细胞自动机与有限群
Cellular Automata and Finite Groups
摘要：
对于有限群$G$和有限集$A$，我们研究了位形空间$A^G$上元胞自动机的各种代数问题。在这种情况下，$A^G$上所有元胞自动机的集合$\text{CA}(G;A)$是一个有限幺半群，其基本代数性质仍然未知。首先，我们研究了$\text{CA}(G;A)$的单元群$\text{ICA}(G;A)$的结构。我们得到一个分解的$\text{ICA}(G;A)$到一个直接的产品圈产品的群，取决于数量$α_{[H]}$的周期配置的共轭类$[H]$的$G$的子群。我们展示了如何使用$G$的子群格的莫比乌斯函数来计算数字$α_{[H]}$，并且我们使用它来改进最近由高，杰克逊和苏厄德发现的关于$A^G$的非周期配置的数目的下界。进一步研究了$\text{CA}(G;A)$的生成集，特别证明了$\text{CA}(G;A)$不能由小记忆集的元胞自动机生成，并且当$G$的所有子群都正规时，确定了$\text{ICA}(G;A)$在$\text{CA}(G;A)$上的相对秩，即集合$V \subseteq \text{CA}(G;A)$使得$\text{CA}(G;A) = \langle \text{ICA}(G;A) \cup V \rangle$ .
英文摘要：
For a finite group $G$ and a finite set $A$, we study various algebraic aspects of cellular automata over the configuration space $A^G$. In this situation, the set $\text{CA}(G;A)$ of all cellular automata over $A^G$ is a finite monoid whose basic algebraic properties had remained unknown. First, we investigate the structure of the group of units $\text{ICA}(G;A)$ of $\text{CA}(G;A)$. We obtain a decomposition of $\text{ICA}(G;A)$ into a direct product of wreath products of groups that depends on the numbers $α_{[H]}$ of periodic configurations for conjugacy classes $[H]$ of subgroups of $G$. We show how the numbers $α_{[H]}$ may be computed using the Möbius function of the subgroup lattice of $G$, and we use this to improve the lower bound recently found by Gao, Jackson and Seward on the number of aperiodic configurations of $A^G$. Furthermore, we study generating sets of $\text{CA}(G;A)$; in particular, we prove that $\text{CA}(G;A)$ cannot be generated by cellular automata with small memory set, and, when all subgroups of $G$ are normal, we determine the relative rank of $\text{ICA}(G;A)$ on $\text{CA}(G;A)$, i.e. the minimal size of a set $V \subseteq \text{CA}(G;A)$ such that $\text{CA}(G;A) = \langle \text{ICA}(G;A) \cup V \rangle$.
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