# On Finite Monoids of Cellular Automata

Abstract : For any group G and set A, a cellular automaton over G and A is a transformation $\tau :{A}^{G}\to {A}^{G}$ defined via a finite neighbourhood $S\subseteq G$ (called a memory set of $\tau$) and a local function $\mu :{A}^{S}\to A$. In this paper, we assume that G and A are both finite and study various algebraic properties of the finite monoid $\mathrm{C}\mathrm{A}\left(G,A\right)$ consisting of all cellular automata over G and A. Let $\mathrm{I}\mathrm{C}\mathrm{A}\left(G;A\right)$G and A. In the first part, using information on the conjugacy classes of subgroups of G, we give a detailed description of the structure of $\mathrm{I}\mathrm{C}\mathrm{A}\left(G;A\right)$ in terms of direct and wreath products. In the second part, we study generating sets of $\mathrm{C}\mathrm{A}\left(G;A\right)$. In particular, we prove that $\mathrm{C}\mathrm{A}\left(G,A\right)$ cannot be generated by cellular automata with small memory set, and, when G is finite abelian, we determine the minimal size of a set $V\subseteq \mathrm{C}\mathrm{A}\left(G;A\right)$ such that $\mathrm{C}\mathrm{A}\left(G;A\right)=⟨\mathrm{I}\mathrm{C}\mathrm{A}\left(G;A\right)\cup V⟩$.
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Alonso Castillo-Ramirez, Maximilien Gadouleau. On Finite Monoids of Cellular Automata. 22th International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA), Jun 2016, Zurich, Switzerland. pp.90-104, ⟨10.1007/978-3-319-39300-1_8⟩. ⟨hal-01435036⟩

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