初等、有限和线性vN-Regular元胞自动机
Elementary, Finite and Linear vN-Regular Cellular Automata
摘要：
设$G$是一个群，$A$是一个集合。$A^G$上的元胞自动机（CA）$τ$是von Neumann正则的（vN-正则的），如果存在$A^G$上的CA $σ$使得$τστ= τ$，在这种情况下，$σ$称为$τ$的广义逆.本文研究了各种CA的vN-正则性。首先，我们建立，在任何非平凡的配置空间，总是存在CA是不VN-正则的。然后，我们得到一个部分分类的初等VN-正规CA在$\{ 0,1\}^{\mathbb{Z}}$，特别是，我们表明，规则128和254是VN-正规的（和实际上广义逆的对方），而其他人，如著名的规则$90$和$110$，是不是VN-正规的。其次，当$A$和$G$都是有限的时，我们得到了$A^G$上vN-正则CA的一个完整刻画.最后，我们研究了VN-正则线性CA当$A= V$是一个向量空间在一个域$\mathbb{F}$ ;我们表明，每一个VN-正则线性CA是可逆的$V= \mathbb{F}$和$G$是无挠初等服从（例如，当$G=\mathbb{Z}^d, \ d \in \mathbb{N}$），并且每一个线性CA是VN-正则当$V$是有限维和$G$是局部有限的$Char(\mathbb{F}) \nmid o(g)$为所有$g \in G$。
英文摘要：
Let $G$ be a group and $A$ a set. A cellular automaton (CA) $τ$ over $A^G$ is von Neumann regular (vN-regular) if there exists a CA $σ$ over $A^G$ such that $τστ= τ$, and in such case, $σ$ is called a generalised inverse of $τ$. In this paper, we investigate vN-regularity of various kinds of CA. First, we establish that, over any nontrivial configuration space, there always exist CA that are not vN-regular. Then, we obtain a partial classification of elementary vN-regular CA over $\{ 0,1\}^{\mathbb{Z}}$; in particular, we show that rules like 128 and 254 are vN-regular (and actually generalised inverses of each other), while others, like the well-known rules $90$ and $110$, are not vN-regular. Next, when $A$ and $G$ are both finite, we obtain a full characterisation of vN-regular CA over $A^G$. Finally, we study vN-regular linear CA when $A= V$ is a vector space over a field $\mathbb{F}$; we show that every vN-regular linear CA is invertible when $V= \mathbb{F}$ and $G$ is torsion-free elementary amenable (e.g. when $G=\mathbb{Z}^d, \ d \in \mathbb{N}$), and that every linear CA is vN-regular when $V$ is finite-dimensional and $G$ is locally finite with $Char(\mathbb{F}) \nmid o(g)$ for all $g \in G$.
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