Elementary, Finite and Linear vN-Regular Cellular Automata

Let $G$ be a group and $A$ a set. A cellular automaton (CA) $\tau$ over $A^G$ is von Neumann regular (vN-regular) if there exists a CA $\sigma$ over $A^G$ such that $\tau \sigma\tau = \tau$, and in such case, $\sigma$ is called a generalised inverse of $\tau$. In this paper, we investigate vN-regularity of various kinds of CA. First, we establish that, over any nontrivial configuration space, there always exist CA that are not vN-regular. Then, we obtain a partial classification of elementary vN-regular CA over $\{ 0,1\}^{\mathbb{Z}}$; in particular, we show that rules like 128 and 254 are vN-regular (and actually generalised inverses of each other), while others, like the well-known rules $90$ and $110$, are not vN-regular. Next, when $A$ and $G$ are both finite, we obtain a full characterisation of vN-regular CA over $A^G$. Finally, we study vN-regular linear CA when $A= V$ is a vector space over a field $\mathbb{F}$; we show that every vN-regular linear CA is invertible when $V= \mathbb{F}$ and $G$ is torsion-free elementary amenable (e.g. when $G=\mathbb{Z}^d, \ d \in \mathbb{N}$), and that every linear CA is vN-regular when $V$ is finite-dimensional and $G$ is locally finite with $Char(\mathbb{F}) \nmid o(g)$ for all $g \in G$.