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,某些蜂窝自动自动式自动式计算机的时空图有一个分形结构:例如,帕斯卡的三角模模L2产生一个Sierpi\'nski三角形。已经表明,当字母与一个Abelian组的结构相配时,这种模式可以发生,只要细胞自动图与这个结构是形态。那么,空间时图有一个与$k$-自动性有关的属性。我们表明,这个条件可以从一个Abelian组向一个通货单体,在这种情况下,空间时图仍然表现出相同的规律性。
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