Title: 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 $\text{CA}(G;A)$ 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 $\tau \in \text{CA}(G;A)$ is von
Neumann regular (or simply regular) if there exists $\sigma \in \text{CA}(G;A)$
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
$\text{CA}(G;A)$ itself is regular if all its elements are regular. We
establish that $\text{CA}(G;A)$ is regular if and only if $\vert G \vert = 1$
or $\vert A \vert = 1$, and we characterise all regular elements in
$\text{CA}(G;A)$ 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 elementary amenable (e.g. when $G=\mathbb{Z}^d, \ d \in
\mathbb{N}$) and $V=\mathbb{F}$, and that every linear CA is regular when $V$
is finite-dimensional and $G$ is locally finite with $\text{Char}(\mathbb{F})
\nmid o(g)$ for all $g \in G$.