线性元胞自动机的可逆性和闭象性
On the reversibility and the closed image property of linear cellular automata
摘要：
当$G$是任意群且$V$是有限维向量空间时，已知每个双射线性元胞自动机$τ\colon V^G \to V^G$都是可逆的，并且每个线性元胞自动机$τ\colon V^G \to V^G$的像对于前离散拓扑在$V^G$中是闭的.本文利用集合的射影序列的Mittag-Leffler引理给出了这两个结果的一个新证明。证明了若$G$是非周期群，$V$是无限维向量空间，则对于预离散拓扑，存在双射但不可逆的线性元胞自动机$τ_1 \colon V^G \to V^G$和像在$V^G$中不闭的线性元胞自动机$τ_2 \colon V^G \to V^G$.
英文摘要：
When $G$ is an arbitrary group and $V$ is a finite-dimensional vector space, it is known that every bijective linear cellular automaton $τ\colon V^G \to V^G$ is reversible and that the image of every linear cellular automaton $τ\colon V^G \to V^G$ is closed in $V^G$ for the prodiscrete topology. In this paper, we present a new proof of these two results which is based on the Mittag-Leffler lemma for projective sequences of sets. We also show that if $G$ is a non-periodic group and $V$ is an infinite-dimensional vector space, then there exist a linear cellular automaton $τ_1 \colon V^G \to V^G$ which is bijective but not reversible and a linear cellular automaton $τ_2 \colon V^G \to V^G$ whose image is not closed in $V^G$ for the prodiscrete topology.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
