论文标题
细胞自动机对组的概括
A generalization of cellular automata over groups
论文作者
论文摘要
让$ g $为一个组，让$ a $为有限的设置，至少有两个要素。 $ a^g $上方的蜂窝自动机（CA）是通过有限内存集$ s \ subseteq g $和本地函数$μ：a^s \ to a $定义的函数$τ：a^g \ to a^g $。本文的目的是通过集体同构$ ϕ：h \ t to g $介绍广义蜂窝自动机（GCA）$τ：a^g \ to a^h $的定义。我们的定义保留了CA的本质，因为我们证明了CA理论中三个关键结果的类似版本：GCA的广义柯蒂斯·赫德隆德定理，GCA的组成定理以及GCA的不可逆转定理。当$ g = h $时，我们证明，超过$ a^g $的可逆gca组是$ \ text {aut}（aut}（g）^{op} $的半导向产品的同构和一组可逆的ca。最后，我们将结果应用于研究Monoid $ \ text {ca}（g; a）$的自动形态，由所有CA组成，超过$ a^g $。 In particular, we show that every $ϕ\in \text{Aut}(G)$ defines an automorphism of $\text{CA}(G;A)$ via conjugation by the invertible GCA defined by $ϕ$, and that, when $G$ is abelian, $\text{Aut}(G)$ is embedded in the outer automorphism group of $ \ text {ca}（g; a）$。
Let $G$ be a group and let $A$ be a finite set with at least two elements. A cellular automaton (CA) over $A^G$ is a function $τ: A^G \to A^G$ defined via a finite memory set $S \subseteq G$ and a local function $μ:A^S \to A$. The goal of this paper is to introduce the definition of a generalized cellular automaton (GCA) $τ: A^G \to A^H$, where $H$ is another arbitrary group, via a group homomorphism $ϕ: H \to G$. Our definition preserves the essence of CA, as we prove analogous versions of three key results in the theory of CA: a generalized Curtis-Hedlund Theorem for GCA, a Theorem of Composition for GCA, and a Theorem of Invertibility for GCA. When $G=H$, we prove that the group of invertible GCA over $A^G$ is isomorphic to a semidirect product of $\text{Aut}(G)^{op}$ and the group of invertible CA. Finally, we apply our results to study automorphisms of the monoid $\text{CA}(G;A)$ consisting of all CA over $A^G$. In particular, we show that every $ϕ\in \text{Aut}(G)$ defines an automorphism of $\text{CA}(G;A)$ via conjugation by the invertible GCA defined by $ϕ$, and that, when $G$ is abelian, $\text{Aut}(G)$ is embedded in the outer automorphism group of $\text{CA}(G;A)$.
