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1.4.
POWERFUL DIGRAPHS
1.4.2. Transitive closures of powerful digraphs
101
Recall that a partially ordered set
(
upward)directed
z ≤ x
and
z ≤ y
if for each
(respectively,
x, y ∈ X
x ≤ z
hX; ≤i
is said to be
, there exists
and
y ≤ z
).
z ∈ X
downward
such that
Let us list the main possibilities that exhaust the structures
of the transitive closures of powerful digraphs obtained from the
x)
structures of nonprincipal powerful types
x)
ber of nonprincipal
1.4.2.1. Theorem.
hX; Qi
sitive closure
in which
TC(Γ) =¿X;
acl({a}) ∩ 5(a) = {a}
-t
l(
ypes is nite.
Given a saturated powerful digraph
S
n∈ω
À
n
Q
is isomorphic to a downward
,
p(
for each
for which the num-
Γ =
a ∈ X
, the tran-
directed set with a transitive automorphism group and one of the
following orders:
(1α)
a dense partial order with maximal antichains containing
α
elements,
(2)
α ∈ (ω + 1) \ {0}
;
a partial order with innitely many covering elements for
each element.
Proof.
the digraph
≤
follows from the acyclicity ofΓ. The existence in the partially
ordered set
The reexivity and transitivity of the relation
TC(Γ) = hX; ≤i
TC(Γ)
of the meet of two arbitrary elements follows
are obvious. The antisymmetry of
≤
of
from the pairwise intersection property. If the order≤is not dense
then the existence of innitely many covering elements for each
a ∈ X
situation the consistent formula
does not belong to an algebraic type overa.
follows from the relation
acl({a})∩5(a) = {a}
, since in this
(a < x) ∧ ¬∃y ((a < y) ∧ (y < x))
¤
Note that the dense partial orders with maximal antichains
of cardinalityα, that we mentioned, are realized by replacing each
element in a dense linear order without endpoints by an equivalence
class containingαpairwise incomparable elements.
Note that a partial order with innitely many covering elements
for each element comes only from the powerful digraphs for which
the formula
Q(x, y)
is not principal.
102
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
Indeed, if
of
|= Q(a, b) ∧ Q(a, c) ∧ Q(c, b)
Q(x, y)
is a principal formula then the truth
for eachaand some
b, c
and the
existence of automorphisms, xingaand connecting arbitrary el-
ements in
in
Q(a, Γ) ∩ Q(Γ, b)
Q(a, Γ)
, imply that for eachbin
. Consequently, in the graph
Q(a, Γ)
TC(Γ)
there exists
, between
each pair of distinct elements there is another element. Therefore,
we have
1.4.2.2. Corollary.
a principal formula
If
Γ = hX; Qi
Q(x, y)
is a powerful digraph with
then the relation
S
Qnis a dense
n∈ω
partial order.
Note that if the relation≤in the transitive closure of a sat-
urated powerful digraph
Γ = hX; Qi
is not denable by formulas
in the language ofΓ(i. e., if the lengths of the shortest paths are un-
bounded) then, by Compactness Theorem, in
there is an innite antichain belonging to
TC(Γ)
Q(a, Γ)
, over eacha,
. Therefore, the
theorem 1.4.2.1 implies
1.4.2.3. Corollary.
hX; Qi
with unbounded lengths of shortest paths such that
Given a saturated powerful digraph
Γ =
c
acl({a}) ∩ 5(a) = {a}
for each
a ∈ X
, its transitive closure
TC(Γ)
is isomorphic to
a downward directed set with a transitive automorphism group that
has one of the following orders:
(1)
a dense partial order with innite antichains;
(2)
a partial order with innitely many covering elements
for each element.
Note that while the necessity of the local presence of powerful
digraphs in the structures of nonprincipal powerful types is proved,
the question remains open of suciency, i. e., the possibility of ex-
pansion of each powerful digraph to the structure of a powerful
type.
1.4.
POWERFUL DIGRAPHS
103
1.4.3. Forking. Simple theories. Weight for powerful di-
graphs
Recall several notions of Stability Theory related to the class
of simple theories [23, 47, 59].
1.4.3.1. Denition.
saturated model
a
theory
of
type
isk-inconsistent
formula
T k
-divides
tp(a/A)
V
ϕ(
n∈w
such that the set
, i. e., for every
x, an)
A partial type
x)
implied
type
divides
by
whic
π(
overAif theyk-divide for some
A partial type
x),
ϕ0(
o
. . . , ϕn(
verA.
If
p ∈ S(A),q ⊃ p
x)
is called to be a (
q ⊃fp(q ⊃nfp
).
A theoryTis called (
does not fork over a subsetAofBwith
1.4.3.2. Remark.
Let
k ∈ ω,A
M
of a theoryT,
over a setAif there are tuples
is
inconsistent inT.
x) k
π(
-divides
be a set in a suciently large
a ∈ M
.
A formula
ϕ(
an,
x, an) | n ∈ ω}
{ϕ(
w ⊂ ω
o
verAif there is a formula
of the cardinalitykthe
of
formulas
h isk-divides overA. A formula or a partial
k ∈ ω
.
x)
forks
π(
suc
h that
o
verAif there are
W
π(
x) `
ϕi(x)
i≤n
n ∈ ω
,
and each
and formulas
x)
ϕi(
, andqforks (does not fork) overAthen
non-)forking extensionofp
super)simple
if for any type
and it is denoted by
p ∈ S(B),p
|A| ≤ |T|(|A| < ω
By the denition, a supersimple theory is sim-
x, a)
in
n ∈ ω
x)
ϕ(
divides
).
ple. Moreover,Tis supersimple if and only if there do not exist
A0⊆ A1⊆ . . . ⊆ Ai. . .
for each
i ∈ ω
.
and
pi∈ S(Ai),i ∈ ω
, such that
p
i+1⊃fpi
,
q
1.4.3.3. Remark.
Every (super)stable theory is (super)simple.
The following properties of non-forking in simple theories are
shown by B. Kim [256].
1. (
S(B)
2. (
only if
Extension
.
Symmetry
tp(c/A
) For any
) A type
b)
do
es not fork overA.
p ∈ S(A)
tp(b/A
c)
and
A ⊆ B,p
do
es not fork overAif and
has
q ⊃nfp
in
104
3. (
Chapter
Transitivity
1. CHARACTERIZATION OF EHRENFEUCHTNESS
) If
A ⊆ B ⊆ C
and
p ∈ S(C)
, thenpdoes not
fork overAif and only ifpdoes not fork overBand
restriction ofptoB, does not fork overA.
p ¹ B
, the
aisdep
not dependent of
a
If
b
o
A tuple
is
verA.
1.4.3.4. Remark.
endent
In view of forking symmetry, the (in)dependence
b
o
of
verAif
b
o
verA, one say that
tp(a/A
b)
divides
aisindep
overA.
endent
of
in simple theories is symmetric too.
a
1.4.3.5. Denition.
o
verAwe may say that
being (in)dependent over
By Remark 1.4.3.4, if
a
andbare
∅
are called simply (in)
sequence (set) of tuples is said to be
(in)
dependent
independent
is
dependent of
overA. Tuples
dependent
(overA) if each
. A
tuple of this sequence (set) is independent (overA) with every tuple
formed by coordinates of other elements of the sequence (set).
a
A theoryThas the innite weight
A
, and an innite independent sequence
a
and
the tuples
A type
a realization
of
realizations of
for each
n ∈ ω
anare
x) ∈ S(∅)
p(
aofp(x)
p(
.
dependent overAfor each
has
the innite own weight
and
an innite independent sequence
x)
suc
h that the tuples
if there exist a tuple
an)
(
o
verAsuch that
n∈ω
n ∈ ω
if there exists
a
and
anare
,
a set
.
an)
(
n∈ω
dependent
The following proposition shows that the powerful digraphs do
not occur in the models of known classes of simple theories that
do not include Ehrenfeucht theories.
b
1.4.3.6. Proposition.IfT = Th(Γ)
powerful digraph
Γ = hX; Qi
then the(unique)type
is a simple theory of some
p ∈ S1(∅)
the innite own weight.
Proof.
a
andbare dependent. For that it suces to establish that the
formula
m ∈ ω
m
elements
all
1 ≤ i < j ≤ m
acyclicity of the digraphΓ, there is an innite sequence
At rst we show that if
Qk(a, x)
is copied over∅. Indeed, there exists a number
|= Qk(a, b)
such that for each element
a1, . . . , amsatisfying the conditions
, because otherwise, by compactness and the
for some
k > 0
a0there is at most
Qk(ai, aj)
(an)
has
, then
for
n∈ω
1.5.
THE TSUBOI AND KIM THEOREMS
105
with the condition
|= Qk(ai, aj) ⇔ i < j,
which contradicts the simplicity ofT.
Dene a sequence
ement
a0inX. If
satisfying the condition
maximal number of sets
imply that the set
(an)
a0, . . . , a
Γ |= Qk(a
inductively. Pick an arbitrary el-
n∈ω
have been chosen then pick
n−1
, an)
n−1
Qk(ai, Γ),i < n
{Qk(an, x) | n ∈ ω}ism
a
, and belonging to the
. The remarks above
-inconsistent. Since ev-
ery two elements are connected by an automorphism, the formula
Qk(a, x)
is copied over∅.
Notice now that, by the pairwise intersection property, for all
elements
a1, . . . , an∈ X
, there exists
a ∈ Q(Γ,a1) ∩ Q2(Γ, a2) ∩ .. . ∩ Qn(Γ, an),
and, in particular, everynelements comprising an independent se-
quence depend on somea. Since every two elements are connected
by an automorphism and the integernis unbounded, there exist
innitely many elements that form an independent sequence and
depend ona.
¤
n
1.5. The Tsuboi and Kim theorems
1.5.1. Dense orders and unions ofω-categorical theories
Recall [52, 53] that a (
transitive, antisymmetric relation such that any two distinct com-
parable elements
a, b
ment greater thanaand less thanb, or less thanaand greater than
b
. An order is
identical
elements are comparable.
Obviously, a non-identical dense order, on a set containing at
least two comparable elements, has an innite
many pairwise comparable elements. There are also innitely many
pairwise comparable elements for any strict dense order, connecting
at least two elements.
strict)dense order
is an (ir)reexive,
have an intermediate element, i. e., an ele-
, if, relative to that order, only coincident
chain
, i. e., innitely
106
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
An order≤(respectively a strict order<) on a setAof tuples
in a structure
x, y),l(x)
ϕ(
M
= l(
is (formula)
y)
,
such that for any tuples
denable
if there exists a formula
a, binA
,
a ≤ b ⇔
Clearly
M |= ϕ(
if a theoryT(i. e., some model ofT) has a denable
a, b)¡a<b ⇔
M |= ϕ(
a, b)¢.
order with an innite chain on a denable set, thenThas the strict
order property. A theoryThas also the strict order property if
T
has a denable set with a denable strict order having innitely
many pairwise comparable elements.
Now we present the Tsuboi theorem on non-representability of
Ehrenfeucht theories without denable non-identical dense orders
and, in particular, without the strict order property, as unions of
ω
-categorical theories.
1.5.1.1. Theorem.
theory and is a union ofω-categorical theories
that
which
Proof.
Tn⊆ T
denes a non-identical dense order on a denable set.
,
n+1
By Lemmas 1.1.1.2 and 1.1.1.10, the Ehrenfeucht the-
oryThas a nonprincipal powerful type
satisfying
•
the following conditions:
the formula
symmetric on the set of realizations ofpin a model
•
for any realization
Consider a theory
T
. We set
n
0
z(ϕk(x, z) ∧ ϕ(z,y)),k ∈ ω \
∃
of
T
, and
n
0
T
ϕ0(
n
0
pairwise non-equivalent, in
(A. Tsuboi [461]).
n ∈ ω
, then there exists a formula
x, y)
ϕ(
witnesses
aofp
T
such that
n
0
,
the formula
x, y) (x ≈ y),ϕ1(x, y) ϕ(x,y),ϕ
{0}
. Since all
IfTis an Ehrenfeucht
Tn,
x)
and
p(
a formula
that the relation
M |= T
a, y)
is
is
a formula of
x, y)
ϕ(
ϕ(
x, y)
ϕk(
n ∈ ω
ψ(
, such
x, y)
ϕ(
SIpis non-
;
principal.
k+1
(x, y)
are
formulas
of
x, y)
T
isω-categorical, there are at most nitely many
T
, formulas
n
0
ϕ0(
x, y),
ϕ1(
x, y),
. . . , ϕ
m−1
x, y)
(
among
the set
Using
all formulas
ϕk(
{k < m | ` ∃
thatF, dene
Dl,
D0 m \ {0}; D
x, y),k ∈ ω
.
For each
i, j ∈ ω
let
z(ϕi(x, z) ∧ ϕj(z,y)) ↔ ϕk(x, y)}.
l ∈ ω
, by the following induction:
[{F (i, j) | i, j ∈ Dl}.
l+1
F (i, j)
be
1.5.
THE TSUBOI AND KIM THEOREMS
107
It is clear that for each
is nite,
D
T
l∈ω
l ∈ ω,D
⊆ Dland
l+1
Dl6= ∅
Dlis a nonempty subset of cardinal
. Since
m
which
m
contains some nonzero cardinal. For thisD, we put
_
x, y)
ψ(
By
compactness is suces to show that
i∈D
ϕi(x, y).
ψ(
x, y)
denes
a non-
identical dense order on the set of realizations ofpinM, i. e., the
set
P {(
is
that order.
Since
is reexive. The relationPis antisymmetric; indeed, if
a
then
By
x) ` (ψ(x, y)∧¬x ≈ y) ↔∃z(ψ(x, z)∧ψ(z,y)∧¬x ≈ z∧¬z ≈ y).
p(
Th
usPis transitive and dense.
a, b) | M |= ψ(a, b) ∨ (a ≈ b), M |= p(a), M |= p(b)}
D \ {0} 6= ∅,P
semi-isolatesb,
is non-identical. By the denition,
b
and
can
semi-isolate
a
only
the denitions ofFandDwe have
¤
for
a, b) ∈ P
(
a = b
.
P
Notice that, in the conditions of previous theorem, to prove
the strict order property for the theory
weak version of Tsuboi Theorem [460]) it suces to consider the
set
χ(
a, M)
Φ(
a, M)
for
der property holds for the formula
x)
p(
.
of
Indeed,
exists a realization
a, M)
χ(
tions
ai,
.
Since
i ∈ ω
S
ϕk(a, M)
k∈ω
some formula
ϕ(
x, y)
since
bofpinM
a
andbrealize
,
ofpin
whic
h is an
x, y)
χ(
,
and to prove that the strict or-
χ(
witnesses
suc
that
h that
the same typep, there are realiza-
M
such that the following holds:
T
(it is asserted in a
a
x, y)
-formula
on
a set of realizations
,
i. e., equals
SIpis non-symmetric, there
b ∈ χ(a,M)
and
χ(b, M) ⊂
M |= ∀
y(χ(ai, y) → χ(aj, y)) ⇔ i ≤ j
.
It means thatThas the strict order property.
Since the strict order property for a theoryTimplies thatTis
unstable, the following corollary holds.
108
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
1.5.1.2. Corollary.
(A. Tsuboi [460, 461]).
IfTis a countable
stable theory without non-identical denable dense orders on den-
able sets(in particular, ifTis stable)andTis obtained from an
ω
-categorical theory by addition of axioms for new constants, then
I(T, ω) = 1orI(T, ω) ≥ ω
.
1.5.2. Pseudo-superstable and pseudo-supersimple theo-
ries
1.5.2.1. Denition
pseudo-superstableifT
Similarly we say that a simple theoryTis
(A. Tsuboi [461]). A stable theory
fails to have the innite weight.
pseudo-supersimple
T
is
ifTfails to have the innite weight.
1.5.2.2. Remark.
Every supersimple theory is pseudo-
supersimple. Actually the proof of the Kim theorem [257] implies
that any pseudo-supersimple theory is not an Ehrenfeucht theory.
1.5.2.3. Example
ing equivalence relations
is divided into innitely many
since
|S(M)| = |M|
(A. Tsuboi [461]). LetTbe the theory of ren-
En(x, y),n ∈ ω
E
n+1
for any model
, such that each
En-class
-classes. The theoryTis stable
M |= T
of cardinality
2ω. The
theoryTis pseudo-supersimple since the dependence of elements
a
andbmeans thataandbbelong to some common
T
is non-supersimple by Remark 1.4.3.2 since each type
that
is realized by elementsbbeing non-
a
ements of
by elementscbeing
elements of
,
has a forking extension
En-equivalent and non-
ab.¤
En-equivalent to the el-
E
ab)
n+1
q ∈ S(
En-class. And
p ∈ S(
,
that is realized
a)
-equivalent to the
Recall
is the theory of countable
disjoint innite sets
that for any nite disjoint subsets
that
an example of a supersimple unstable theory [257]: this
bipartite random graphM, consisting of
U, V
with the relationRbetween
(a, c) ∈ R
for
a ∈ A
and
A, BofU
(b, c) 6∈ R
for
there is
b ∈ B
, and vice versa.
U, V
c ∈ V
such
such
Taking a disjoint union of a union of pseudo-superstable theo-
ries (similar to Example 1.5.2.3) and of a theory of random graph
we get a theoryTwhich is the union of pseudo-supersimple theories
1.5.
THE TSUBOI AND KIM THEOREMS
109
Tnwith
theories
1.5.2.4. Denition
A nonempty set
tp(a/A) ∈ S
class
Tn⊆ T
0
T
with
n
such thatTis not a union of pseudo-superstable
n+1
0
n
⊆ T
0
n+1
,
n ∈ ω
.
T
(A. Tsuboi [461]). LetSbe a subset of
R ⊆ S(A)
and
tp(b/A) ∈ S
, consisting of some types
, is said to be a
(onS) if the following conditions hold:
tp(
tp(
ab/
ab/
A) ∈ R
A) ∈ R
then
and
(a) if
(b) if
∅
-classes are called simply
tp(a/A
tp(
b) ⊃ftp(a/
bc/
A) ∈ R
classes
.
ab/
tp(
transitive forkingA-
A)
;
tp(
ac/
A) ∈ R
then
LetTbe a simple theory,Ra transitive forkingA-class, and
p
a type in
cardinalκsuch that for every
and
tuples
abi/
tp(
1.5.2.5. Remark.IfT
1.5.2.6. Proposition
b
b
e realizations of a type
then
tp(
Proof.
c
b
Let
W
e claim that
b/ac)
so
tp(
Indeed, let
aibici)
(
cia
tp(
i+1
i ≤ j
. It suces to show that
If not, then there is
Then
plies
to
(
a
(
b, a) 6∈ SIp.
(
No
w if
non-forking, we can nd a tuple
a, b, c0}
{
a/b)
tp(
S(A)
. TheR-weight
bi,
i ≤ λ
,
such that
A) ∈ R
for each
i ≤ λ
is pseudo-supersimple then
(B. Kim [256]).
p ∈ S(T )
a/b)
forks
over∅.
Take a formula
e
any
tuple such that
x, a, c) ϕ(a, x) ∧ ϕ(x, c)
ψ(
forks
over∅.
a0= a,b0= b,c0= c
suc
i∈ω
h that for all
)
.
By transitivity of semi-isolation,
d
aj, d), (d, ci) ∈ SIp,
, ci) ∈ SIp.
i+1
a
andbare
is
independent. This contradicts the claim above. Thus,
forks
over∅.
i ∈ ω,tp(
suc
h that
so
But since
independent (over∅), then, by properties of
¤
wR(p)ofpinR
λ < κ
, there is a realization
(
bi)
i≤λ
is
independent overAand
.
LetTbe a simple theory,
a, b) ∈ SIpand
. If
(
x, y)
ϕ(
tp(
{ψ(
aj, ci) ∈ SIp,
(
witnessing
ab)
bc)
= tp(
.
There exists a sequence
aibici)
= tp(
x, ai, ci) | i ∈ ω}is2
aj, d) ∧ϕ(d, ci)
|= ϕ(
and
ab)
tp(
c0suc
= tp(
h that
is the maximal
wR(p) ≤ ω
(b, a) 6∈ SI
that
a, b) ∈ SIp.
(
.
forks
over∅, and
abc)
and
ai, aj) ∈ SIpfor
(
-inconsisten
for
some
a
, aj) ∈ SIpim-
(
i+1
cia
)
,
it contradicts
i+1
ab)
tp(
= tp(
tp(ab)
S(A)
A)
aofp
bc0)
with
.
.
a
every
j > i
and
.
,
p
=
t.
.
110
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
Lemma 1.1.3.7 and Proposition 1.5.2.6 imply
1.5.2.7. Corollary
b
b
e realizations of a type
a/b)
then
tp(
forks
(B. Kim [256]).
p ∈ S(T )
over∅.
LetTbe a simple theory,
. If
a, b) ∈ Ipand
(
(b, a) 6∈ I
a
,
p
1.5.2.8. Corollary.
R {tp(
LetTbe a simple theory,
ab) | (a,b) ∈ SIpand
ThenRis a transitive forking class on
(b, a) 6∈ SIp}
{p}
.
x) ∈ S(T )
p(
6= ∅.
Repeating the proof of [461, Proposition 3.3] we obtain
1.5.2.9. Proposition.
R
be a transitive forkingA-class onS. Then there is a type
with
wR(p) = 1
Proof.
wR(p) ≥ 2
realizations
(1)
both
a
(2) {
aj)
Let
(
a
.
Since
2i+2
two realizations
(1)0both
0
b
(2)
.
By way of a contradiction, assume that for any
. We shall construct by induction a sequence
of types inSsuch that
a2ia
tp(
| j ≤ i}
2j+1
b
j≤2i
e already dened. We have to dene
a2irealizes
b
andcof
a2ib/
tp(
andcare
LetTbe a pseudo-supersimple theory and
2i+1
/
∪ {
A)
a
and
2i+2
a2ia
tp(
}
are
independent overA.
2i+2
/
A)
belong toR;
a type inS, by the assumption, there are
types inSsuch that
A)
and
tp(
a2ic/
A)
belong toR;
independent overA.
(
a
ai)
2i+1
,
p ∈ S
p ∈ S
i∈ω
and
,
of
Now we choose tuples
a
2i+1a2i+2
.
e prove that these
above. By (3) and non-forking symmetry, we have
a
tp((
2j+1)j
a2i})
{
and
(3) tp(
W
(2)
<i
/
A ∪ {
/A
a
a2ia
a
and
2i+1
a
2j+1
and
2i+1
2i+1a2i+2
a
2i+2
| j
< i} ∪ {
a
2i+2
) ⊃nftp((a
so
that
a2i}) ⊃nftp(bc/
satisfy
the conditions
2j+1)j
<i
/A
A ∪
(1)
a2i).
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