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- W3016427363 abstract "For a group $G$ and a set $A$, let $text{CA}(G;A)$ be the monoid of all cellular automata over $A^G$, and let $text{ICA}(G;A)$ be its group of units. By establishing a characterisation of surjunctuve groups in terms of the monoid $text{CA}(G;A)$, we prove that the rank of $text{CA}(G;A)$ (i.e. the smallest cardinality of a generating set) is equal to the rank of $text{ICA}(G;A)$ plus the relative rank of $text{ICA}(G;A)$ in $text{CA}(G;A)$, and that the latter is infinite when $G$ has an infinite decreasing chain of normal subgroups of finite index. Moreover, we study the monoid $text{LCA}(G;V)$ of all linear cellular automata over $V^G$ and its group of units $text{ILCA}(G;V)$. For any infinite abelian group $G$ and any vector space $V$, we show that $text{LCA}(G;V)$ is not finitely generated, while for any finitely generated torsion-free nilpotent group $H$ and any field $mathbb{F}$, we show that $text{ILCA}(H;mathbb{F})$ is finitely generated if and only if $mathbb{F}$ is finite." @default.
- W3016427363 created "2020-04-24" @default.
- W3016427363 creator A5035086666 @default.
- W3016427363 date "2020-04-15" @default.
- W3016427363 modified "2023-09-27" @default.
- W3016427363 title "Generating infinite monoids of cellular automata" @default.
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