Von Neumann Regular Cellular Automata

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 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 is an element of CA(G; A) is von Neumann regular (or simply regular) if there exists sigma is an element of CA(G; A) such that tau circle sigma circle tau = tau and sigma circle tau circle sigma = sigma, where circle is the composition of functions. Such an element s is called a generalised inverse of tau. The monoid CA(G; A) itself is regular if all its elements are regular. We establish that CA(G; A) is regular if and only if vertical bar G vertical bar = 1 or vertical bar A vertical bar = 1, and we characterise all regular elements in 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 F; in particular, we show that every regular linear CA is invertible when G is torsion-free (e.g. when G = Z(d), d >= 1), and that every linear CA is regular when V is finite-dimensional and G is locally finite with char(F) inverted iota circle (g) for all g is an element of G.