Von Neumann Regular Cellular Automata

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10248)

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 $$g \in G$$.

Keywords

Cellular automata Linear cellular automata Monoids von Neumann regular elements Generalised inverses

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