amenability and linear cellular automata over semisimple modules of finite length

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Amenability and Linear CellularAutomata Over Semisimple Modules ofFinite LengthTullio Ceccherini-Silberstein a & Michel Coornaert ba Dipartimento di Ingegneria , Universitá del Sannio , Benevento,Italyb Institute de Recherche Mathématique Avancée, Université LouisPasteur and CNRS , Strasbourg, FrancePublished online: 20 Jun 2008.

To cite this article: Tullio Ceccherini-Silberstein & Michel Coornaert (2008) Amenability and LinearCellular Automata Over Semisimple Modules of Finite Length, Communications in Algebra, 36:4,1320-1335, DOI: 10.1080/00927870701864015

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Communications in Algebra®, 36: 1320–1335, 2008Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927870701864015

AMENABILITY AND LINEAR CELLULAR AUTOMATAOVER SEMISIMPLE MODULES OF FINITE LENGTH

Tullio Ceccherini-Silberstein1 and Michel Coornaert21Dipartimento di Ingegneria, Universitá del Sannio, Benevento, Italy2Institute de Recherche Mathématique Avancée,Université Louis Pasteur and CNRS, Strasbourg, France

Let M be a semisimple left module of finite length over a ring R and let G bean amenable group. We show that an R-linear cellular automaton � � MG → MG issurjective if and only if it is pre-injective.

Key Words: Amenable group; Artinian module; Linear cellular automaton; Module of finite length;Noetherian module; Pre-injectivity; Semisimple module.

2000 Mathematics Subject Classification: 43A07; 37B15; 20F65; 37B15; 20F65; 13E05; 13E10; 16P70;68Q80.

1. INTRODUCTION

Let G be a group and M be a left module over a ring R. We consider the setMG consisting of all maps x � G → M . Note that MG is a left module over R in anatural way. Moreover, the group G acts on MG on the right by �x� g� �→ xg, wherexg�g′� = x�gg′� for all x ∈ MG and g� g′ ∈ G.

An R-linear cellular automaton over G with values in M is a map � � MG → MG

satisfying the following condition: there exist a finite subset S ⊂ G and an R-linearmap � � MS → M such that

��x��g� = ��xg �S� for all x ∈ MG� g ∈ G� (1.1)

where xg �S denotes the restriction of xg to S. Such a set S is called a memory set and� is called a local defining map for �. Observe that every linear cellular automaton� � MG → MG is R-linear and G-equivariant (see Section 2 for the general notion ofa cellular automaton and some examples).

Two elements x� x′ ∈ MG are said to be almost equal if the set �g ∈ G � x�g� �=x′�g�� is finite. Equivalently, denoting by MG the submodule of MG consisting ofall maps from G to M with finite support, two elements x� x′ ∈ MG are almost equalif and only if x − x′ ∈ MG.

Received September 4, 2006; Revised March 23, 2007. Communicated by A. Yu. Olshanskii.Address correspondence to Tullio Ceccherini-Silberstein, Via Goffredo Mameli 40, Roma 00153,

Italy; Fax: +39-47-70-10-07; E-mail: [email protected]

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AMENABILITY AND LINEAR CELLULAR AUTOMATA 1321

Following terminology introduced in Gromov (1999a), one says that a mapf � MG → MG is pre-injective if, for any x� x′ ∈ MG which are almost equal, if f�x� =f�x′�, then x= x′. Every injective map MG → MG is pre-injective but the converse isnot true in general. Observe also that every R-linear cellular automaton � � MG →MG induces by restriction an R-linear map �0 � MG → MG, and that � is pre-injective if and only if �0 is injective.

The aim of this article is to establish the following theorem (see Section 2 forthe definition of the notions involved in the statement).

Theorem 1.1. Let M be a semisimple left module of finite length over a ring R and letG be an amenable group. Let � � MG → MG be an R-linear cellular automaton. Then �is surjective if and only if it is pre-injective.

We note that when R is a field, the preceding theorem reduces to Theorem 1.2in Ceccherini-Silberstein and Coornaert (2006). It can also be viewed as a linearanalog of the classical “Garden of Eden Theorem” for cellular automata with finitealphabet. The latter was proved by Moore and Myhill for G = �2 (Moore, 1963;Myhill, 1963) and then extended to groups of sub-exponential growth by Machì andMignosi (1993) and to countable amenable groups by Ceccherini-Silberstein et al.(1999b).

The article is organized as follows. In Section 2 we introduce the notation andpresent some background material. In particular we recall some basic facts in thetheory of modules (length, semisimplicity), of cellular automata (with values in anarbitrary alphabet), and of amenable groups. In the subsequent section we define themean length �̃�X� of a submodule X ⊂ MG. It plays the role of entropy used in theclassical framework, namely, for cellular automata with finite alphabet. In Section 4we establish Theorem 1.1 by showing that both surjectivity and pre-injectivity of �are equivalent to the fact that �̃��MG�� is equal to the length of M (Theorem 4.1).We also consider linear cellular automata over nonamenable groups. We providetwo examples of R-linear cellular automata over the free group F2 with values in asemisimple module of length 2: the first one is pre-injective but not surjective, thesecond one is surjective but not pre-injective. Finally, we show that the hypothesisof finite length for the left module M is essential in the statement of Theorem 1.1.

2. PRELIMINARIES

2.1. Modules of Finite Length

Let us briefly review the notions and results from linear algebra that we shallneed hereafter. We refer to the books by Atiyah and Macdonald (1969), Bourbaki(1958), Golan and Head (1991), Hungerford (1987), or Zariski and Samuel (1975)for more details and proofs.

Let R be a ring. Let M be a left module over R.One says that M is Noetherian (or that it satisfies the ascending chain condition)

if every increasing sequence of submodules of M

M0 ⊂ M1 ⊂ · · · ⊂ Mn ⊂ Mn+1 ⊂ · · ·

eventually stabilizes (i.e., there exists n0 ∈ � such that Mn = Mn0for all n ≥ n0).

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1322 CECCHERINI-SILBERSTEIN AND COORNAERT

One says that M is Artinian (or that it satisfies the descending chain condition)if every decreasing sequence of submodules of M

M0 ⊃ M1 ⊃ · · · ⊃ Mn ⊃ Mn+1 ⊃ · · ·

eventually stabilizes.The length ��M� ∈ � ∪ �� of the module M is defined as being the supremum

of the integers n such that M admits a strictly increasing sequence of submodules

M0 � M1 � M2 � · · · � Mn

of length n.Thus ��M� = 0 if and only if M = �0�.One shows that M has finite length if and only if M is both Noetherian and

Artinian.The ring R is said to be left Noetherian (resp., left Artinian) if it is Noetherian

(resp., Artinian) as a left module over itself.

Proposition 2.1. Every left Artinian ring is also left Noetherian. Thus, every leftArtinian ring is a left module of finite length over itself.

Note that there are left Noetherian rings which are not left Artinian (e.g., thering � of integers) and that there are Artinian modules which are not Noetherian(e.g., the subgroup of �/� consisting of elements whose order is a power of a fixedprime p, viewed as a �-module).

Proposition 2.2. Let M be a left R-module and let N be a submodule of M . Then Mis Noetherian (resp., Artinian, resp., of finite length) if and only if N and M/N are bothNoetherian (resp., Artinian, resp., of finite length). Moreover, one has

��M� = ��N�+ ��M/N�� (2.1)

Corollary 2.3. Let M and M ′ be left R-modules. Suppose that there exists a surjectiveR-linear map u � M → M ′. Then ��M ′� ≤ ��M�.

Corollary 2.4. Let M be a left R-module and let N be a submodule of M . Then onehas ��N� ≤ ��M�. In particular, if M has finite length so does N . Moreover, if M hasfinite length and ��N� = ��M�, then N = M .

Corollary 2.5. Let M1 and M2 be left R-modules. Then

��M1 ×M2� = ��M1�+ ��M2��

In particular, if M1 and M2 have finite length, then so does M1 ×M2.

Let M be a left R-module.One says that M is simple if M �= �0� and the only submodules of M are �0�

and M . Equivalently, M is simple if it has length ��M� = 1.

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One says that M is semisimple if it can be written as a direct sum, M = ⊕j∈J Nj

(with J possibly infinite), with each submodule Nj simple. It follows from thisdefinition that every direct sum of semisimple modules is semisimple.

A submodule N of a module M is called a direct summand if there exists asubmodule N ′ ⊂ M such that M = N ⊕ N ′.

Proposition 2.6. Let M be a left module over a ring R. Then the following conditionsare equivalent:

(a) M is semisimple;(b) every submodule of M is a direct summand.

Corollary 2.7. Every submodule of a semisimple module is semisimple.

The following theorem characterizes the semisimple modules which have finitelength.

Theorem 2.8. Let M be a semisimple left module over a ring R. Let �Nj�j∈J be afamily of simple submodules of M such that M = ⊕

j∈J Nj . Then the following conditionsare equivalent:

(a) M is Artinian;(b) M is Noetherian;(c) M has finite length;(d) J is finite.

Moreover, if these conditions are satisfied, then ��M� = �J �.

Remark 2.9. Let R be a nonzero ring. By Zorn’s lemma, we can find a maximalleft ideal N ⊂ R. Then M0 = R/N is a simple left R-module and, for any positiveinteger n, the left R-module M = ⊕n

i=1 M0 is semisimple and has length ��M� = n.

2.2. Cellular Automata

Let G be a group, called the universe and A be a set, called the alphabet orthe set of states. We denote by AG = �x � G → A� the set of all maps from G withvalues in A. Such maps are called configurations. The group G acts on the right onAG by setting xg�g′� = x�gg′� for all x ∈ AG and g� g′ ∈ G. A cellular automaton overG with values in A is a map � � AG → AG such that there exist a finite subset S ⊂ G,called a memory set, and a map � � AS → A, called a local defining map, such that��x��g� = ��xg �S� for all x ∈ AG and g ∈ G.

Example 2.10 (Conway’s Game of Life). The universe is G = �2, the free abeliangroup of rank two: the group elements are referred to as “cells.” The set of statesis A = �0� 1�: state 0 corresponds to “absence of life” while state 1 corresponds to“life” and, for a given cell, passing from state 0 to state 1 (resp., from 1 to 0)

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is interpreted as “birth” (resp., “death”). The cellular automaton with memory setS= �−1� 0� 1�2 ⊂ �2 and local defining map � � AS → A given by

��y� =

1 if

∑s∈S

y�s� = 3

or∑s∈S

y�s� = 4 and y��0� 0�� = 1�

0 otherwise

for all y ∈ AS , is called the Game of Life. It was introduced by John Horton Conway(Berlekamp et al., 1982).

Suppose that A = M is a left module over a ring R. Then we say that a cellularautomaton � � MG → MG is R-linear provided it is an R-linear endomorphism of theleft R-module MG into itself. Note that this is equivalent to the R-linearity of thelocal defining map � � MS → M .

Example 2.11 (The Laplace Operator). Let G be a group, S a finite subset of Gcontaining 1G (the identity element of the group), and A = M , a left module over aring R. The R-linear map S � M

G → MG defined by

S�x��g� = ��S� − 1�x�g�− ∑s∈S\�1G�

x�gs�

is a cellular automaton over G with memory set S and local defining map� �MS →M given by ��y� = ��S� − 1�y�1G�−

∑s∈S\�1G� y�s�, for all y ∈ MS . It is

called the Laplace operator or discrete Laplacian over G with coefficients in M .

Let � � AG → AG be a cellular automaton with memory set S and local definingmap � � AS → A.

Note that any finite subset S′ ⊂ G containing S is also a memory set for �.The corresponding local defining map �′ � AS′ → A is given by �′ = � � �S′�S where�S′�S � A

S′ → AS is the restriction map.Let now H be a subgroup of G containing S and consider the map

�H � AH →AH defined by �H�x��h� = ��xh�S� for all x ∈ AH and h ∈ H . Then �H isa cellular automaton over H , which is called the restriction of � to H (Ceccherini-Silberstein and Coornaert, 2006).

Proposition 2.12. Let G be a group, A an alphabet and � � AG → AG a cellularautomaton with memory set S ⊂ G. Let also H be a subgroup of G containing S. Thenthe cellular automaton � is pre-injective (resp., surjective) if and only if its restriction �His pre-injective (resp., surjective). Moreover, if A is a left R-module, then � is R-linearif and only if �H is R-linear.

We shall use the following Closure Lemma whose proof can be found inCeccherini-Silberstein and Coornaert (2007c, Lemma 3.2.) (see also Lemma 3.1 inCeccherini-Silberstein and Coornaert, 2006 in the particular case when R is a field).

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Lemma 2.13. Let G be a countable group and let M be an Artinian left module overa ring R. Let � � MG → MG be an R-linear cellular automaton. Suppose that x ∈ MG

is such that, for every finite subset � ⊂ G there exists y ∈ MG such that the restrictionsx �� and ��y� �� coincide. Then x ∈ ��MG�.

In other words, ��MG� is closed in MG for the Tychonov product topology inducedby the discrete topology on M .

2.3. Groups

Interiors, neighborhoods, and boundaries. Let G be a group. Givensubsets � and E of G, we define the E-interior �−E , the E-neighborhood �+E andthe E-boundary �E�� of �, respectively, by

�−E = �g ∈ G � gE ⊂ ��

�+E = �g ∈ G � gE ∩� �= ��

�E�� = �+E\�−E�

Example 2.14. Let G = �2, E = �−1� 0� 1�2 and � = a� b× c� d, wherea� b� c� d ∈ � and a� b = �z ∈ � � a ≤ z ≤ b�. We then have

�−E = a+ 1� b − 1× c + 1� d − 1 and �+E = a− 1� b + 1× c − 1� d + 1

as in Figure 1.

Figure 1 ��−E��+E and �E�.

Remarks. The following properties can be easily deduced from the definitions (seeCeccherini-Silberstein and Coornaert, 2006 for more details):

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1) One has G\�+E = �G\��−E;2) For any e0 ∈ E, one has �−Ee0 ⊂ � ⊂ �+Ee0. In particular, one has �−E ⊂

�⊂�+E if 1G ∈ E;3) Suppose that � � AG → AG is a cellular automaton and S ⊂ G is a memory set

for �. If x� x′ ∈ AG coincide on � (resp., outside �), then ��x� and ��x′� coincideon �−S (resp., outside �+S).

Amenability. We now recall the definition of amenability for groups andsome of its basic properties (see Ceccherini-Silberstein et al., 1999a; Greenleaf, 1969;Gromov, 1999a,b; Paterson, 1988).

Let G be a group and denote by ��G� the set of all subsets of G. Thegroup G is said to be amenable if there exists a right-invariant mean that is a map� ��G�→ 0� 1 such that the following conditions are satisfied:

(1) ��G� = 1 (normalization);(2) ��A ∪ B� = ��A�+ ��B� for all A�B ∈ ��G� such that A ∩ B = � (finite

additivity);(3) ��Ag� = ��A� for all g ∈ G and A ∈ ��G� (right-invariance).

It can be proved that if G is amenable, namely, such a right-invariant mean exists,then also left-invariant and in fact even bi-invariant means do exist. The classof amenable groups includes: finite groups, solvable groups (in particular abeliangroups), and finitely-generated groups of subexponential growth. Moreover, it isclosed under the operations of taking subgroups, taking factors, taking extensionsand taking directed limits.

The free group F2 of rank two and therefore all groups containing nonabelianfree subgroups are nonamenable.

Følner (1955) gave the following combinatorial characterization ofamenability: a countable group G is amenable if and only if it admits a Følnersequence, i.e., a sequence ��n�n∈� of nonempty finite subsets of G such that

limn→�

��E��n��n�

= 0 for all finite subsets E ⊂ G� (2.2)

(We use � · � to denote cardinality of finite sets.)

Nets. Let G be a group. Let E and F be subsets of G. A subset N ⊂ G iscalled an �E� F�-net if it satisfies the following conditions:

(i) The subsets �gE�g∈N are pairwise disjoint, i.e., gE ∩ g′E = � for all g� g′ ∈ N suchthat g �= g′;

(ii) G = ⋃g∈N gF .

Note that if N is an �E� F�-net, then it is also an �E′� F ′�-net for all E′, F ′ suchthat E′ ⊂ E and F ⊂ F ′ ⊂ G.

Using Zorn’s lemma one easily proves the following existence result fromCeccherini-Silberstein and Coornaert (2006, Lemma 2.2).

Lemma 2.15. Let G be a group. Let E be a nonempty subset of G and let F =EE−1 =�xy−1 � x� y ∈ E�. Then G contains an �E� F�-net.

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A proof of the following lemma is given in Ceccherini-Silberstein andCoornaert (2006, Lemma 4.3).

Lemma 2.16. Let G be a countable amenable group and let ��n�n∈� be a Følnersequence for G. Let E and F be finite subsets of G and let N ⊂ G be an �E� F�-net.Let N−

n ⊂ N denote the set of g ∈ N such that gE ⊂ �n. Then there exist � ∈ �0� 1 andn0 ∈ � such that �N−

n � ≥ ��n� for all n ≥ n0.

3. MEAN LENGTH FOR SUBSHIFTS OVER MODULESOF FINITE LENGTH

In this section, G is a countable amenable group, R is a ring, and M is a leftR-module of finite length ��M� < �.

It is assumed that a Følner sequence ��n�n∈� for G has been chosen once andfor all.

Given a submodule X of MG and a subset � ⊂ G, we set X� = �x�� x ∈ X�,where x�� denotes the restriction of x to �. Note that X� is a submodule of theproduct module M�. If the subset � is finite, it follows from Corollaries 2.4 and 2.5that ��X�� ≤ ��M�� = ��M� < �.

Definition 3.1. Let X be a submodule of MG. The mean length of X (with respectto the Følner sequence ��n�n∈�) is the non-negative real number �̃�X� given by

�̃�X� = lim infn→�

��X�n�

��n�� (3.1)

Observe that �̃�X� ≤ ��M� for every submodule X of MG and that equalityholds if X = MG. Note also that �̃�X� ≤ �̃�Y� if X and Y are submodules of MG suchthat X ⊂ Y .

The following statement can be deduced from a result of Ornstein and Weiss(1987). It is proved in Lindenstrauss and Weiss (2000, Th. 6.1) and Gromov (1999b,1.3.A) (see also Krieger, 2007).

Lemma 3.2 (Ornstein–Weiss Lemma). Let G be a countable amenable group andlet � �G� denote the set of all finite subsets of G. Let � � � �G� → 0�� be a functionsatisfying the following properties:

(a) �� ≤ ��′� for all ��′ ∈ � �G� such that � ⊂ �′ (monotonicity);(b) �� ∪�′� ≤ ��+ ��′� for all ��′ ∈ � �G� such that � ∩�′ = �

(subadditivity);(c) ��g�� = �� for all g ∈ G and � ∈ � �G� (left-invariance).

Then there is a real number � = ��G�� ≥ 0, depending only on G and �, such that

limn→�

��n�

��n�= �

for any Følner sequence ��n�n∈� of G.

The following shows that in the definition of the mean length �̃�X� one cantake a true limit instead of the liminf in (3.1) if the submodule X is G-invariant.

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Proposition 3.3. Suppose that X is a G-invariant submodule of MG (e.g., X= ��MG�where � � MG → MG is an R-linear cellular automaton). Then one has

�̃�X� = limn→�

��X�n�

��n��

Moreover, �̃�X� does not depend on the choice of the Følner sequence ��n�n∈�.

Proof. Let � �G� denote the set of finite subsets of G. Let us verify that thefunction � � � �G� → � defined by �� = ��X�� satisfies the assumptions ofLemma 3.2.

Let ��′ ∈ � �G� such that � ⊂ �′. Then there is a surjective R-linear mapX�′ → X� given by restriction. It follows that ��X�� ≤ ��X�′� by Corollary 2.3. Thisshows (a).

Assume now that ��′ ∈ � �G� satisfy � ∩�′ = �. Then there is a naturalembedding of left R-modules X�∪�′ ⊂ X� × X�′ . Hence we have ��X�∪�′� ≤ ��X��+��X�′� by Corollaries 2.4 and 2.5. This gives (b).

Finally, Property (c) follows from the G-invariance of X which implies thatthe map x �→ xg induces an isomorphism of left R-modules Xg� → X�. �

4. PROOF OF THE MAIN RESULT

In order to prove Theorem 1.1 we first observe that, in virtue ofProposition 2.12 we can reduce to the case where G is countable and, in fact, thatG = �S�, that is G is generated by the memory set of the cellular automaton.

We actually prove the following more precise statement.

Theorem 4.1. Let M be a left module of finite length over a ring R and let G bea countable amenable group. Let � � MG → MG be an R-linear cellular automaton.Consider the following conditions:

(a) � is pre-injective;(b) �̃��MG�� = ��M�;(c) � is surjective.

Then (a) ⇒ (b) and (b) ⇔ (c). If in addition M is semisimple, then all these conditionsare equivalent.

Before presenting the proof of this theorem we give some preliminary results.Let G be a countable amenable group and fix once and for all a Følner

sequence ��n�n∈� in G.Let also M be a nontrivial left module of finite length over a ring R. For the

moment we do not need M to be semisimple.

Proposition 4.2. Let X be a submodule of MG. Suppose that there exist finite subsetsE and F of G, and an �E� F�-net N ⊂ G such that XgE � MgE for all g ∈ N . Then�̃�X� < ��M�.

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Proof. Let us set, as in Lemma 2.16, N−n = �g ∈ N � gE ⊂ �n� and let �∗

n =�n\

∐g∈N−

ngE. The modules MgE , g ∈ N , have finite length by Corollary 2.5. Thus,

using Corollary 2.4, we deduce from our hypothesis that ��XgE� ≤ ��MgE�− 1 forall g ∈ N . It follows that

��X�n� ≤ ��X�∗

n�+ ∑

g∈N−n

��XgE�

≤ ��M�∗n �+ ∑

g∈N−n

��MgE�− 1�

= ��M�n�− �N−n �

= ��n��M�− �N−n ��

Since, by Lemma 2.16, we can find � > 0 and n0 ∈ � such that �N−n � ≥ ��n� for all

n ≥ n0, this gives us �̃�X� ≤ ��M�− � < ��M�. �

Corollary 4.3. Let X be a G-invariant submodule of MG. Suppose that there exists afinite subset � ⊂ G such that X� � M�. Then �̃�X� < ��M�.

Proof. Let us set E = � and F = EE−1. By Lemma 2.15, there exists an �E� F�-net N ⊂ G. Since X is G-invariant and X� � M�, we have XgE � MgE for all g ∈ N .This implies �̃�X� < ��M� by Proposition 4.2. �

Lemma 4.4. Let � � MG → MG be an R-linear cellular automaton. Suppose that � isnot surjective. Then �̃��MG�� < ��M�.

Proof. Let X = ��MG� and consider an element y ∈ MG\X. By Lemma 2.13, wecan find a finite subset � ⊂ G such that y �� X�. Therefore we have X� � M�.This implies �̃�X� < ��M� by Corollary 4.3. �

Proposition 4.5. Let � � MG → MG be an R-linear cellular automaton and let X bea submodule of MG. Then �̃��X�� ≤ �̃�X�.

Proof. Let us set Y = ��X�. Let S ⊂ G be a memory set for �. We can assume1G ∈ S. For each subset � ⊂ G, the automaton � induces a surjective R-linear mapfrom X� onto Y�−S . As Y� is a submodule of Y�−S × Y�\�−S ⊂ Y�−S ×M�\�−S

, thisyields, by Corollaries 2.3–2.5,

��Y�� ≤ ��Y�−S �+ ��\�−S��M�

≤ ��X��+ ��S��M��

Therefore, we have

��Y�n�

��n�≤ ��X�n

�

��n�+ ��S��n��

��n��M�

for all n ∈ �. Taking the limit as n → � and invoking (2.2), we get �̃�Y� ≤ �̃�X�.�

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Lemma 4.6. Let � � MG → MG be an R-linear cellular automaton. If � ispre-injective, then �̃��MG�� = ��M�.

Proof. Let us set Y = ��MG�. Let S be a memory set for � such that1G ∈ S. Suppose, by contradiction, that �̃�Y� < ��M�. As Y�+S

nis a submodule of

Y�n×M�+S

n \�n , we have, again by Corollaries 2.4 and 2.5,

��Y�+Sn� ≤ ��Y�n

�+ ��+Sn \�n��M�

≤ ��Y�n�+ ��S��n��M��

It follows from (2.2) that we can find an n0 ∈ � such that ��Y�+Sn0� < ��n0

��M�. LetZ denote the submodule of MG consisting of the elements of MG whose supportis contained in �n0

. Note that Z is of finite length: in fact Z � M�n0 as R-modules.Also observe that ��x� vanishes outside �+S

n0for every x ∈ Z. Thus we have

��Z�� = ��Z��+Sn0� ≤ ��Y�+S

n0� < ��n0

��M� = ��Z��

This shows that the restriction of � to Z is not injective. Therefore � is notpre-injective. �

Lemma 4.7. Suppose in addition that the module M is semisimple. Let � � MG → MG

be an R-linear cellular automaton. If � is not pre-injective, then �̃��MG�� < ��M�.

Proof. Suppose that � is not pre-injective. Then we can find an element x0 ∈ MG

with nonempty finite support � ⊂ G such that ��x0� = 0. Let S be a memory set for� such that 1G ∈ S and S = S−1. Let E = �+S2 . By Lemma 2.15, we can find a finitesubset F ⊂ G such that G contains an �E� F�-net N . Note that for each g ∈ G, thesupport of xg0 is g−1� ⊂ g−1E.

For each g ∈ N , let M ′g = R�x

g0�g−1�� denote the submodule of Mg−1� generated

by the restriction to g−1� of xg0. Every direct sum of semisimple modules is

semisimple. SinceM is semisimple, it follows that the moduleMg−1� is also semisimple.Thus, by Proposition 2.6, the submoduleM ′

g ⊂ Mg−1� is a direct summand, that is, wecan find a submoduleM ′′

g ⊂ Mg−1� such thatMg−1� = M ′g ⊕M ′′

g .Consider now the submodule X ⊂ MG consisting of all x ∈ MG such that the

restriction of x to g−1� belongs to M ′′g for each g ∈ N .

We claim that ��MG� = ��X�. Indeed, let z ∈ MG. Since Mg−1� = M ′g ⊕M ′′

g , wecan find an element rg ∈ R such that the restriction to g−1� of z+ rgx

g0 belongs to

M ′′g for each g ∈ N . Consider the element z′ ∈ MG such that z′�g−1� = �z+ rgx

g0��g−1�

for each g ∈ N and z′ = z outside∐

g∈N g−1�. We have z′ ∈ X by construction.On the other hand, since z′ and z coincide outside

∐g∈N g−1�, we have ��z′�= ��z�

outside∐

g∈N g−1�+S . Now, if h ∈ g−1�+S for some g ∈ N , then we have hS ⊂g−1�+S2 = g−1E, and therefore ��z′��h� = ��z+ rgx

g0��h� = ��z��h� since xg0 lies in the

kernel of �. Thus ��z� = ��z′� and the claim follows.Thus we have

�̃��MG�� = �̃��X�� ≤ �̃�X� < ��M�

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where the first inequality follows from Proposition 4.5 and the second one fromProposition 4.2. �

We are now able to present the proof of Theorem 4.1 (and therefore ofTheorem 1.1).

Proof of Theorem 4.1. Condition (a) implies (b) by Lemma 4.6.That (c) implies (b) immediately follows from the fact that �̃�MG� = ��M�. The

converse implication follows from Lemma 4.4.Finally, if M is semisimple, (b) implies (a) by Lemma 4.7. �

Corollary 4.8 (Myhill-Type Theorem). Let G be an amenable group and M aleft module of finite length over a ring R. Then every pre-injective R-linear cellularautomaton � � MG → MG is surjective. In particular, every injective R-linearcellular automaton � �MG → MG is surjective.

Remark 4.9. When G is a residually finite group and M is a left ArtinianR-module, it is shown in Ceccherini-Silberstein and Coornaert (2007c) thatevery injective R-linear cellular automaton � � MG → MG is surjective. The sameconclusion holds when G is sofic (a condition weaker than residual finiteness) andthe module M has finite length (a stronger condition than being Artinian): this isshown in Ceccherini-Silberstein and Coornaert (2007a,b).

Let F2 denote the free group on two generators a and b. Set S= �a� a−1� b� b−1�and consider the Cayley graph of F2 with respect to S, that is, the graph whose

Figure 2 The elements of F2 at distance ≤2 from 1.

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vertex set is F2 and g� g′ ∈ F2 are joined by an edge if and only if g−1g′ ∈ S, see, forinstance, Fig. 2.

We now provide, for any nonzero ring R, examples of R-linear cellularautomata over the free group F2 with memory set S and taking values in asemisimple left R-module of length 2.

The first cellular automaton is pre-injective but not surjective. This shows thatCorollary 4.8 fails to hold, when the group G is the (nonamenable) free group F2.Similar automata appear in Machì and Mignosi (1993, p. 55) and Ceccherini-Silberstein et al. (1999b, p. 682) in the case when R is equal to �/2�, and inCeccherini-Silberstein and Coornaert (2006) when R is an arbitrary field.

Example 4.10. Let G = F2 be the free group on two generators a� b. Let R bea nonzero ring, M0 a nontrivial simple left R-module and consider the semisimplemodule M = M0 ×M0 (cf. Remark 2.9). Also let pi � M → M , i = 1� 2, be theR-linear maps respectively defined by p1�m1�m2� = �m1� 0� and p2�m1�m2� = �m2� 0�for all m1�m2 ∈ M0.

Finally, let � � MG → MG be the R-linear cellular automaton, with memory setS = �a� b� a−1� b−1�, given by

��x��g� = p1�x�ga��+ p1�x�ga−1��+ p2�x�gb��+ p2�x�gb

−1��

for all x ∈ MG, g ∈ G. The image of � is contained in �M0 × �0��G. Therefore � isnot surjective.

Let us show that � is pre-injective. Assume that there is an element x0 ∈ MG

with nonempty finite support � ⊂ G such that ��x0� = 0. Consider a vertex g0 ∈ �at maximal distance n0 from the identity in the Cayley graph of G. The vertex g0has (at least) three adjacent vertices at distance n0 + 1 from the identity. It followsfrom the definition of � that ��x0� does not vanish at (at least) one of these threevertices. This gives a contradiction. Thus � is pre-injective.

Similarly, we now present an example of an R-linear cellular automata overthe free group F2 with values in a semisimple left R-module of length 2, which issurjective but not pre-injective.

Example 4.11. Let G, R, M0 and M be as in Example 4.10. Consider theR-linear maps qi � M → M , i = 1� 2, respectively, defined by q1�m1�m2� = �m1� 0�and q2�m1�m2� = �0�m1� for all m1�m2 ∈ M0.

Let � � MG → MG be the R-linear cellular automaton, with memory setS= �a� b� a−1� b−1�, given by

��x��g� = q1�x�ga��+ q1�x�ga−1��+ q2�x�gb��+ q2�x�gb

−1��

for all x ∈ MG, g ∈ G.Let m′ be a non-zero element in M0 and consider the configuration x ∈ MG

defined by

x0�g� ={�0�m′� if g = 1G�0� 0� otherwise.

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Then, x0 is almost equal to 0 and ��x0� = 0 = ��0�. This shows that � is not pre-injective.

However, � is surjective. To see this, let z = �z1� z2� ∈ MG0 ×MG

0 = MG: weshow that there exists x ∈ MG such that ��x� = z. We define x�g� by induction onthe graph distance, which we denote by �g�, of g ∈ G from 1G in the Cayley graphof G. We first set x�1G� = �0� 0�.

Then, for s ∈ S we set

x�s� =

�z1�1G�� 0� if s = a

�z2�1G�� 0� if s = b

�0� 0� otherwise.

Suppose that x�g� has been defined for all g ∈ G with �g� ≤ n, for some n≥ 1.For g ∈ G with �g� = n, let g′ ∈ G and s′ ∈ S be the unique elements such that�g′� = n− 1 and g = g′s′. Then, for s ∈ S with s′s �= 1G, we set

x�gs� =

�z1�g�− x1�g′�� 0� if s′ ∈ �a� a−1� and s = s′

�z2�g�� 0� if s′ ∈ �a� a−1� and s = b

�z1�g�� 0� if s′ ∈ �b� b−1� and s = a

�z2�g�− x2�g′�� 0� if s′ ∈ �b� b−1� and s = s′

�0� 0� otherwise.

Then one easily cheks that ��x� = z. This shows that � is surjective.

We end the article by showing that both implications in Theorem 1.1 fail tohold, in general, when the module M is not of finite length.

Example 4.12. Let R be a nonzero ring and M0 a simple left R-module (cf.Remark 2.9). Consider the left semisimple R-module M = ⊕

n∈� M0. We denote bym = �mn�n∈� the elements in M .

Consider the R-linear maps �� M → M defined by ��m�n = mn+1 for alln∈�, and

��m�n ={0 if n = 0

mn−1 otherwise,

for all m ∈ M .Given any group G, the product maps �G� �G � MG → MG are R-linear cellular

automata with memory set S = �1G� and local defining maps � and �, respectively.Clearly, �G is surjective but not pre-injective and �G is injective but not

surjective.

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ACKNOWLEDGMENTS

We wish to thank Claudio Procesi for a stimulating conversation on the notionof length for R-modules. The first author expresses his gratitude to IRMA Strasbourgfor financial support and for the warmest hospitality. Finally, we are very gratefulto the referee for her/his most careful reading of the manuscript and for the severalsuggestions which undoubtedly improved the presentation of our article.

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amenability and linear cellular automata over semisimple modules of finite length

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