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 Sierpinski 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 and the initial configuration has finite support. The spacetime diagram then has a property related to k-automaticity. We show that these conditions can be relaxed: the Abelian group can be a commutative monoid, the initial configuration can be k-automatic, and 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的帕斯卡三角形会生成谢尔宾斯基三角形。已有研究表明，当字母表具有阿贝尔群结构时，只要细胞自动机是该结构的态射且初始构形具有有限支撑，此类模式便可能出现。此时时空图具有与k-自劢性相关的性质。本文证明这些条件可以放宽：阿贝尔群可推广为交换幺半群，初始构形可以是k-自劢的，而时空图仍会展现出相同的规律性。