Amenability of groups and semigroups characterized by Configuration

0.15 MB·English

by  Ali Tavakoli

AMENABILITY OF GROUPS AND SEMIGROUPS
CHARACTERIZED BY CONFIGURATION
5
1 ALI TAVAKOLI∗ ANDALI REJALI
0
2

n
a
J Abstract. In 2005, Abdollahi and Rejali, studied the relations be-

5 tween paradoxical decompositions and configurations for semigroups.
2 In the present paper, we introduce another concept of amenability on
semigroups and groups which includes amenability of semigroups and

] inner-amenability of groups. We have the previous known results to
A
semigroups and groups satisfying this concept.
F Keywords: Amenability,configuration,paradoxicaldecomposition,semi-
h. group
t MSC(2010): Primary: 22A05; Secondary: 43A07.
a
m
1. Introduction
[

1 The notion of an amenable group was introduced by von Neumann in
v 1929 in relation with his studies of the Banach-Tarski paradox. Tarski in
2
1929 proved the well known alternative theorem: a group is either amenable
5
1 or paradoxical. The theory of amenability was extended in the semigroup
6 setting by Day in the 1950s. Nowadays it plays a major role not only in
0
Geometric Group Theory, but also in Functional and Harmonic Analysis, in
.
1 Ergodic Theory and Dynamical Systems, and in Operator Algebras. The
0
notion of a configuration for groups was first introduced by Rosenblatt and
5
1 Willis in [6], but here, the definition is changed to another form.
: Let G be a finitely generated group and F be a non-empty subset of the
v
i set S(G) of all bijective maps on G. Let ϕ = (ϕ1,...,ϕn) be a sequence
X
in F such that the subgroup < F > generated by F in S(G), is equal to
r
a < ϕ1,...,ϕn > and let E = {E1,...,Em} be a partition of G. An (n+1)-
tuple C = (c ,...,c ), where c ∈ {1,...,m} for each i ∈ {0,1,...,n}, is
0 n i
called an F- configuration corresponding to the configuration pair (ϕ,E),
if there exist an element x ∈ G with x ∈ E such that ϕ (x) ∈ E , for
c0 i ci
each i ∈ {1,...,n}. The set of all F-configurations corresponding to the
configuration pair (ϕ,E) will be denoted by Con (ϕ,E).
F
Let x (C) = E ∩ϕ−1(E )∩...∩ϕ−1(E ) and x (C) = ϕ (x (C)), for
0 c0 1 c1 n cn j j 0
C ∈ Con (ϕ,E). Then the F-configuration equation corresponding to the
F
configuration pair (ϕ,E) is the system of equations
(1.1) {f | x (C)⊆ E } = {f | x (C)⊆ E },
C 0 i C j i
X X
∗
Corresponding author.
1
2 A.TAVAKOLIANDA.REJALI
where f is the variable corresponding to the configuration C. This system
C
ofequationswillbedenotedbyEq (ϕ,E). Inthiscase, thisequationsystem
F
is equivalent to a matrix equation as
(1.2) AX = 0,
where A is an nm×|Con (ϕ,E)| matrix whose entries are 0, 1 or -1 and X
F
is the vector [f ], where C runs over Con (ϕ,E).
C F
A solution [f ] to Eq (ϕ,E) satisfying {f | C ∈ Con (ϕ,E)} = 1 and
C F C C F
fC ≥ 0, for all C ∈ ConF(ϕ,E) will bePcalled a normalized solution of the
equations system (1.1). The corresponding matrix form whose solution is
normalized, has the form AX = B, where A is an (nm+1)×|Con (ϕ,E)|
F
matrix whose entries are 0, 1 or -1 and all entries of the last row of A are 1.
X is the vector [f ] and B is the vector whose last entry is 1 and all others
C
are 0. It is well known that, if A = [a ], then a = 1 [resp. a = −1] if
i,j i,j i,j
and only if x (C) ⊆ E and x (C)* E [resp. x (C)* E and x (C)⊆ E ],
i j 0 j i j 0 j
for some C ∈ Con (ϕ,E); otherwise a = 0.
F i,j
Byanon-zerosolutionofEq (ϕ,E),wemeanasolution{f |C ∈ Con (ϕ,E)}
F C F
of Eq (ϕ,E) such that f 6= 0 for some C ∈ Con (ϕ,E). We show that (see
F C F
proposition 2.1) the equation in matrix form has a non-zero solution if and
only if the latter has a normalized solution. It is easy to see that a matrix
equation AX = 0 has a non-zero solution if and only if rank(A) is less than
the number of columns of A. Therefore the matrix equation (1.2) has no
non-zero solution if and only if rank(A) ≤ |Con (ϕ,E)|.
F
The relation between amenability and configuration of a group was studied
in [6] and [7]. Here, we introduce the concept of F-amenability of a group.
Definition 1.1. A group G is called F-amenable, if there exist an F-
invariant mean M on ℓ∞(G) that is M(f ◦ϕ) = M(f), for all f ∈ ℓ∞(G)
and ϕ ∈ F, where ℓ∞(G) denotes the set of all real valued bounded functions
on G.
Now let G be a finitely generated group and L(G) ={λ : x ∈ G}, where
x
λ : G → G is the left translation y 7→ xy for each y ∈ G, and I(G) =
x
{I : x ∈ G} where I : G → G is the inner automorphism y 7→ x−1yx.
x x
Then, according to our terminology, G is L(G)-amenable [I(G)-amenable]
if and only if G is amenable [resp. inner amenable]. In general, inner
amenability is much weaker than amenability. So, F-amenability does not
imply amenability.
The configuration which introduced in [6] can be obtained as an important
special case of our notion. In fact, Rosenblatt and Willis studied
Con(G)= {Con (ϕ,E)|F is a finite subset of L(G) s.t. λ(G) =<F >}.
F
Remark 1.2. Let F =< ϕ ,...,ϕ >, for some ϕ ∈ S(G). Then each
1 n i
ϕ∈ F is a finite product of ϕ and ϕ−1. Let M be a {ϕ ,...,ϕ }-invariant
j j 1 n
mean on ℓ∞(G). Then
M(f)= M((f ◦ϕ−1)◦ϕ )= M(f ◦ϕ−1),
j j j
AMENABILITY OF GROUPS AND SEMIGROUPS CHARACTERIZED BY CONFIGURATION3
for all f ∈ ℓ∞(G) and j ∈ {1,2,...,n}. Therefore M is an F-invariant
mean on ℓ∞(G).
Now suppose that F is a non-empty subset of S(G), not necessary finite.
We have the following two facts.
(1) if M is an F-invariant mean on ℓ∞(G), then M is an < F >-invariant
mean on G.
(2) if F ⊆ F are non-empty subsets of S(G), then F -amenability of G
1 2 2
implies F -amenability of G.
1
Lemma 1.3. Let F be a non-empty subset of S(G), not necessary finite.
The following statements are equivalent.
(1) G is F-amenable.
(2) G is < ϕ ,...,ϕ >-amenable, for all finite subset {ϕ ,...,ϕ } of
1 n 1 n
F.
(3) G is {ϕ ,...,ϕ }-amenable, for all finite subset {ϕ ,...,ϕ } of F.
1 n 1 n
Proof. Due to the remark 1.2, it is sufficient to prove (3)⇒(1).
Let T be the family of all finite non-empty subsets of F. Then for every
C ∈ T, there exists a C-invariant mean MC on ℓ∞(G). If T is partially
∗
ordered by set inclusion, then, every M ∈ w −cl{M } is an F-invariant
C
mean on ℓ∞(G), where w∗−cl means the weakly-∗ closure. (cid:3)
In[6]itisprovedthatafinitelygeneratedgroupGisamenableifandonly
if each configuration equation associated to a configuration pair in Con(G)
has a normalized solution. The link between amenability and normalized
solutionisseenin[2]andcertaingrouppropertieswhichcanbecharacterized
by configurations is also studied. In [2] it is asked whether the normalized
solution can be replaced by a non-zero solution in the latter. In section 2
we not only give a positive answer to this question, but also we generalize
it for F-amenability.
Definition 1.4. Let {A ,...,A ;B ,...,B } be a partition of G such that
1 n 1 m
thereexisttwosubsets{ϕ ,...,ϕ }and{ψ ,...,ψ }ofF withthefollowing
1 n 1 m
property:
G = A ∪A ∪...∪A ∪B ∪B ∪...B
1 2 n 1 2 m
= ϕ (A )∪ϕ (A )∪...∪ϕ (A )
1 1 2 2 n n
= ψ (B )∪ψ (B )∪...∪ψ (B ).
1 1 2 2 m m
Then we say that G has an F-paradoxical decomposition (ϕ ,ψ ;A ,B ). In
i j i j
this case, the F-Tarski number of a group G is the minimum of m+n, over
all possible F-paradoxical decompositions of G and we denote it by τ (G).
F
If G has no F-paradoxical decomposition, we put τ (G) = ∞.
F
4 A.TAVAKOLIANDA.REJALI
Insection 3,westudytherelationbetween non-F-amenability andhaving
an F-paradoxical decomposition for a group.
A dynamical system is a triple (G,X,α), where α : G → S(X) is an action
of a group G on a set X. The dynamical system (G,X,α) is amenable if
there exists a finitely additive probability measure µ defined on the power
set P(X) of the space X which is α-invariant, i.e. µ(α (A)) = µ(A), for
g
all A ⊂ X and g ∈ G. We know that the dynamical system (G,X,α) is
amenable if and only if X has no paradoxical decomposition (see [4]). Let
F = {α |g ∈ G} and X = G. Then the dynamical system (G,X,α) is
g
amenable if and only if G if F-amenable.
2. F-Amenability of Groups
Throughout this section G is a finitely generated group and F is a non-
empty subset of all bijective maps on G such that < F >=< ϕ ,...,ϕ >,
1 n
where ϕ ∈ F, for i = 1,2,...,n.
i
Proposition 2.1. The following statements are equivalent.
(1) G is F-amenable.
(2) Each F-configuration equation Eq (ϕ,E) has a normalized solution.
F
(3) Each F-configuration equation Eq (ϕ,E) has a non-zero solution.
F
Proof. (1)⇒ (2) Let M be an F-invariant mean on ℓ∞(G). Then fC =
M(χ ), for C ∈ Con (ϕ,E), is a normalized solution of Eq (ϕ,E).
x0(C) F F
(2)⇒(1) Let (f ) be a normalized solution of Eq (ϕ,E).
C F
Choose x ∈ x (C) and define:
C 0
f if x = x ,
f(ϕ,E)(x) = (cid:26) C0 otherwiCse .
Theneach M ∈ w∗−cl{fˆ(ϕ,E)} satisfies M(f◦ϕ) = M(f), for all f ∈ ℓ∞(G)
and ϕ ∈ F.
(3)⇒(2) Let f ∈ ℓ (G) be a non-zero solution of Eq (ϕ,E). Define Φ ∈
1 F
∗
ℓ∞(G) by Φ(h) = x∈Gf(x)h(x), for h ∈ ℓ∞(G). There exist positive
linear functionals Φ+Pand Φ− such that Φ = Φ+ −Φ− and kΦk = kΦ+k+
kΦ−k. Since kΦk = kfk 6= 0, so we can assume Φ+ 6= 0, say. By definition,
1
Φ+(g) = sup{Φ(h) :0 ≤ h ≤g},
for any non-negative function g. Furthermore,
Φ(χEioϕj) = Φ(χϕ−j1(Ei)) = XC {Φ(χx0(C)) :xj(C)⊆ Ei}
= {f : x (C)⊆ E }
C j i
XC
= {f : x (C)⊆ E }= Φ(χ ),
C 0 i Ei
XC
AMENABILITY OF GROUPS AND SEMIGROUPS CHARACTERIZED BY CONFIGURATION5
for all i and j. Thus Φ(h◦ϕ ) = Φ(h), for all h ≥ 0. Therefore:
j
Φ+(χϕ−j1(Ei)) = sup{Φ(h◦ϕj) :0 ≤ h◦ϕj ≤ χEi ◦ϕj}
= sup{Φ(h) : 0≤ h ≤ χ } = Φ+(χ ).
Ei Ei
Letk = Φ+(χ )/kΦ+k,then(k )isanormalizedsolutionofEq (ϕ,E).
C x0(C) C F
(2)⇒(3) This is trivial. (cid:3)
Corollary2.2. LetG andG befinitelygeneratedgroupssuchthatCon (G )=
1 2 F1 1
Con (G ). Then G is F -amenable if and only if G is F -amenable.
F2 2 1 1 2 2
3. F-Paradoxical Decomposition of Groups
In this section, we generalize Tarski’s theorem on amenability for F-
amenability of groups. For the special case, set F = L(G).
Let F be a subgroup of S(G) under composition operation and A,B ⊆ G.
So A and B are F-equidecomposable if there exist partitions {A ,...,A }
1 m
and {B ,...,B } of A and B, respectively, and elements ϕ ∈ F such that
1 m i
ϕ (A ) = B for all i = 1,...,m. If A and B are F-equidecomposable, then
i i i
we write A ∼= B. We say that A ≤ B, if A ∼= C for some subset C of B. It
is routine to show that ”∼=” is an equivalence relation on power set P(G).
Also a standard Cantor-Bernstein argument shows that A ≤ B and B ≤ A
implies A ∼= B.
LetS bethesetofallbijectivemapsonN. Define(ϕ,p)(x,n) =(ϕ(x),p(n)),
N
for ϕ∈ F and p ∈ S . Let
N
N = {C ⊆ G×N : C ⊆ B ×F for some B ⊆ G and finite set F ⊆ N}.
n
Then each N ∈ N can be written uniquely in the form N = C ×{j },
i=1 i i
where 1 ≤ j1 < j2 < ... < jn and ∅ =6 Ci ⊆ G. S
n n
Let N = C ×{j } and N = D ×{k } be elements of N. Then
1 i=1 i i 2 i=1 i i
N1 ∼= N2 ifSand only if there exist ϕiS∈F and pi ∈ SN such that ϕi(Ci) = Di
and pi(ji) = ki, for i ∈ {1,2,...n}. Define = N∼= = {N∼ : N ∈ N},
where N∼ is the equivalence class of N. ChPoose h ∈ F × SN such that
h(N )∩N = ∅. Then is an abelian semigroup under addition operation
1 2
N1∼+N2∼ := (h(N1)∪NP2)∼.
Define α = (G×{1})∼, so 2α = α+α = (G×{1}∪G×{2})∼.
In the following, we show that, G is F-amenable if and only if α 6= 2α.
A finitely additive probability measure µ of the power set P(G) is called
F-invariant, if µ(φ(A)) = µ(A) for all A ⊆ G and φ ∈F.
Lemma 3.1. The following statements are equivalent.
(1) G is F-amenable.
(2) There exist an F ×S -invariant measure µ on N such that µ(G×
N
{1}) = 1.
6 A.TAVAKOLIANDA.REJALI
(3) There exist a homomorphism f : → [0,∞) such that f(α) = 1.
P
(4) α 6= 2α.
Proof. (1)⇒(2) let ν be an F-invariant measure on P(G). Define µ(N) =
n ν(C ), for each N = ∪n C × {j } in N. Since ν(G) = 1, we have
i=1 i i=1 i i
Pµ(G×{1}) = 1 and
n n n
µ(ϕ×p(N)) = µ( ϕ(C )×{p(j )}) = ν(ϕ(C )) = ν(C ) = µ(N).
i i i i
i[=1 Xi=1 Xi=1
Hence µ is an F ×S -invariant measure on N.
N
(ii)⇒(i)LetµbeanF×S -invariantmeasureonN. Thenν(A) = µ(A×{1})
N
is an F-invariant measure on P(G). Thus G is F-amenable.
(3)⇒(2) Let ν(A) = f(A×{1})∼, for A⊆ G. Then
ν(G) = f(G×{1})∼ = f(α) = 1,
and
ν(A ∪A ) = f((A ×{1})∪(A ×{1}))∼ = ν(A )+ν(A ),
1 2 1 2 1 2
for A ,A ⊆ G such that A ∩A = ∅.
1 2 1 2
(4)⇒(3) Let T = {nα : n ∈ N} and F : T → [0,∞) defined by F(nα) = n.
Then by a similar argument as is used in [5], p. 119, α 6= 2α if and only if
kα 6= lα whenever k 6= l. T is a sub-semigroup of the abelian semigroup
and F(α) = 1; also s ≤ t in T (i.e. s = t or there exist w ∈ T such thPat
s+w = t)implies F(s) ≤ F(t); thus F can beextended to a homomorphism
f : → [0,∞) so that f(α) = 1 by [5], p. 117.
(1)⇒P(3)Letν beanF-invariantmeasureinG. Definef(N∼)= ni=1ν(Ci),
for N = ∪ni=1Ci ×{ji}. Let N1 = ∪ni=1Ci ×{ji} and N2 = ∪ni=P1Di ×{ki}
and N∼ = N∼. Then N∼ ∼= N∼, so there exist ϕ ∈ F and p ∈ S
1 2 1 2 i i N
such that ϕ (C ) = D and p (j ) = k . Hence f(N∼) = n ν(C ) =
i i i i i i 1 i=1 i
ni=1ν(ϕ(Ci)) = f(N2∼), so f is well-defined. P
LPet h = ϕ×p ∈ F ×SN, such that h(N1)∩N2 = ∅. Then:
n
f(N∼+N∼) = f( ϕ(C )×{p(j )}∪D ×{k })
1 2 i i i i
i[=1
n n n n
= ν(ϕ(C ))+ ν(D ) = ν(C )+ ν(D )
i i i i
Xi=1 Xi=1 Xi=1 Xi=1
= f(N∼)+f(N∼).
1 2
So f is a homomorphism. Clearly, f(α)= f((G×{1})∼)= ν(G) = 1.
(3)⇒(4) Since f(α) = 1, so f(2α) = 2. Thus α 6= 2α. Hence the proof is
complete. (cid:3)
We now state the main result of this section.
Theorem 3.2. The following statements are equivalent.
AMENABILITY OF GROUPS AND SEMIGROUPS CHARACTERIZED BY CONFIGURATION7
(1) G is F-amenable.
(2) There exist no F-paradoxical decomposition for G.
Proof. (1)⇒(2) Suppose not! Let ν be an F-invariant measure for G and
(ϕ ,ψ ;A ,B ) be an F-paradoxical decomposition for G. Then:
i j i j
n n n
1= ν(G) = ν( ϕ (A )) = ν(ϕ (A )) = ν(A ).
i i i i i
i[=1 Xi=1 Xi=1
Similarly, m ν(B ) = 1. Hence,
j=1 j
P n m
1= ν(G) = ν( A )+ν( B )= 1+1 = 2,
i j
i[=1 j[=1
which is a contradiction.
(2)⇒(1) Suppose not! so by lemma 3.1, α = 2α. Then G× {1} ∼= (G ×
{1})∪(G×{2}). Thus there exist a partition {A ×{1},...,A ×{1};B ×
1 n 1
{1},...,B × {1}} of G × {1} and (ϕ ,p ),(ψ ,q ) ∈ F × S such that
m i i j j N
p (1) = 1 and q (1) = 2 for all i and j, so that:
i j
n m
G×{1} G×{2} =( ϕ ×p (A ×{1})) ( ψ ×q (B ×{1}))
i i i j j j
[ i[=1 [ j[=1
Thus G×{1} = ∪n ϕ (A )×{1} and G×{2} = ∪m ψ (B )×{2}. Hence
i=1 i i j=1 j j
G = ∪ϕ (A ) = ∪ψ (B ). So G has an F-paradoxical decomposition, which
i i j j
is a contradiction. (cid:3)
Similar to [7], we are interested to construct an F-paradoxical decompo-
sition for non-F-amenable groups by using F-configuration equations and
conversely.
Let (ϕ ,ψ ;A ,B ) be an F-paradoxical decomposition of G and f ∈ ℓ+(G).
i j i j 1
Then:
kfk = {f : C ∈ Con (ϕ,E)}
1 C F
XC
n m
= {f : x (C) ⊆ A }+ {f :x (C)⊆ B }
C 0 i C 0 j
XC Xi=1 XC Xj=1
n m
= {f : x (C) ⊆ A }+ {f :x (C)⊆ B }
C i i C j j
XC Xi=1 XC Xj=1
= 2 {f :C ∈ Con (ϕ,E)} = 2kfk ,
C F 1
XC
whereϕ = (ϕ ,...,ϕ ;ψ ,...,ψ )andE = {A ,...,A ;B ,...,B }. There-
1 n 1 m 1 n 1 m
fore Eq (ϕ,E) has no non-zero solutions.
F
SupposeEq (ϕ,E)hasnonon-zerosolutions. SupposeCon (ϕ,E) = {D ,...,D }
F F 1 s
such that E = ∪r1 x (D ),E = ∪r1+r2 x (D ) and so on. Define E′ =
1 i=1 0 i 2 i=r1+1 0 i
8 A.TAVAKOLIANDA.REJALI
′ ′ ′
{E : i = 1,...,s} where E = x (D ) for each i. Then Eq (ϕ,E ) has a
i i 0 i F
′
non-zero solutions. Similarly, if Eq (ϕ,E ) has a non-zero solutions then
F
Eq (ϕ,E) has a non-zero solutions.
F
Question 3.3. Let Eq (ϕ,E) be a system of equations having no non-zero
F
solution for some configuration pair (ϕ,E). How can ”explicitly” construct
an F-paradoxical decomposition from Eq (ϕ,E)?
F
It is to be noted that G =< g ,g ,...,g > is non-amenable if and only
1 2 l
if the equation |g−1E ∩X| = |E ∩X|, for 1 ≤ i ≤ l,1 ≤ j ≤ m, has no
i j i
non-empty finite solution X in G, for some partition E = {E ,...,E }.
1 m
Example 3.4. [6]. Let G =< g ,g > be the free group on two (free)
1 2
generators g ,g and E be the set of all reduced words starting g , for i =
1 2 i i
1,2, and E = G−(E ∪E ). Then Eq(ϕ,E) has no non-zero solution. In
3 1 2
compare to the above notations, let
(ϕ )= (1,λ ,λ ),(ψ ) = (1,1,1,λ ,λ )
i g1 g1 j g2 g2
and
′ ′ ′ ′ ′ ′
(A )= (E ,E ,E ),(B )= (E ,E ,A,E ,B),
i 1 2 5 j 2 3 7
′ ′
for some A ⊆ E and B = E −A. Then (ϕ ,ψ ;A ,B ) is a paradoxical
6 6 i j i j
decomposition of G.
4. F-Amenability of Semigroups
In this section, a new type of amenability for semigroups is introduced.
Also the notion of an F-paradoxical decomposition for semigroups which
was asked by Paterson in special case in [5] p. 120, is defined. We find the
relation between the existence of F-paradoxical decompositions and non-
F-amenability for semigroups. The definition is almost similar to that of
groups, we bring it for completeness.
Let S be a discrete semigroup and A ⊆ S. For any map f : S → S
(not necessary invertible), recall that f−1(A) = {t ∈ S : f(t) ∈ A}. Let
ϕ = (ϕ ,...,ϕ ) be an n-tuple of the functions (not necessary invert-
1 n
ible) on S and E = {E ,...,E } be a partition of S. An (n + 1)-tuple
0 1 m
C = (c ,...,c ), where c ∈ {1,...,m} for each i ∈{0,1,...,n}, is called a
0 n i
configuration corresponding to the configuration pair (ϕ,E ), if there exist
0
anelementx ∈ S withx ∈ E suchthatϕ (x) ∈ E ,foreachi∈ {1,...,n}.
c0 i ci
The set of all configurations corresponding to the configuration pair (ϕ,E )
0
will be denoted by Con(ϕ,E ). Let E = {ϕ−1(E ) : j ∈ {1,...,m}}, for
0 i i j
each i∈ {1,...,n}. Then E is a partition of S for each i = 1,2,...,n. (We
i
remove empty elements from these collections.)
Let x (C)= E ∩ϕ−1(E )∩...∩ϕ−1(E ) and x (C)= ϕ (x (C)).
0 c0 1 c1 n cn j j 0
Let F be a non-empty subset of the set of all maps SS on S. For n-tuples
ϕ= (ϕ ,...,ϕ ) in F such that the semigroup < F > generated by F in SS
1 n
AMENABILITY OF GROUPS AND SEMIGROUPS CHARACTERIZED BY CONFIGURATION9
is equal to < ϕ ,...,ϕ >, we call a configuration correspondingto the con-
1 n
figuration pair (ϕ,E) is denoted by Con (ϕ,E). Then the F-configuration
F
equations corresponding to the configuration pair (ϕ,E) are defined simi-
larly to the previous case. (equations (1.1))
A semigroup S is called F-amenable, if there exists an F-invariant mean M
on ℓ∞(S), that is M(f ◦ϕ) = M(f), for all f ∈ ℓ∞(S) and ϕ ∈ F, where
ℓ∞(S) denoted the set of all real valued bounded functions on S.
Adler and Hamilton, [3], showed that S is left amenable if and only if S
satisfies the following left invariant condition:
for any sequence (s ,...,s ) in S and for all sequences (A ,...,A ) of sub-
1 n 1 n
sets in S there exists a non-empty finite set X ⊆ S such that |s−1A ∩X| =
i i
|A ∩X| for all i∈ {1,2,...,n}.
i
We prove that S is F-amenable if and only if S satisfies the F-invariant
condition.
Definition 4.1. Let {A ,...,A ;B ,...,B } be a partition of semigroup
1 n 1 m
S and there exist two subsets {ϕ ,...,ϕ } and {ψ ,...,ψ } of F such that
1 n 1 m
the sets {ϕ−1(A ),...,ϕ−1(A )} and {ψ−1(B ),...,ψ−1(B )} are two par-
1 1 n n 1 1 m m
titions of S. Then we say that S admits an F-paradoxical decomposition
(ϕ ,ψ ;A ,B ). In this case, the F-Tarski number of a semigroup S is the
i j i j
minimum of m+n, over all possible F-paradoxical decompositions of S.
We show that the F-Tarski number for semigroups can be 2; however the
correspondingnumberforgroupsisatleast4. Atfirst,byasimilarargument
as in used in proposition 2.1, the following proposition is immediate.
Proposition 4.2. The following statements are equivalent.
(1) S is F-amenable.
(2) Each F-configuration equation Eq (ϕ,E) has a normalized solution.
F
(3) Each F-configuration equation Eq (ϕ,E) has a non-zero solution.
F
Lemma 4.3. The following statements are equivalent.
(1) S is F-amenable.
(2) For any sequence (ϕ ,...,ϕ ) in F and for all sequence (A ,...,A )
1 k 1 k
of subsets in S, there exist a finite non-empty subset X ⊆ S such
that,
|ϕ−1(A )∩X| = |A ∩X|, for all i= 1,...,k.
i i i
(3) Foranysequence(ϕ ,...,ϕ )inF andforeachpartition {E ,...,E }
1 n 1 m
of S, there exist a non-empty finite subset X ⊆ S such that,
|ϕ−1(E )∩X|= |E ∩X|, for all i,j.
i j j
10 A.TAVAKOLIANDA.REJALI
Proof. (2)⇒(3) Let (ϕ ,...,ϕ ) be a sequence in F and {E ,...,E } be a
1 n 1 m
partition of S. Put
A = E ,A = E ,...,A = E for all j = 1,...,m.
j j m+j j (n−1)m+j j
Put also,
′ ′ ′
ϕ = ϕ ,ϕ = ϕ ,...,ϕ = ϕ for all j = 1,...,m.
j 1 m+j 2 (n−1)m+j n
′ ′
Then for (ϕ ,...,ϕ ) and (A ,...,A ), there exists a non-empty subset
1 mn 1 mn
X ⊆ S such that,
|ϕ−1(E )∩X| = |E ∩X|,
i j j
for i∈ {1,2,...,n} and j ∈ {1,2,...,m}.
(3)⇒(2) Let (ϕ ,...,ϕ ) bea sequence in F and (A ,...,A )bea sequence
1 k 1 k
of subsets in S. Let E = {A ,Ac}, for i = 1,...,k and E be the family of
i i i
all n-tuple intersections on E . Clearly, the cardinality of E is 2n and it is
i
a partition of S. By (3), There exist a finite, non-empty subset X ⊆ S so
that |ϕ−1(E)∩X| = |E ∩X| for all i = 1,...,k and E ∈ E. Then one can
i
show that easily |ϕ−1(A )∩X| = |A ∩X|, for all i= 1,...,k.
i i i
For example, if k = 1, then E = {A ,Ac} and there exist a finite, non-
1 1
empty subset X ⊆ S such that |ϕ−1(A )∩X| = |A ∩X|. Also, if k = 2,
1 1 1
then E = {A ∩A ,A ∩Ac,Ac ∩A ,Ac ∩Ac}. Hence, there exist a finite
1 2 1 2 1 2 1 2
non-empty subset X ⊆ S such that,
|ϕ−1(E)∩X| = |E ∩X|,
i
for all i ∈{1,2} and E ∈ E. Now we have:
|ϕ−1(A )∩X| = |ϕ−1(A ∩A )∩X|+|ϕ−1(A ∩Ac)∩X|
1 1 1 1 2 1 1 2
= |(A ∩A )∩X|+|(A ∩Ac)∩X| = |A ∩X|.
1 2 1 2 1
Similarly, |ϕ−1(A )∩X| = |A ∩X|.
2 2 2
This completes the proof of (2).
(3)⇒(1) SupposeE = {E ,...,E } is a partition of S and ϕ= (ϕ ,...,ϕ )
1 m 1 n
is a sequence in F. Then there exist a non-empty finite subset X ⊆ S such
that,
|ϕ−1(E )∩X| = |E ∩X|, for all i,j.
i j j
Let
1
f = |X ∩x (C)|, for all C ∈ Con (ϕ,E).
C 0 F
|X|
Therefore, [f ] is a normalized solution. In fact:
C
1
{f : x (C)⊆ E } = |ϕ−1(E )∩X|
C i j |X| i j
X
1
= |E ∩X| = {f :x (C)⊆ E }.
j C 0 j
|X|
X
Hence, S is F-amenable.
(1)⇒(2) See [3]. (cid:3)