广义线性元胞自动机下的伊甸园定理
The Garden of Eden Theorem over Generalized Linear Cellular Automata
刘诗雨 1黄小军1
作者信息
- 1. 重庆大学数学与统计学院，重庆401331
- 折叠
摘要
设 $G$ 为一个群,且 $V$ 是域 $\mathbb{K}$ 上的向量空间。在配置空间 $V^G$ 上的线性元胞自动机（LCA）是一个线性映射 $\tau: V^G \to V^G$,它由有限记忆集 $S \subseteq G$ 和局部映射 $\mu: V^S \to V$ 定义。本文旨在通过群同态 $\varphi: H \to G$（其中 $H$ 是另一个群）,引入广义线性元胞自动机（$\varphi$-LCA）$\tau: V^G \to V^H$。这一对经典理论的扩展使我们能够建立 $\varphi$-LCA 的伊甸园定理（Garden of Eden theorem）,并验证了 Moore 和 Myhill 定理在广义线性背景下依然成立。这些研究结果显著丰富了伊甸园定理的理论基础,展示了其在更广泛的群及其同态像中的适用性。总而言之,本研究对 $\varphi$-LCA 进行了深入探讨,有效地扩大了伊甸园定理的范围,丰富了离散动力系统与线性元胞自动机的理论体系,使其涵盖了更大类别的线性自动机。
Abstract
Let $G$ be a group and let $V$ be a vector space over a field $\mathbb{K}$. A linear cellular automaton (LCA) over the configuration space $V^G$ is a linear map $\tau: V^G \to V^G$ defined via a finite memory set $S \subseteq G$ and a local map $\mu: V^S \to V$. The goal of this paper is to introduce Generalized Linear Cellular Automata ($\varphi$-LCA) $\tau: V^G \to V^H$, where $H$ is another group, via a group homomorphism $\varphi: H \to G$. This extension of the classical theory enables us to establish the Garden of Eden theorem for $\varphi$-LCA by validating that the Moore and Myhill theorems remain true in this generalized linear context. These findings significantly enrich the theoretical infrastructure of the Garden of Eden theorem, demonstrating its relevance across a wider variety of groups and homomorphic images. In summary, this work provides a thorough investigation of $\varphi$-LCA, which effectively expands the applicable scope of the Garden of Eden theorem, enriches the theoretical framework of discrete dynamical systems and linear cellular automata, and extends the theorem to cover a broader class of linear automata.关键词
动力系统/$\varphi$-线性元胞自动机/伊甸园定理/顺从群/满射性/预单射性Key words
Dynamical Systems/$\varphi$-Linear Cellular Automata/The Garden of Eden Theorem/Amenable Group/Surjectivity/Pre-injectivity引用本文复制引用
刘诗雨,黄小军.广义线性元胞自动机下的伊甸园定理[EB/OL].(2026-04-01)[2026-04-04].http://www.paper.edu.cn/releasepaper/content/202604-10.学科分类
数学
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