For any group G and any set A, a cellular automaton (CA) is a
transformation of the configuration space AG defined via a finite memory set
and a local function. Let CA(G;A) be the monoid of all CA over AG.
In this paper, we investigate a generalisation of the inverse of a CA from the
semigroup-theoretic perspective. An element ΟβCA(G;A) is von
Neumann regular (or simply regular) if there exists ΟβCA(G;A)
such that ΟβΟβΟ=Ο and ΟβΟβΟ=Ο, where β is the composition of functions. Such an
element Ο is called a generalised inverse of Ο. 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 β£Gβ£=1
or β£Aβ£=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 elementary amenable (e.g. when G=Zd,Β dβN) and V=F, and that every linear CA is regular when V
is finite-dimensional and G is locally finite with Char(F)β€o(g) for all gβG.Comment: 10 pages. Theorem 5 corrected from previous versions, in A.
Dennunzio, E. Formenti, L. Manzoni, A.E. Porreca (Eds.): Cellular Automata
and Discrete Complex Systems, AUTOMATA 2017, LNCS 10248, pp. 44-55, Springer,
201