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 τ ∈ CA(G; A) is von Neumann regular (or simply regular) if there exists σ ∈ CA(G; A) such that τ • σ • τ = τ and σ • τ • σ = σ, where • is the composition of functions. Such an element σ is called a generalised inverse of τ. 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 |G| = 1 or |A| = 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) ∤ o(g) for all g ∈ G.