Let $G$ be a group. Let $X$ be a connected algebraic group over an algebraically closed field $K$. Denote by $A=X(K)$ the set of $K$-points of $X$. We study a class of endomorphisms of pro-algebraic groups, namely algebraic group cellular automata over $(G,X,K)$. They are cellular automata $\tau \colon A^G \to A^G$ whose local defining map is induced by a homomorphism of algebraic groups $X^M \to X$ where $M\subset G$ is a finite memory set of $\tau$. Our first result is that when $G$ is sofic, such an algebraic group cellular automaton $\tau$ is invertible whenever it is injective and $\text{char}(K)=0$. As an application, we prove that if $G$ is sofic and the group $X$ is commutative then the group ring $R[G]$, where $R=\text{End}(X)$ is the endomorphism ring of $X$, is stably finite. When $G$ is amenable, we show that an algebraic group cellular automaton $\tau$ is surjective if and only if it satisfies a weak form of pre-injectivity called $(\bullet)$-pre-injectivity. This yields an analogue of the classical Moore-Myhill Garden of Eden theorem. We also introduce the near ring $R(K,G)$ which is $K[X_g: g \in G]$ as an additive group but the multiplication is induced by the group law of $G$. The near ring $R(K,G)$ contains naturally the group ring $K[G]$ and we extend Kaplansky's conjectures to this new setting. Among other results, we prove that when $G$ is an orderable group, then all one-sided invertible elements of $R(K,G)$ are trivial, i.e., of the form $aX_g+b$ for some $g\in G$, $a\in K^*$, $b\in K$. This allows us to show that when $G$ is locally residually finite and orderable (e.g. $\mathbb{Z}^d$ or a free group), and $\text{char}(K)=0$, all injective algebraic cellular automata $\tau \colon \mathbb{C}^G \to \mathbb{C}^G$ are of the form $\tau(x)(h)= a x(g^{-1}h) +b$ for all $x\in \mathbb{C}^G, h \in G$ for some $g\in G$, $a\in \mathbb{C}^*$, $b\in \mathbb{C}$.

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关于苏菲群、卡普兰斯基猜想和前代数群的自同态

让 $G$ 成为一个组。令 $X$ 是代数闭域 $K$ 上的连通代数群。用$A=X(K)$表示$X$的$K$-点集合。我们研究了一类前代数群的自同态，即 $(G,X,K)$ 上的代数群元胞自动机。它们是元胞自动机 $\tau \colon A^G \to A^G$ 其局部定义映射是由代数群 $X^M \to X$ 的同态引起的，其中 $M\subset G$ 是一个有限的记忆集$\tau$。我们的第一个结果是，当 $G$ 是 sofic 时，这样的代数群元胞自动机 $\tau$ 是可逆的，只要它是单射的并且 $\text{char}(K)=0$。作为一个应用，我们证明如果$G$是sofic且群$X$是可交换的，那么群环$R[G]$，其中$R=\text{End}(X)$是$X$, 是稳定有限的。当 $G$ 适合时，我们证明代数群元胞自动机 $\tau$ 是满射的，当且仅当它满足称为 $(\bullet)$-pre-injectivity 的弱形式的预射性。这产生了经典的摩尔-迈希尔伊甸园定理的类似物。我们还引入了近环 $R(K,G)$，即 $K[X_g: g \in G]$ 作为一个加性群，但是乘法是由 $G$ 的群定律引起的。近环 $R(K,G)$ 自然包含群环 $K[G]$，我们将 Kaplansky 的猜想扩展到这个新设置。在其他结果中，我们证明了当 $G$ 是一个可排序群时，$R(K,G)$ 的所有单边可逆元素都是平凡的，即对于某些 $g\，形式为 $aX_g+b$以 G$、$a\in K^*$、$b\in K$。这使我们能够证明，当 $G$ 是局部残差有限且可排序的（例如 $\mathbb{Z}^d$ 或一个自由群）时，