元胞自动机的通用群
Universal groups of cellular automata
AI总结：
本研究证明任意字母表上的可逆元胞自动机（RCA）群存在由三个对合生成的通用子群，含所有有限生成RCA群的同构副本，并刻画了不同阶字母表下相关RCA群的结构性质。
摘要：
我们证明，在任意字母表$A$上的可逆元胞自动机（reversible cellular automata, RCA）群中，存在一个由三个对合生成的子群，该子群包含任意字母表$B$上所有有限生成RCA群的同构副本。这一结果源自对RCA群的案例研究：这类群由符号置换与部分移位（等价于分块元胞自动机）基于字母表的固定笛卡尔积分解生成。对于素数阶字母表，我们证明该群是虚循环群；对于合数阶字母表，该群不可顺从。字母表大小为4时，该群是线性群。对于非素数且大小不为4的字母表，该群包含所有有限生成RCA群的副本。我们还证明，在字母表足够大的任意全移位上，由双半径为1的RCA生成的群也具备该性质，部分完美有限生成RCA群同样满足。
英文摘要：
We prove that the group of reversible cellular automata (RCA), on any alphabet $A$, contains a subgroup generated by three involutions which contains an isomorphic copy of every finitely generated group of RCA on any alphabet $B$. This result follows from a case study of groups of RCA generated by symbol permutations and partial shifts (equivalently, partitioned cellular automata) with respect to a fixed Cartesian product decomposition of the alphabet. For prime alphabets, we show that this group is virtually cyclic, and that for composite alphabets it is non-amenable. For alphabet size four, it is a linear group. For non-prime non-four alphabets, it contains copies of all finitely generated groups of RCA. We also prove this property for the group generated by RCA of biradius one on any full shift with large enough alphabet, and also for some perfect finitely generated groups of RCA.
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