Von Neumann Regular Cellular Automata

Abstract : For any group G and any set A, a cellular automaton (CA) is a transformation of the configuration space $$A^G$$ defined via a finite memory set and a local function. Let $$\mathrm {CA}(G;A)$$ be the monoid of all CA over $$A^G$$. In this paper, we investigate a generalisation of the inverse of a CA from the semigroup-theoretic perspective. An element $$\tau \in \mathrm {CA}(G;A)$$ is von Neumann regular (or simply regular) if there exists $$\sigma \in \mathrm {CA}(G;A)$$ such that $$\tau \circ \sigma \circ \tau = \tau $$ and $$\sigma \circ \tau \circ \sigma = \sigma $$, where $$\circ $$ is the composition of functions. Such an element $$\sigma $$ is called a generalised inverse of $$\tau $$. The monoid $$\mathrm {CA}(G;A)$$ itself is regular if all its elements are regular. We establish that $$\mathrm {CA}(G;A)$$ is regular if and only if $$\vert G \vert = 1$$ or $$\vert A \vert = 1$$, and we characterise all regular elements in $$\mathrm {CA}(G;A)$$ when G and A are both finite. Furthermore, we study regular linear CA when $$A= V$$ is a vector space over a field $$\mathbb {F}$$; in particular, we show that every regular linear CA is invertible when G is torsion-free (e.g. when $$G=\mathbb {Z}^d, d \ge 1$$), and that every linear CA is regular when V is finite-dimensional and G is locally finite with $$\mathrm {char}(\mathbb {F}) \not \mid o(g)$$ for all $$g \in G$$.
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Alonso Castillo-Ramirez, Maximilien Gadouleau. Von Neumann Regular Cellular Automata. 23th International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA), Jun 2017, Milan, Italy. pp.44-55, ⟨10.1007/978-3-319-58631-1_4⟩. ⟨hal-01656360⟩

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