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2.5.
UNIFORMd-AMALGAMATION PROPERTY
The following theorem generalizes Theorem 2.28 in J. T. Bald-
win and N. Shi [96].
141
2.5.1.2. Theorem.If(D0; 6)
uniformd-amalgamation property and the class
sures, then the
over, any nite set
A ⊆ M
diagram
ω
,
the type
Φ(A)
Pr
oof.
-saturated model of
andNare
between
M
(D0; 6)
A ⊆
A)
tp(
L
,
Let
∞,ω
Φ(Y ) ` tp(A)
and
M
b
-equiv
andNthere
-generic structure
M
c
ontains the type
e a
(D0; 6)
Th(
alent. To do this, it suces to establish that
are nite partial isomorphismsf:
is a self-sucient class having the
K
has nite clo-
Misω
is
extendable to its self-sucient closure
Φ(Y )
-satur
ated. More-
for
a self-sucient
.
-generic structure and
M)
.
We show that the structures
which are mutually extendable for any self-sucient sets
and
A0⊆ N
Φ0(A0)
, where
Letf:
conditions above. Consider a self-sucient diagram
with
Φ0(A0) 6 Ψ0(B0)
generic structure,
is realized overAand is such that
and
Φ(A) 6 Ψ(B)
that the required extensiong:
f
exists.
Consider a self-sucient diagram
Φ(A) 6 Ψ(B)
corresp
Φ0(A0) = Φ(A0)
∼
A
→
onding to self-sucient diagrams
.
A0be a nite partial isomorphism satisfying the
Ψ0(B0) ∈ D
M
and
M
N |= Ψ0(B0)
has
an isomorphic copyBof
. Since
M
|= Ψ(B),Ψ(B) = Ψ0(B)
, for a self-sucient diagram
∼
B
B0of the partial isomorphism
→
is
B0over
Ψ(B)
. This means
Ψ(B) ∈ D0with
M
and
|= Ψ(B)
. Since the formulas
a
N
be an
∼
A
→
A ⊆
Φ(A)
(D0; 6)
A0which
M
A
M
and
0
0
-
,
∀
are true in
x ((χ
M
(X) ∧ ϕ(X)) →
Φ
,
they are also true inN. The set
∃Y (χ
(X
, Y ) ∧ ψ(X, Y )))
Ψ
{(χΨ(A0, Y )} ∪ {ψ(A0, Y ) | ψ(A, B \ A) ∈ Ψ(B)}
is locally satisable and the structure
N
isω-saturated; so the
set at hand is satisable inN, that is, there exists a set
satisfying
a self-sucient diagram
the required extensiong:
N |= Ψ0(B0),Ψ0(B0) = Ψ(B0)
Ψ0(B0)
. Hence again we are faced up to
∼
B
B0off.
→
, and
Φ0(A0) 6 Ψ0(B0)
B0⊆ N
, for
142
Chapter
2. GENERIC CONSTRUCTIONS
In view of the possibility for extending isomorphismsf:
A
∼
→
A0,
so as to preserve formulas of corresponding self-sucient diagrams
Φ(A)
we conclude that the structure
and
Φ0(A0)
, on the basis of the back-and-forth argument,
M
with
a constantly distinguished
setAis isomorphic to a countable elementary substructure of the
structureNwith a constantly distinguished set
A
and
A0are chosen arbitrarily and the structure
we see that any type over a nite set in
M
is
a saturated structure.
M
A possibility for extending partial isomorphismsf:
A0. Since the sets
N
is saturated,
is
realized in, that is,
∼
B
→
B0so
as to preserve formulas of corresponding self-sucient diagrams
Ψ(B)
an automorphism of
tweenBand
Φ(A)
orem 2.3.0.11 above), any nite set
self-sucient closure
type
and
Ψ0(B0)
, implies that if
M
extending
B0. Hence,
tp
M
Ψ(B) = Ψ0(B)
a given partial isomorphism be-
(B)
= tp
M
(B0)
.
can be extended to its self-sucient closure
M
Φ(Y )
A ⊆ M
for
a self-sucient diagram
so
A ⊆
that the type
Φ(A)
,
is
and
then there exists
Since any diagram
Φ(A)
(see
The-
extendable to its
A)
tp(
con
tains the
Φ(Y ) ` tp(A)
¤
Theorem
Lemma 1.2.1.7, implies that for any self-sucient class
with uniformd-amalgamation, if the class
then the
∆(D0)
istentially quantifying over conjunctions
ϕi(X) ∈ Φ(X),i = 1, . . . , n
2.5.1.2, combined with Compactness Theorem and
(D0; 6)
K
has nite closures,
(D0; 6)
-generic theory
T = Th(
M)is∆(D0)
-based,
where
is a set consisting of all possible formulas obtained by ex-
n
V
, where
ϕi(X)
i=1
Φ(A) ∈ D0for some setA.
of formulas
Thus Theorem 2.5.1.2 gives rise to the following:
.
2.5.1.3. Corollary.
served under Boolean combinations of the formulas,
If
P
is some property of formulas pre-
(D0; 6)
is a
self-sucient class possessing the uniformd-amalgamation prop-
erty, and the class
(D0; 6)
∆(D0)
-generic theory possessesPif and only if any formula in
has the propertyP.
K
has nite closures, then any formula of
2.5.
UNIFORMd-AMALGAMATION PROPERTY
143
In V. Harnik and L. Harrington [194], it was stated that any
Boolean combination of stable formulas is itself a stable formula.
Based on Corollary 2.5.1.3 we derive
2.5.1.4. Corollary.If(D0; 6)
is a self-sucient class possessing
the uniformd-amalgamation property, and the classKhas nite
closures, then the
any formula in the set
(D0; 6)
∆(D0)
{generic theory is stable if and only if
is stable.
2.5.2. Examples
Saturated models of Examples 1.2.3.5, 1.3.3.3, and 1.4.1.7 can
be naturally obtained by corresponding generic constructions using
Theorem 2.5.1.2 guaranteing required∆-basednesses. Below we il-
lustrate the mechanism constructing saturated model for Example
1.2.3.5 (for Examples 1.3.3.3 and 1.4.1.7 similar constructions cor-
respond to the schema below).
2.5.2.1. Denition.
M
be anω-saturated model of
structureAof language
binary relations
Q0,
restriction of the relations on
is thec-graphifW
(1)
shortest
ρ(ai, aj)
(Q0∪ Q
describing lengths, colors of arcs and vertices in
−1
0
Let
T0be the theory in Example 1.2.3.5,
T0,Abe a nite set inM. The
Σ(T0)
, consisting of the universeA, the
Q1and unary relations
M
to the setA, and the recordW,
Coln,
n ∈ ω
consists of the following formulas:
∪ Q1∪ Q
−1
)
-paths between elements
1
, being the
aiand
a
inA;
(2)
¬ρ(al, am)
\saying" that elements
aland
amofAbelong to
distinct connected components.
Thec-graphAis denoted by
pose that the empty structure (with
hA, Q0, Q1, Coln, W i
A = ∅
) is also ac-graph.
n∈ω
.
We sup-
j
Note that eachc-graphAcan be syntactically represented by
diagram
These formulas are written as conjunctions of formulas
where
Φ(A)
consisting of formulas described in items (1) and (2).
ϕ(x, y) ∈ ∆,δ ∈ {0, 1}
ϕδ(ai, aj)
.
,
144
Chapter
2. GENERIC CONSTRUCTIONS
IfAandBarec-graphs (in the modelM), then sets
A ∪ B
and
ofM, whereas the value of
lows, we drop the index
are universes of thec-graphs, which we denote by
A ∪MB
Obviously, the value of
, respectively.
A ∩MB
A∪MB
does not depend on the choice
may vary withM. In what fol-
·Mfrom the above-mentioned denotations
A∩ B
A ∩MB
and
if there is clarity as to which digraph is being spoken of.
For ac-graph
minimal, by inclusion, graph
arcs and vertices, containing the shortest
(a, b) ∈ A2connected by a path in accordance with the record
A = hA, Q0, Q1, Coln, W i
Γ ⊇ hA;Q0, Q1, Colni
(a, b)
,
n∈ω
cc(A)
n∈ω
denotes a
with colored
-path for each pair
W
(since the graph is acyclic, if the shortest path exists then it is
unique).
By the denition, any structure
cc(A)
is nite and represented
a disjoint union of connected acyclic graphs. Moreover, elements in
cc(A)
and
nected componentsCof
ponents forming
2.5.2.2. Denition.
c
c
A ⊆cB
n ∈ ω
(2) on shortest paths linking elements ofAin
form algebraic (and simultaneously denable) closure
acl(A)
is a union of algebraic closures
M
having elements inA. Connected com-
acl(A)
are said to be
components generated byA.
acl(C ∩ A)
We dene the relation
-graphs. Ac-graph
-subgraph
,) if
, and
of ac-graph
A ⊆ B,Q
WAis the record correspondent the conditions (1) and
A = hA,Q
B = hB, Q
= Q
i,A
i,B
, Q
0,A
0,B
∩ A2,
, Col
1,A
, Q
1,B
n,A
, Col
i = 0,1,Col
n,B
acl(A)
for all con-
⊆con the class of
, WAi
, WBi
n,A
cc(B)
n∈ω
n∈ω
= Col
.
is called a
(written
∩ A
n,B
,
,
Clearly, for the diagrams
B
, respectively, the conditions
equivalent.
2.5.2.3. Denition.
called
8
Ac-graph
closedifA = cc(A)
written inW. We denote by
8
Keeping
as semantic objects with some superstructures realizing paths. We realize the
syntactic approach explicitly in Section 4.10.
in mind the type syntax forc-graphs, we shall considerc-graphs
Φ(A)
and
Ψ(B)
Φ(A) ⊆ Ψ(B)
ofc-graphsAand
and
A = hA, Q0, Q1, Coln, W i
, i. e.,
K
A
contains all shortest paths
the class of all closedc-graphs.
0
A ⊆cB
n∈ω
are
is
2.5.
UNIFORMd-AMALGAMATION PROPERTY
IfA,
B = hB, Q
Col
C
n,
, WCi
n∈ω
amalgamofc
-graphsBandCover
, Q
0,B
1,B
, Col
n,B
, WBi
are closedc-graphs,
, and
n∈ω
A = B ∩ C
A
(denoted by
C = hC, Q
, then a
B ∗AC
, Q
0,C
freec-
) is a
minimal, by inclusion, closedc-graph containing the structure
145
1,C
,
hB ∪ C, Q
An injective mapf:
A = hA,Q
(writtenf:
to the structure
W
of thec-subgraph ofBwith the universe
f(A)
the record obtained from
a ∈ A
by the elements
c
-GraphsAandBare calledc-isomorphic
embeddingf:
c
-isomorphism
c
-isomorphic copies
2.5.2.4. Lemma
∪ Q
0,B
, Q
0,A
A →cB
, Q
0,C
1,B
A → B
, Col
1,A
n,A
B = hB, Q
∪ Q
, WAi
0,B
1,C
, Col
n,B
∪ Col
, WB∪ WCi
n,C
is called ac-embeddingofc
to ac-graph
n∈ω
, Q
1,B
, Col
n,
, WBi
B
n∈ω
), iffis an embedding of the structure
hA; Q
hB; Q
0,B
0,A
, Q
, Q
1,B
1,A
, Col
, Col
n,B
n,Ain∈ω
, i
n∈ω
such that the record
f(A)
WAwith the replacement of all elements
f(a)
.
if there exists ac-
A →cB
with
f(A) = B
. The mapfis called a
betweenAandBand thec-graphsAandBare
.
(Amalgamation Lemma).
The class
closedc-graphs satises thec-amalgamation property(c
for anyc-embeddings
C ∈ K0, there is ac-graph
and
g1:
C →cD
f0:
such that
A →cB
and
g0:
A →cC
D ∈ K0andc-embeddings
f0◦ f1= g0◦ g1.
, whereA,B,
n∈ω
-graph
is equal to
K
of all
0
-AP)
, i. e.,
f1:
B →cD
.
Proof.
and
A ⊆cC
forD.
Denote by
in
K
. Using Lemma 2.5.2.4 one can immediately check that the
0
class
D
Without loss of generality we may assume that
. Clearly, the closedc-graph
B ∗AC
¤
D
the class of all types correspondent toc-graphs
0
with the relation6, where
0
Φ(A) 6 Ψ(B) ⇔ A ⊆cB
A ⊆cB
can be taken
, is a
self-sucient generative class having the uniformd-amalgamation
property.
146
Chapter
2. GENERIC CONSTRUCTIONS
Hence, there is a
(D0; 6)
-generic structureN, its theory coin-
cides with the theoryT, and Theorem 2.5.1.2 implies
2.5.2.5. Theorem.
rated. Any nite set
A =
sure
cc(A) ⊆ N
The
(D0; 6)
A ⊆ N
is extensible to its self-sucient clo-
, and the type
-generic structure
A)
tp(
is
implied by its subtype
N
is satu-
consisting of quantier free formulas describing connected compo-
A
nentsCcomposed by elements of
,
as well as of formulas with two
free variables describing that any two elements in distinct compo-
nentsCare not linked by paths.
The generic structure
N
consists of countably many pairwise
isomorphic connected components, each of which is acyclic.
As already noticed in Example 1.2.3.5, the theory
T0isω-stable.
Moreover, any connected component ofNhas a proper elementary
extension which is also a connected component, and some count-
able elementary extension of a countable model
realizations of all types in
S1(M)
.
Besides, by Theorem 2.5.2.5 the equality
holds, where
ϕn(x, y)
is a formula \saying" that the distance be-
tweenxandyis equal ton. Since each connected componentCof
a model of
have
RM(x ≈ x) = ω
T0is represented as the union
n∈ω
.
MofT0contains
RM(ϕn(x, a)) = n
S
ϕn(C, a),a ∈ C
, we
Thus we obtain
2.5.2.6. Theorem.
2.5.2.7. Remark.
The theory
T0isω-stable of Morley rankω.
The mechanism above, generating the
∆0-
baseness by formulas with at most two free variables, describing
lengths of shortest paths and types of elements in this paths, is
valid for any theory of language consisting of unary predicate sym-
bols
Pi,
i ∈ ω
, and binary predicate symbols
graph with the relation
S
(Qi∪ Q
i∈ω
−1
i
)
Qi,
i ∈ ω
, where the
is acyclic. The set
∆0of
formulas is obtained from the set∆of formulas in Example 1.2.3.5
by all possible replacements of symbols
by symbols
Qi,
i ∈ ω
. The detailed exposition for the proof of this
Q0,
Q1in these formulas
baseness is presented in the paper [394].
2.6.
FINITE CLOSURE PROPERTY IN FUSIONS
147
2.6. On the nite closure property in fusions
of generative classes
E. Hrushovski [223] dened a mechanism of fusion of two generic
theories for obtaining a strongly minimal theory, having a struc-
ture with elds of two distinct characteristics. His technique has
developed essentially last time by questions of fusions of elds and
of fusions of vector spaces having various required properties (see
A. Baudisch, A. Martin-Pizarro, M. Ziegler [117, 118]; A. Hasson,
M. Hils [200]; K. Holland [217, 218]; M. Ziegler [478]). Consid-
ered in Section 1.2 combinations and colorings of structures, when
these structures are countable and homogeneous, are interpretable
as partial cases of fusions for corresponding generative classes.
As shown in Section 2.3, any self-sucient generative class gen-
erates an operation of self-sucient closure on its generic structure.
After fusion of generic closures, these operations, via a transitive
closure, are extended to the operation of self-sucient closure on
a generic structure of this fusion. Thus, the system of nite clo-
sures arises, generating new and more general operation of nite
closures.
Since a saturation of generic structure is conditioned by formula-
A
denability of operation of self-sucient closure
setA, a natural question arises on possibility of creating for fusion
of generic structures, having formula-denable operations of self-
sucient closures, with formula-denability of resulted operation
of self-sucient closure.
In this Section, we shall formulate exactly this problem on fu-
sion of generative classes and shall propose sucient conditions
for the existence of such fusions with that structures possess nite
closures.
of
each nite
2.6.0.1. Denition.
that
(D; 6)
generic theory has nite closures.
The following theorem proposes a characterization of the nite
closure property for generative classes in terms of domination re-
lation.
has the
Let
(D; 6)
nite closure property
be a generative class. We say
if any model of
(D; 6)
-
148
Chapter
2. GENERIC CONSTRUCTIONS
2.6.0.2. Theorem.
the nite closure property if and only if
a generative class
A generative class
(D0; 60)
of languageΣ, satisfying the following
(D; 6)
(D; 6)
of languageΣhas
is dominated by
conditions:
(1)
each diagram
Φ(A)ofD0contains a description of some its
minimal self-sucient extension and has a restriction to a diagram,
inD, overA;
(2)
each type
extension
y) ∈ S(∅)
q(
x) ∈ S(∅)
p(
,
containing a type
set of coordinates of tuple
Pr
oof.
Suppose that the class
property. Then each nite set in a model of
y
of
and
Φ(A) ∈ D0.
(D0; 60)
(D; 6)
-generic
[Φ(A)]
theory has an
A
, whereYis the
Y
has the nite closure
(D; 6)
-generic the-
oryTcan be extended to a self-sucient set. Since the set of all
types
[Φ(A)]
all possible pairwise inconsistent extensions of diagrams
diagrams
tensions, form required generative class
A
corresponding to diagrams
X
Ψ(A)
, containing descriptions of their self-sucient ex-
Φ(A) ∈ D
(D0; 60)
is countable,
, where
6
Φ(A)
0
to
inherits
the relation6.
Conversely, suppose that the generative class
nated by a generative class
that any diagram
Φ(A)ofD0contains a description of some its
(D0; 60)
of the same language and such
(D; 6)
is domi-
minimal self-sucient extension and has a restriction to a diagram,
inD, overA, and also each type
theoryThas
an extension
y) ∈ S(∅)
q(
x) ∈ S(∅)of(D0; 60)
p(
,
containing a type
whereYis the set of coordinates of tuple
any
(D0; 60)
p ∈ S(T )
-generic structure is
(D; 6)
contains an information on the existence of self-sucient
y
and
Φ(A) ∈ D0.
-generic. Since any type
-generic
[Φ(A)]
Then
A
Y
extensions of its realizations, we have the nite closure property
for the generative class
(D; 6).¤
,
In view of the condition (2) in Theorem 2.6.0.2, if
the class
countably many restrictions of types
2.6.0.3. Denition.
Theorem 2.6.0.2, is said to be a
nite closure property
(D; 6)
has the nite closure property, then there are
y) ∈ S(∅)
q(
The generative class
to
(D0; 60)
types
[Φ(A)]
, described in
A
Y
generative class, witnessing the
for the class
(D; 6)
. Similar addition of
external information on properties of diagrams of given generative
class
a
witness
(D; 6)
, forming an
expansion
of generative class, is said to be
for the corresponding property.
.
2.6.
FINITE CLOSURE PROPERTY IN FUSIONS
149
2.6.0.4. Denition.
generative classes of languages
Σ1∩Σ2,
60= 61∩ 6
Σ1∪Σ2, such that
be a
fusion
case, a
and
(D2; 62)
Fusions of
of classes
(D3; 63)
-generic structure (theory) is a
-generic structures (theories).
(D1; 61)
Let
(D0; 60),(D1; 61)
Σ0,
Σ1, and
. A generative class
2
, and
Σ2respectively,
(D3; 63)
(D3; 63) ¹ Σi= (Di; 6i), i ∈ {0, 1, 2}
(D1; 61)
and
(D2; 62)
over
(D0; 60)
fusionof(D1; 61)
and
(D2; 62)
over
(D0; 60)
denoted by
A fusion of
(D2; 62)
structure
T1F
-generic structure
M0(theory
T2).
T
0
Notice that for
(D1; 61)
(D1; 61) F
M1F
M
(D1; 61) F
(D0;60)
-generic structure
M2(theory
T0) is denoted by
i ∈ {0,1, 2}
(D0;60)
M2¹ Σi= Mi, T1F
0
(D2; 62) ¹ Σi= (Di; 6i),
(D2; 62).
M1(theory
T2) over
M1F
(D0; 60)
M2(respectively
M
0
the following equalities hold:
T2¹ Σi= Ti.
T
0
Notice also that the following associativity property holds:
¡
D1; 61) F
(D0;60)
(D2; 62)¢F
(D0;60)
(D3; 63) =
¡
= (D1; 61) F
(D0;60)
(D2; 62) F
(D0;60)
(D3; 63)¢.
(D2; 62)
be
Σ0=
of language
, is said to
. In this
will be
T1) and
-generic
-
Having that identity we denote (sequential) fusions of theories
T2, and
T3over a theory
T0by
T1F
Clearly, fusions of classes
T2F
T
0
(D1; 61)
T3.
T
0
and
(D2; 62)
T1,
are
allowed to not exist (if, for instance, a chain of self-sucient
closures for a set relative to
6
and to
1
6
is not stabilized), and if it
2
exists, then it can be not unique. Moreover, a presence of the nite
closure property or the uniformd-amalgamation for the classes
(D1; 61)
and
(D2; 62)
does not imply corresponding properties
hold for fusions.
Besides notice that the nite closure property may be satised
both with uniform bounds for cardinalities of closures, depending
on cardinalities of initial nite sets, and without these bounds. In
the case of the absence of such bounds we assume that cardinalities
and structures of closures are described in diagrams of given nite
150
Chapter
2. GENERIC CONSTRUCTIONS
sets. Generative classes with bounds for cardinalities are called
PE
-classes
classes
and generative classes without such bounds are NPE
.
All examples of generative classes are PE-classes, that sim-
ilar to Hrushovski example generated non-negative predimension
functionsδand have saturated generic structures (see the surveys
by J. T. Baldwin [97, 99] and by B. P. Poizat [351]). Generative
classes of free acyclic and of cubic theories, being NPE-classes, are
described in [54, 403, 421].
In view of Theorem 2.6.0.2, the nite closure property for fu-
sions of generative classes is obviously characterized in terms of
expansions of generative classes.
Let
M3be a
(Di; 6i)
Mibe a
(D1; 61) F
and
(D1; 61) F
(Di; 6i)
(D0;60)
-generic structure,
(D2; 62)
(D0;60)
-generic structure, where
(D2; 62)
i = 0, 1, 2
are self-sucient generative classes.
Clearly,
and in
embeddable in
suppose that
of
M2¹ Σ0;
and of
M3¹ Σ2respectively. Moreover,
Denote the operations of self-sucient closures in
i = 1,2, 3
Clearly, for any nite set
S
Anholds, where
n∈ω
since the set
lized, starting with somen. That number is said to be an
number
If
Cl3(A) =
the operation
Cl3= hCl1, Cl2i
M0is elementarily embeddable in
M2¹ Σ0; the structures
M3¹ Σ1and in
M1and
M3¹ Σ2respectively. So we shall
M0is an elementary substructure of
M1and
M2are elementary substructures of
M3= M1F
.
A ⊆ M3, the inclusion
A0= A,A
Cl3(A)
and it is denoted by
is nite, the chain of sets
S
nA(M3)
= Cl1(Cl2(An))
n+1
or by
Anfor any nite set
n∈ω
Cl3is
generated
by
Cl1and
.
M1¹ Σ
M2are elementarily
M1¹ Σ0and
M3¹ Σ
M2.
M
0
Miby
Cli,
Cl3(A) ⊇
. Moreover,
An,
n ∈ ω
, is stabi-
iterative
nA.
A ⊆ M3, we say that
Cl2and write
Note that conditions of coincidence or non-coincidence of oper-
ators
Cl3and
hCl1, Cl2i
are witnessed by expansions of fusions of
generative classes.
Hrushovski style fusions of generative classes (
fusions
) (see E. Hrushovski [223]; A. Baudisch, A. Martin-Pizarro,
Hrushovski
-
,
0
1
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