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 \(g \in G\).