周知，某些细胞自动机的时空图具有分形结构：例如模2的帕斯卡三角形生成谢尔宾斯基三角形。已有研究表明，当字母表赋予阿贝尔群结构且细胞自动机是该结构的同态时，此类模式便会出现。此时时空图具有与$k$自动机性相关的性质。我们证明该条件可从阿贝尔群放宽至交换幺半群，且在此情形下时空图仍呈现相同规律性。