AMENABILITY AND LINEAR CELLULAR AUTOMATA OVER - IRMA

16 Pages·2007·0.22 MB·English

AMENABILITY AND LINEAR CELLULAR
AUTOMATA OVER SEMISIMPLE MODULES OF
FINITE LENGTH
TULLIO CECCHERINI-SILBERSTEIN AND MICHEL COORNAERT
Abstract. Let M be a semisimple left module of finite length
over a ring R and let G be an amenable group. We show that an
R-linearcellularautomatonτ: MG →MG issurjectiveifandonly
if it is pre-injective.
1. Introduction
Let G be a group and M be a left module over a ring R. We consider
the set MG consisting of all maps x: G → M. Note that MG is a left
module over R in a natural way. Moreover, the group G acts on MG
on the right by (x,g) 7→ xg, where xg(g0) = x(gg0) for all x ∈ MG and
g,g0 ∈ G.
An R-linear cellular automaton over G with values in M is a map
τ: MG → MG satisfying the following condition: there exist a finite
subset S ⊂ G and an R-linear map µ: MS → M such that
(1.1) τ(x)(g) = µ(xg| ) for all x ∈ MG,g ∈ G,
S
where xg| denotes the restriction of xg to S. Such a set S is called
S
a memory set and µ is called a local defining map for τ. Observe
that every linear cellular automaton τ: MG → MG is R-linear and G-
equivariant(seeSection2forthegeneralnotionofacellularautomaton
and some examples).
Two elements x,x0 ∈ MG are said to be almost equal if the set
{g ∈ G : x(g) 6= x0(g)} is finite. Equivalently, denoting by M[G]
the submodule of MG consisting of all maps from G to M with finite
support,twoelementsx,x0 ∈ MG arealmostequalifandonlyifx−x0 ∈
M[G].
Following terminology introduced in [Gro-1], one says that a map
f: MG → MG is pre-injective if, for any x,x0 ∈ MG which are almost
equal, if f(x) = f(x0), then x = x0. Every injective map MG → MG is
pre-injective but the converse is not true in general. Observe also that
Date: 23 March 2007.
2000 Mathematics Subject Classification. 43A07, 37B15, 20F65, 37B15, 20F65,
13E05, 13E10, 16P70, 68Q80.
Key words and phrases. Linear cellular automaton, pre-injectivity, amenable
group, Artinian module, Noetherian module, module of finite length, semisimple
module.
1
2 TULLIO CECCHERINI-SILBERSTEIN AND MICHEL COORNAERT
every R-linear cellular automaton τ: MG → MG induces by restriction
an R-linear map τ : M[G] → M[G], and that τ is pre-injective if and
0
only if τ is injective.
0
The aim of this paper is to establish the following theorem (see Sec-
tion 2 for the 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 let G be an amenable group. Let τ: MG → MG be an
R-linear cellular automaton. Then τ is surjective if and only if it is
pre-injective.
We note that when R is a field, the preceding theorem reduces to
Theorem 1.2 in [CC1]. It can also be viewed as a linear analogue of the
classical “Garden of Eden Theorem” for cellular automata with finite
alphabet. ThelatterwasprovedbyMooreandMyhillforG = Z2 [Moo],
[Myh]andthenextendedtogroupsofsub-exponentialgrowthbyMach`ı
and Mignosi [MaMi] and to general amenable groups by Ceccherini-
Silberstein, Mach`ı and Scarabotti [CMS].
The paper is organized as follows. In Section 2 we introduce the
notation and present some background material. In particular we re-
call some basic facts in the theory of modules (length, semisimplicity),
of cellular automata (with values in an arbitrary alphabet), and of
amenable groups. In the subsequent section we define the mean length
‘e(X) of a submodule X ⊂ MG. It plays the role of entropy used in the
classical framework, namely, for cellular automata with finite alphabet.
In Section 4 we establish Theorem 1.1 by showing that both surjectiv-
ity and pre-injectivity of τ are equivalent to the fact that ‘e(τ(MG)) is
equal to the length of M (Theorem 4.1). We also consider linear cel-
lular automata over non-amenable groups. We provide two examples
of R-linear cellular automata over the free group F with values in a
2
semisimple module of length 2: the first one is pre-injective but not
surjective, the second one is surjective but not pre-injective. Finally,
we show that the hypothesis of 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 shall need hereafter. We refer to
the books by Atyiah and Macdonald [AtM], Bourbaki [Bou], Golan and
Head [GH], Hungerford [Hun], or by Zariski & Samuel [ZS] 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
M ⊂ M ⊂ ··· ⊂ M ⊂ M ⊂ ···
0 1 n n+1
AMENABILITY AND LINEAR CELLULAR AUTOMATA 3
eventually stabilizes (i.e., there exists n ∈ N such that M = M for
0 n n0
all n ≥ n ).
0
One says that M is Artinian (or that it satisfies the descending chain
condition) if every decreasing sequence of submodules of M
M ⊃ M ⊃ ··· ⊃ M ⊃ M ⊃ ···
0 1 n n+1
eventually stabilizes.
The length ‘(M) ∈ N∪{∞} of the module M is defined as being the
supremum of the integers n such that M admits a strictly increasing
sequence of submodules
M ( M ( M ( ··· ( M
0 1 2 n
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 Noe-
therian 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 left Artinian ring is a left module of finite length over
itself. (cid:3)
Note that there are left Noetherian rings which are not left Artinian
(e.g. the ring Z of integers) and that there are Artinian modules which
are not Noetherian (e.g. the subgroup of Q/Z consisting of elements
whose order is a power of a fixed prime p, viewed as a Z-module).
Proposition 2.2. Let M be a left R-module and let N be a submodule
of M. Then M is Noetherian (resp. Artinian, resp. of finite length) if
and only if N and M/N are both Noetherian (resp. Artinian, resp. of
finite length). Moreover, one has
(2.1) ‘(M) = ‘(N)+‘(M/N).
(cid:3)
Corollary 2.3. Let M and M0 be left R-modules. Suppose that there
exists a surjective R-linear map u: M → M0. Then ‘(M0) ≤ ‘(M). (cid:3)
Corollary 2.4. Let M be a left R-module and let N be a submodule of
M. Then one has ‘(N) ≤ ‘(M). In particular, if M has finite length
so does N. Moreover, if M has finite length and ‘(N) = ‘(M), then
N = M. (cid:3)
Corollary 2.5. Let M and M be left R-modules. Then
1 2
‘(M ×M ) = ‘(M )+‘(M ).
1 2 1 2
In particular, if M and M have finite length, then so does M ×
1 2 1
M . (cid:3)
2
4 TULLIO CECCHERINI-SILBERSTEIN AND MICHEL COORNAERT
Let M be a left R-module.
One says that M is simple if M 6= {0} and the only submodules of
M are {0} and M. Equivalently M is simple if it has length ‘(M) = 1.
One says that M is semisimple if it can be written as a direct sum,
M = ⊕ N (with J possibly infinite), with each submodule N sim-
j∈J j j
ple. It follows from this definition that every direct sum of semisimple
modules is semisimple.
A submodule N of a module M is called a direct summand if there
exists a submodule N0 ⊂ M such that M = N ⊕N0.
Proposition 2.6. Let M be a left module over a ring R. Then the
following conditions are equivalent:
(a) M is semisimple;
(b) every submodule of M is a direct summand.
(cid:3)
Corollary 2.7. Every submodule of a semisimple module is semisim-
ple. (cid:3)
The following theorem characterizes the semisimple modules which
have finite length.
Theorem 2.8. Let M be a semisimple left module over a ring R. Let
(N ) be a family of simple submodules of M such that M = ⊕ N .
j j∈J j∈J j
Then the following conditions are 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|. (cid:3)
Remark 2.9. Let R be a nonzero ring. By Zorn’s lemma we can find a
maximal left ideal N ⊂ R. Then M = R/N is a simple left R-module
0
and, for any positive integer n, the left R-module M = ⊕n M is
i=1 0
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 or the set of states. We denote by
AG = {x: G → A} the set of all maps from G with values in A. Such
maps are called configurations. The group G acts on the right on AG
by setting xg(g0) = x(gg0) for all x ∈ AG and g,g0 ∈ G. A cellular
automaton over G with values in A is a map τ: AG → AG such that
there exist a finite subset S ⊂ G, called a memory set, and a map
µ: AS → A, called a local defining map, such that τ(x)(g) = µ(xg| )
S
for all x ∈ AG and g ∈ G.
Example 2.10 (Conway’s Game of Life). The universe is G = Z2, the
free abelian group of rank two: the group elements are referred to
AMENABILITY AND LINEAR CELLULAR AUTOMATA 5
as “cells”. The set of states is 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) is interpreted
as “birth” (resp. “death”). The cellular automaton with memory set
S = {−1,0,1}2 ⊂ Z2 and local defining map µ: AS → A given by
 P
y(s) = 3
  s∈S

1 if or
µ(y) =
P
 y(s) = 4 and y((0,0)) = 1,
 s∈S

0 otherwise
for all y ∈ AS, is called the Game of Life. It was introduced by John
Horton Conway [BCG].
Suppose that A = M is a left module over a ring R. Then we
say that a cellular automaton τ: MG → MG is R-linear provided
it is an R-linear endomorphism of the left R-module MG into itself.
Note that this is equivalent to the R-linearity of the local defining map
µ: MS → M.
Example 2.11 (The Laplace operator). Let G be a group, S a finite
subset of G containing 1 (the identity element of the group), and
G
A = M, aleftmoduleoveraringR. TheR-linearmap∆ : MG → MG
S
defined by
X
∆ (x)(g) = (|S|−1)x(g)− x(gs)
S
s∈S\{1G}
is a cellular automaton over G with memory set S and local defining
map µ: MS → M given by µ(y) = (|S|−1)y(1 ) − P y(s),
G s∈S\{1G}
for all y ∈ MS. It is called the Laplace operator or discrete Laplacian
over G with coefficients in M.
Let τ: AG → AG be a cellular automaton with memory set S and
local defining map µ: AS → A.
Note that any finite subset S0 ⊂ G containing S is also a memory
set for τ. The corresponding local defining map µ0: AS0 → A is given
by µ0 = µ◦π where π : AS0 → AS is the restriction map.
S0,S S0,S
Let now H be a subgroup of G containing S and consider the map
τ : AH → AH defined by τ (x)(h) = µ(xh| ) for all x ∈ AH and
H H S
h ∈ H. Then τ is a cellular automaton over H, which is called the
H
restriction of τ to H ([CC1]).
Proposition 2.12. Let G be a group, A an alphabet and τ: AG → AG
a cellular automaton with memory set S ⊂ G. Let also H be a sub-
group of G containing S. Then the cellular automaton τ is pre-injective
(resp. surjective) if and only if its restriction τ is pre-injective (resp.
H
surjective). Moreover, if A is a left R-module, then τ is R−linear if
and only if τ is R-linear. (cid:3)
H
6 TULLIO CECCHERINI-SILBERSTEIN AND MICHEL COORNAERT
We shall use the following Closure Lemma whose proof can be found
in [CC3, Lemma 3.2.] (see also Lemma 3.1 in [CC1] in the particular
case when R is a field).
Lemma 2.13. Let G be a countable group and let M be an Artinian
left module over a ring R. Let τ: MG → MG be an R−linear cellular
automaton. Suppose that x ∈ MG is such that, for every finite subset
Ω ⊂ G there exists y ∈ MG such that the restrictions x| and τ(y)|
Ω Ω
coincide. Then x ∈ τ(MG).
In other words, τ(MG) is closed in MG for the Tychonov product
topology induced by the discrete topology on M. (cid:3)
2.3. Groups.
Interiors, neighborhoods and boundaries. Let G be a group.
Given subsets Ω and E of G, we define the E-interior Ω−E, the E-
neighborhood Ω+E and the E-boundary ∂ (Ω) of Ω respectively by
E
Ω−E = {g ∈ G : gE ⊂ Ω},
Ω+E = {g ∈ G : gE ∩Ω 6= ∅},
∂ (Ω) = Ω+E \Ω−E.
E
Example 2.14. Let G = Z2, E = {−1,0,1}2 and Ω = [a,b] × [c,d],
where a,b,c,d ∈ Z and [a,b] = {z ∈ Z : a ≤ z ≤ b}. We then have
Ω−E = [a+1,b−1]×[c+1,d−1] and Ω+E = [a−1,b+1]×[c−1,d+1]
as in the figure below.
t t t t t t
t t t t t t t t t t
t t t t t t t t t t
t t t t t t
Ω Ω−E
t t t t t t t tt t t t t t t t t
t t t t t t t tt t t t t t t t t
t t t t t t t tt t t t t
t t t t t t t tt t t t t
t t t t t t t tt t t t t t t t t
t t t t t t t tt t t t t t t t t
Ω+E ∂ (Ω)
E
Figure 1. Ω, Ω−E, Ω+E and ∂ Ω
E
AMENABILITY AND LINEAR CELLULAR AUTOMATA 7
Remarks. The following properties can be easily deduced from the def-
initions (see [CC1] for more details).
1) One has G\Ω+E = (G\Ω)−E.
2) For any e ∈ E, one has Ω−Ee ⊂ Ω ⊂ Ω+Ee . In particular, one
0 0 0
has Ω−E ⊂ Ω ⊂ Ω+E if 1 ∈ E.
G
3) Suppose that τ: AG → AG is a cellular automaton and S ⊂ G is
a memory set for τ. If x,x0 ∈ AG coincide on Ω (resp. outside Ω) then
τ(x) and τ(x0) coincide on Ω−S (resp. outside Ω+S).
Amenability. We now recall the definition of amenability for groups
and some of its basic properties (see [CGH], [Gre], [Gro-1], [Gro-2],
[Pat]).
LetGbeagroupanddenotebyP(G)thesetofallsubsetsofG. The
group G is said to be amenable if there exists a right-invariant mean
that is a map µ: P(G) → [0,1] such that the following conditions are
satisfied:
(1) µ(G) = 1 (normalization);
(2) µ(A∪B) = µ(A)+µ(B) for all A,B ∈ P(G) such that A∩B = ∅
(finite additivity);
(3) µ(Ag) = µ(A) for all g ∈ G and A ∈ P(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 class of amenable groups includes: finite groups, solvable
groups (in particular abelian groups) and finitely–generated groups of
subexponential growth. Moreover, it is closed under the operations of
taking subgroups, taking factors, taking extensions and taking directed
limits.
The free group F of rank two and therefore all groups containing
2
non-abelian free subgroups are non-amenable.
Følner[Føl]gavethefollowingcombinatorialcharacterizationofamenabil-
ity: a countable group G is amenable if and only if it admits a Følner
sequence, i.e., a sequence (Ω ) of non-empty finite subsets of G such
n n∈N
that
|∂ (Ω )|
E n
(2.2) lim = 0 for all finite subsets E ⊂ G.
n→∞ |Ωn|
(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 is called an (E,F)-net if it satisfies the following conditions:
(i) the subsets (gE) are pairwise disjoint, i.e., gE ∩g0E = ∅ for
g∈N
all g,g0 ∈ N such that g 6= g0;
S
(ii) G = gF.
g∈N
Note that if N is an (E,F)-net then it is also an (E0,F0)-net for all
E0, F0 such that E0 ⊂ E and F ⊂ F0 ⊂ G.
8 TULLIO CECCHERINI-SILBERSTEIN AND MICHEL COORNAERT
Using Zorn’s lemma one easily proves the following existence result
from [CC1, Lemma 2.2].
Lemma 2.15. Let G be a group. Let E be a non-empty subset of G and
let F = EE−1 = {xy−1 : x,y ∈ E}. Then G contains an (E,F)-net.
(cid:3)
A proof of the following lemma is given in [CC1, Lemma 4.3].
Lemma 2.16. Let G be a countable amenable group and let (Ω )
n n∈N
be a Følner sequence for G. Let E and F be finite subsets of G and
let N ⊂ G be an (E,F)-net. Let N− ⊂ N denote the set of g ∈ N
n
such that gE ⊂ Ω . Then there exist α ∈ (0,1] and n ∈ N such that
n 0
|N−| ≥ α|Ω | for all n ≥ n . (cid:3)
n n 0
3. Mean length for subshifts over modules of finite
length
In this section, G is a countable amenable group, R is a ring, and M
is a left R-module of finite length ‘(M) < ∞.
It is assumed that a Følner sequence (Ω ) for G has been chosen
n n∈N
once and for all.
Given a submodule X of MG and a subset Ω ⊂ G, we set X =
Ω
{x| : x ∈ X}, where x| denotes the restriction of x to Ω. Note
Ω Ω
that X is a submodule of the product module MΩ. If the subset Ω
Ω
is finite, it follows from Corollaries 2.4 and 2.5 that ‘(X ) ≤ ‘(MΩ) =
Ω
|Ω|‘(M) < ∞.
Definition 3.1. Let X be a submodule of MG. The mean length of
X (with respect to the Følner sequence (Ω ) ) is the non–negative
n n∈N
real number ‘e(X) given by
‘(X )
(3.1) ‘e(X) = liminf Ωn .
n→∞ |Ωn|
Observe that ‘e(X) ≤ ‘(M) for every submodule X of MG and that
equality holds if X = MG. Note also that ‘e(X) ≤ ‘e(Y) if X and Y are
submodules of MG such that X ⊂ Y.
The following statement can be deduced from a result of Ornstein
and Weiss in [OrW]. It is proved in [LiW, Th. 6.1] and [Gro-2, 1.3.A]
(see also [Kri]).
Lemma 3.2 (Ornstein-Weiss Lemma). Let G be a countable amenable
group and let F(G) denote the set of all finite subsets of G. Let
φ: F(G) → [0,∞) be a function satisfying the following properties:
(a) φ(Ω) ≤ φ(Ω0) for all Ω,Ω0 ∈ F(G) such that Ω ⊂ Ω0 (monotonic-
ity);
(b) φ(Ω∪Ω0) ≤ φ(Ω)+φ(Ω0) for all Ω,Ω0 ∈ F(G) such that Ω∩Ω0 = ∅
(sub-additivity);
AMENABILITY AND LINEAR CELLULAR AUTOMATA 9
(c) φ(gΩ) = φ(Ω) for all g ∈ G and Ω ∈ F(G) (left-invariance).
Then there is a real number λ = λ(G,φ) ≥ 0, depending only on G and
φ, such that
φ(Ω )
n
lim = λ
n→∞ |Ωn|
for any Følner sequence (Ω ) of G. (cid:3)
n n∈N
The following shows that in the definition of the mean length ‘e(X)
one can take a true limit instead of the liminf in (3.1) if the submodule
X is G-invariant.
Proposition 3.3. Suppose that X is a G-invariant submodule of MG
(e.g. X = τ(MG) where τ: MG → MG is an R-linear cellular automa-
ton). Then one has
‘(X )
‘e(X) = lim Ωn .
n→∞ |Ωn|
Moreover, ‘e(X) does not depend on the choice of the Følner sequence
(Ω ) .
n n∈N
Proof. Let F(G) denote the set of finite subsets of G. Let us verify
that the function φ: F(G) → N defined by φ(Ω) = ‘(X ) satisfies the
Ω
assumptions of Lemma 3.2.
Let Ω,Ω0 ∈ F(G) such that Ω ⊂ Ω0. Then there is a surjective R-
linear map X → X given by restriction. It follows that ‘(X ) ≤
Ω0 Ω Ω
‘(X ) by Corollary 2.3. This shows (a).
Ω0
Assume now that Ω,Ω0 ∈ F(G) satisfy Ω ∩ Ω0 = ∅. Then there is
a natural embedding of left R-modules X ⊂ X ×X . Hence we
Ω∪Ω0 Ω Ω0
have ‘(X ) ≤ ‘(X )+‘(X ) by Corollaries 2.4 and 2.5. This gives
Ω∪Ω0 Ω Ω0
(b).
Finally, Property (c) follows from the G-invariance of X which im-
plies that the map x 7→ xg induces an isomorphism of left R-modules
X → X . (cid:3)
gΩ Ω
4. Proof of the main result
In order to prove Theorem 1.1 we first observe that, in virtue of
Proposition 2.12 we can reduce to the case where G is countable and,
in fact, that G =< 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 be a countable amenable group. Let τ: MG → MG be an
R−linear cellular automaton. Consider the following conditions:
(a) τ is pre-injective;
(b) ‘e(τ(MG)) = ‘(M);
(c) τ is surjective.
10 TULLIO CECCHERINI-SILBERSTEIN AND MICHEL COORNAERT
Then (a) ⇒ (b) and (b) ⇔ (c). If in addition M is semisimple, then
all these conditions are 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 (Ω ) in G.
n n∈N
Let also M be a non-trivial 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 subsets E and F of G, and an (E,F)-net N ⊂ G such that
X ( MgE for all g ∈ N. Then ‘e(X) < ‘(M).
gE
Proof. Let us set, as in Lemma 2.16, N− = {g ∈ N : gE ⊂ Ω }
n n
and let Ω∗ = Ω \ ‘ gE. The modules MgE, g ∈ N, have finite
n n g∈Nn−
length by Corollary 2.5. Thus, using Corollary 2.4, we deduce from our
hypothesis that ‘(X ) ≤ ‘(MgE)−1 for all g ∈ N. It follows that
gE
X
‘(X ) ≤ ‘(X )+ ‘(X )
Ωn Ω∗n gE
g∈Nn−
X
≤ ‘(MΩ∗n)+ (‘(MgE)−1)
g∈Nn−
= ‘(MΩn)−|N−|
n
= |Ω |‘(M)−|N−|.
n n
Since, by Lemma 2.16, we can find α > 0 and n ∈ N such that
0
|N−| ≥ α|Ω |foralln ≥ n , thisgivesus‘e(X) ≤ ‘(M)−α < ‘(M). (cid:3)
n n 0
Corollary 4.3. Let X be a G-invariant submodule of MG. Suppose
that there exists a finite subset Ω ⊂ G such that X ( MΩ. Then
Ω
‘e(X) < ‘(M).
Proof. Let us set E = Ω and F = EE−1. By Lemma 2.15, there
exists an (E,F)-net N ⊂ G. Since X is G-invariant and X ( MΩ,
Ω
we have X ( MgE for all g ∈ N. This implies ‘e(X) < ‘(M) by
gE
Proposition 4.2. (cid:3)
Lemma 4.4. Let τ: MG → MG be an R−linear cellular automaton.
Suppose that τ is not surjective. Then ‘e(τ(MG)) < ‘(M).
Proof. Let X = τ(MG) and consider an element y ∈ MG \ X. By
Lemma 2.13, we can find a finite subset Ω ⊂ G such that y| ∈/ X .
Ω Ω
Therefore we have X ( MΩ. This implies ‘e(X) < ‘(M) by Corol-
Ω
lary 4.3. (cid:3)
Proposition 4.5. Let τ: MG → MG be an R-linear cellular automa-
ton and let X be a submodule of MG. Then ‘e(τ(X)) ≤ ‘e(X).