Elementary, finite and linear vN-regular cellular automata

Abstract

Let G be a group and A a set. A cellular automaton (CA) τ over $A^G$ is von Neumann regular (vN-regular) if there exists a CA σ over $A^G$ such that $\tau \sigma \tau =\tau$, and in such case, σ is called a weak generalised inverse of τ. In this paper, we investigate the vN-regularity of various kinds of CA. First, we establish that, except for trivial cases, there are always CA that are not vN-regular. Second, we obtain a partial classification of elementary vN-regular CA over ${\{0,1\}}^{\mathbb{Z}}$ by taking advantage of some symmetries among them. 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 and characterise them in certain situations.

Introduction

In this paper we follow the general setting for cellular automata (CA) presented in [7]. For any group G and any set A, the configuration space $A^G$ is the set of all functions from G to A. A cellular automaton over $A^G$ is a transformation of $A^G$ defined via a finite memory set and a local function (see Definition 2 for the precise details). Most of the classical literature on CA focus on the case when $G={\mathbb{Z}}^d$, for $d\ge 1$, and A is a finite set (see [13]), but important results have been obtained for larger classes of groups (e.g., see [7] and references therein).

Recall that a semigroup is a set equipped with an associative binary operation, and that a monoid is a semigroup with an identity element. If M is a group, or a monoid, write $K\le M$ if K is a subgroup, or a submonoid, of M, respectively.

Let $\mathrm{CA}(G;A)$ be the set of all CA over $A^G$. It turns out that, equipped with the composition of functions, $\mathrm{CA}(G;A)$ is a monoid. In this paper we apply functions on the right; hence, for $\tau ,\sigma \in \mathrm{CA}(G;A)$, the composition $\tau \circ \sigma$, denoted simply by τσ, means applying first τ and then σ.

A cellular automaton $\tau \in \mathrm{CA}(G;A)$ is invertible, or reversible, or a unit, if there exists $\sigma \in \mathrm{CA}(G;A)$ such that $\tau \sigma =\sigma \tau =\mathrm{id}$. In such case, σ is called the inverse of τ and denoted by $\sigma ={\tau }^{-1}$. When A is finite, it may be shown that $\tau \in \mathrm{CA}(G;A)$ is invertible if and only if it is a bijective function (see [7, Theorem 1.10.2]). Denote the set of all invertible CA over $A^G$ by $\mathrm{ICA}(G;A)$; this is in fact a group, called the group of units of $\mathrm{CA}(G;A)$.

We shall consider the notion of regularity that, as cellular automata, was introduced by John von Neumann, and has been widely studied in both semigroup and ring theory. A cellular automaton $\tau \in \mathrm{CA}(G;A)$ is von Neumann regular (vN-regular) if there exists $\sigma \in \mathrm{CA}(G;A)$ such that $\tau \sigma \tau =\tau$; in this case, σ is called a weak generalised inverse of τ. Equivalently, $\tau \in \mathrm{CA}(G;A)$ is vN-regular if and only if there exists $\sigma \in \mathrm{CA}(G;A)$ mapping every configuration in the image of τ to one of its preimages under τ (see Lemma 1). Clearly, the notion of vN-regularity generalises reversibility.

In general, for any semigroup S and $a,b\in S$, we say that b is a weak generalised inverse of a if

$$aba=a.$$

We say that b is a generalised inverse (often just called an inverse) of a if

$$aba=a\text{ and }bab=b.$$

An element $a\in S$ may have none, one, or more (weak) generalised inverses. It is clear that any generalised inverse of a is also a weak generalised inverse; not so obvious is that, given the set $W(a)$ of weak generalised inverses of a we may obtain the set $V(a)$ of generalised inverses of a as follows (see [8, Exercise 1.9.7]):

$$V(a)=\{ba{b}^{\prime }:b,b^{\prime }\in W(a)\}.$$

Thus, an element $a\in S$ is vN-regular if it has at least one generalised inverse (which is equivalent to having at least one weak generalised inverse). The semigroup S itself is called vN-regular if all of its elements are vN-regular. Many of the well-known types of semigroups are vN-regular, such as idempotent semigroups (or bands), full transformation semigroups, and Rees matrix semigroups. Among various advantages, vN-regular semigroups have a particularly manageable structure which may be studied using Green's relations. For further basic results on vN-regular semigroups see [8, Section 1.9].

Another generalisation of reversible CA has appeared in the literature before [18], [19] using the concept of Drazin inverse [10]. However, as Drazin invertible elements are a special kind of vN-regular elements, our approach turns out to be more general and natural.

In the following sections we study the vN-regular elements in monoids of CA. First, in Section 2 we present some basic results and examples, and we establish that, except for the trivial cases $|G|=1$ and $|A|=1$, the monoid $\mathrm{CA}(G;A)$ is not vN-regular.

In Section 3, we obtain a partial classification of the vN-regular elementary CA in $\mathrm{CA}(\mathbb{Z};\{0,1\})$. We divide the 256 elementary CA into 48 equivalence classes that preserve vN-regularity: this is an extension of the usual division of elementary CA into 88 equivalence classes that preserve dynamical properties. Among various results, 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. Our classification is only partial as the vN-regularity of 11 classes could not be determined (see Table 1).

In Section 4, we study the vN-regular elements of $\mathrm{CA}(G;A)$ when G and A are both finite; in particular, we characterise them and describe a vN-regular submonoid.

Finally, in Section 5, we study the vN-regular elements of the monoid $\mathrm{LCA}(G;V)$ of linear CA, when V is a vector space over a field $\mathbb{F}$. Specifically, using results on group rings, we show that, when G is torsion-free elementary amenable (e.g., $G={\mathbb{Z}}^d$), $\tau \in \mathrm{LCA}(G;\mathbb{F})$ is vN-regular if and only if it is invertible, and that, for finite-dimensional V, $\mathrm{LCA}(G;V)$ itself is vN-regular if and only if G is locally finite and $\mathrm{char}(\mathbb{F})\nmid |〈g〉|$, for all $g\in G$. Finally, for the particular case when $G\cong {\mathbb{Z}}_n$ is a cyclic group, $V:=\mathbb{F}$ is a finite field, and $\mathrm{char}(\mathbb{F})|n$, we count the total number of vN-regular elements in $\mathrm{LCA}({\mathbb{Z}}_n;\mathbb{F})$.

The present paper is an extended version of [5]. Besides improving the general exposition, Theorem 1, Section 3 and Theorem 6 are completely new, and Theorem 7 is corrected (as the proof of Theorem 5 in [5] is flawed).