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2.3.
SELF-SUFFICIENT CLASSES
The following denition generalizes Denition 2.17 in J. T. Bald-
win and N. Shi [96].
131
2.3.0.4. Denition.
class
(D0; 6)
. The pair
and for any diagram
implies
Let
Φ 6 Ψ0.
M
be a structure in the class
the structureM. We say that the setSis
S 6 M
A ⊆ S
if for any minimal pair
implies
B ⊆ S
Now we show that the conditions
LetΦandΨbe diagrams in a self-sucient
(Φ, Ψ)
is a
minimal pairifΦ ⊆ Ψ,Φ 66 Ψ
Ψ0∈ D0, the condition
K(D0)
andSbe a set in
closedinM
(Φ(A), Ψ(B))
with
Φ ⊆ Ψ0$ Ψ
and write
M |= Ψ(B)
.
S 6 M
and
A 6 M
are
compatible for the case whereSis a nite set and some diagram
Φ(S)
belongs to the class
D
0
with
M |= Φ(S)
, i. e., ifSis nite
thenSis closed if and only ifSis self-sucient.
2.3.0.5. Proposition.
and
Φ(A)
be a diagram in the class
Let
M
be a structure in the class
D
with
0
M |= Φ(A)
K(D0)
. The
following conditions are equivalent:
(1)Ais a self-sucient set inM;
(2)
for any minimal pair
A0⊆ A
(Φ0(A0), Ψ(B))
Since
and such that
(ix) we have
Ψ(B) $ Ψ(B)
tain
Φ0(A0) 6 Ψ(B)
pair. Thus,
nite set
grams
implies
Proof.
B ⊆ A
(1) ⇒ (2)
be a minimal pair with
M ∈ K(D0)
M |= Θ(C)
Φ(A) ∩ Ψ(B) 6 Ψ(B)
and by the minimality of
Φ0(A0) 6 Φ(A) ∩ Ψ(B)
that contradicts that
B ⊆ A
(2) ⇒ (1)
. Assume that
B ⊆ M
Φ(A), Ψ(B) ∈ D0, we have
.
. Let
, there is
.
such that
(Φ0(A0), Ψ(B))
A
be a self-sucient set in
M |= Ψ(B)
Θ(C) ∈ D0containing
. Now
Φ(A) 6 Θ(C)
. If
with
M |= Ψ(B)
, and
Φ(A) ∪ Ψ(B)
and by axiom
B 6⊆ A
then
(Φ0(A0), Ψ(B))
M
A0⊆ A
Φ(A) ∩
, we ob-
. Then the transitivity of6implies
A 66 M
A ⊂ B
Φ(A) 66 Ψ(B)
(Φ0(A0), Ψ(B))
, that is, there is a -
, and for corresponding dia-
is a minimal
. Now the nite-
ness ofBand axiom (ix) imply that there exists a minimal pair
(Φ(A), Ψ0(B0))
fact clashes with condition (2).
satisfying
M |= Ψ0(B0)
¤
, with
B06⊆ A
. The last
,
,
,
,
.
132
Chapter
2. GENERIC CONSTRUCTIONS
Notice that axiom (ix) gives rise to the following property of
structures (similar to this axiom): if
N0¹ N
, and
M 6 N
, then
M ∩ N06 N0.
M, N , N0∈ K(D0),M ¹ N
The following statement generalizes Lemma 2.18 in J. T. Bald-
win and N. Shi [96].
,
2.3.0.6. Proposition.
(1)
the class
(D0; 6)
The following conditions are equivalent:
does not contain an innite ascending
chain of minimal pairs;
(2)
the classKhas nite closures;
(3)
everyω-saturated structure inKhas nite closures;
(4)
someω-saturated structure inKhas nite closures.
Proof.
that is, if some nite setAof some structure
extended to a self-sucient set, then (in view of
there exists a nite setBsuch that
Φ(B) ∈ D0, and
(1) ⇒ (2)
B
. If the classKdoes not have nite closures,
M ∈ K
cannot be
K ⊆ K(D0)
A ⊆ B ⊆ M,M |= Φ(B)
can not be extended to a self-sucient set.
Then, by induction, we may construct an innite ascending chain
of minimal pairs in
(2) ⇒ (3)
(4) ⇒ (1)
are
. The fact that there exists an innite ascending chain
of minimal pairs in
ω
-saturated structure inK, which is impossible for structures with
nite closures.
D
(3) ⇒ (4)
(D0; 6)
¤
beginning with some pair
0
obvious.
implies that it is embeddable in any
(Φ(B), Ψ(C))
.
A consequence of Proposition 2.3.0.6 is
)
,
2.3.0.7. Corollary.
If a generic structure
the classKhas nite closures.
2.3.0.8. Denition.
K(D0)
N
, andSa closed set in the structureM. An injectionf:
is called a
strong embeddingofS
inN, and for any diagram
A ⊆ S
, we have
N |= Φ(f(A))
Let
M
Φ(A) ∈ D0such that
We say that a generative class
closed(self-sucient)sets
if for any structures
and
.
M
is
saturated, then
N
be structures in the class
in
Niff(S)
is a closed set
M |= Φ(A)
(D0; 6)
has
amalgamation over
M0, M1∈ K
S →
and
and
2.3.
SELF-SUFFICIENT CLASSES
133
any closed (self-sucient) setSin some structure of the classK,
the existence of strong embeddingsf:
implies that there exist a model
dings
f0:
M0→ N
and
g0:
N |= T
M1→ N
In this instance we also say that the
amalgamation over closed(self-sucient)sets
S → M0andg:
and elementary embed-
such that
(D0; 6)
-generic theoryThas
f ◦ f0= g ◦ g0.
.
S → M
The following theorem generalizes Lemma 2.21 in J. T. Baldwin
and N. Shi [96].
M
2.3.0.9. Theorem.
a
(D0; 6)
of
T = Th(
-generic structure, and
M)
which
Let
(D0; 6)
be a self-sucient class,
K
be the class of all models
has nite closures. The following conditions
b
are equivalent:
(1)
the theoryThas amalgamation over closed sets;
(2)
the theoryThas amalgamation over self-sucient sets;
M
M
is
an
ω1-universal model ofT;
is
anω-saturated model ofT.
(3)
(4)
1
e
Proof.
(1) ⇒ (2)
is obvious.
(2) ⇒ (1)
follows from Com-
pactness Theorem and the condition that the class
closures.
(2) ⇒ (3)
N
is elementarily embeddable in
S
Aiof an ascending6-chain of self-sucient sets
i∈ω
M
Since
M
in
i ∈ ω
via
. Letfbe the embedding
is easy to see that
claim that
an
y tuple
can
be extended to some tuple
some set
of
f(Ai)
amalgamation over self-sucient sets, we have
Consequen
. Let
is
a generic structure, the sets
some strong embeddings
N
be a countable model ofT. We show that
S
i∈ω
N0is a universe of a substructure
M
N04
a ∈ N0.
.
It suces to prove that
Since the classKhas nite closures, the tuple
b
Ai. The self-suciency of
in both of the structures
tly
a) ≡ (M,a)
(N0,
.
M
.
We representNas a union
Ai,
Aiare strongly embeddable
M
fi:
Ai→
fiand
en
N0be the image
umerating the image
,
so that
a) ≡ (M, a)
(N0,
Aiimplies the self-suciency
M
N0and
.
SinceTenjoys
(N0,
K
has nite
i ∈ ω
, inN.
fi⊆ f
f(N )
M
N0of
.
f(Ai)
b) ≡ (M, b)
i+1
. It
We
for
of
,
a
.
134
(3) ⇒ (4)
. Let
M
b
Chapter
e an
ω1-universal structure. SinceKhas -
2. GENERIC CONSTRUCTIONS
nite closures, it is enough to show that all1-types over self-sucient
M
sets in
Φ(A)
in
S1(A)
whic
h the typepis realized by some elementa, lettingfbe an
are
realized in
b
e a diagram in
. Consider a countable elementary extension
elementary embedding of structureNin structure
is self-sucient in
morphismgof
g(f (a))
is the required realization ofpin
(4) ⇒ (2)
M
M
,
mapping
.
We x strong embeddingsf:
g:A → M1, and also a diagram
Φ(f(A))
that
and
M0and
M1|= Φ(g(A))
M1are countable structures. Since
there are elementary embeddings
M
g1:
in
M1→
M
,
and we have
.
Hence
Therefore there is an automorphismhof
g1(g(A))
.
The maps
amalgam of the structures
M
.
D
for which
0
and
M
f(A)toA
. By compactness we may assume
f1(f(A))
M
|= Φ(f1(f(A)))
f1◦ h
and
M0and
LetAbe a self-sucient set in
M
|= Φ(A)
|= Φ(f(A))
, andpbe a type
N
of
M
.
Since
, there exists an auto-
. Consequently the element
M
.
A → M0and
Φ(A) ∈ D0for which
f1:
M0→
and
g1(g(A))
g1witness that
M1.
are self-sucient sets
M
and
M
mapping
M
¤
M
|= Φ(g1(g(A)))
is
M0|=
is
saturated,
M
f1(f(A))
the required
M
M
f(A)
and
in
to
,
.
2.3.0.10. Denition.
M
be a structure inK, andSbe a set in
Let
K
be a class with nite closures,
M
. The least (by
inclusion) closed set inM, containingS, is called an
S
closureofSinM
and is denoted by
iclM(S)
from the context which of the structures
S
set
the setS. A diagram in the class
sucient closure
and
closure
2.3.0.11. Theorem.
any structure
self-sucient closure
is
nite then it is referred to as a
A
of
a setA, is denoted by
M |= Φ(A)
of the diagram
, then the diagram
Φ(A)
.
If the class
M ∈ K
and any nite set
AofA
.
Moreover,
D
, corresponding to the self-
0
Φ(A)
K
has nite closures then for
, or by
M
is in point. If the
self-sucient closure
Φ(A)
is
called a
A ⊆ M
A ⊆ aclM(A)
,
if it is clear
.
If
self-sucient
there exists a
.
intrinsic
Φ(A) ∈ D
of
0
2.4.
GENERICITY OF HOMOGENEOUS STRUCTURES
135
Proof.
By axiom (ix), their intersection
in
M
Let
A1and
A2be self-sucient sets in
M
containingA.
A1∩ A2is also a self-sucient set
containingA. Cardinalities of self-sucient sets are nite;
so there is a unique self-sucient set inMcontainingAand having
a least cardinality.
We show that
A ⊆ aclM(A)
tary extension ofM. Assume that
type. Then there exists a realization
A
from
of
2.3.0.12. Corollary.
generic structure
.
However
iclM(A) = iclN(A) =
A0contradicts the uniqueness of
If the class
Proof.
Let
Mis(
a
andbb
rst
letAandBbe sets consisting of elements of
diagrams
).
Since
f(A)
A ⊆ acl
M
= B
Then the conditions
Φ(B)=Ψ(B)
M|=Ψ(B)
and
A → B
f
:
M.¤
of
for
with
.
LetNbe anω-saturated elemen-
p tp(A/A)
is a non-algebraic
A0ofpinNwhich is distinct
A
.
Hence the existence
iclN(A).¤
K
has nite closures then the
order)homogeneous.
M
ectively.
imply
,
and
that
e two tuples of the same type in
a
andbresp
(A)
and
M
B ⊆ acl
Φ(A), Ψ(B) ∈ D0,
is
generic, taking a partial isomorphism
(B)
M
where
M|=Φ(A)
, we can extendftill an automorphism
Since
countable homogeneous structures realizing a given set
of types are isomorphic, Corollary 2.3.0.12, along with Theorem
2.2.0.13, implies
2.3.0.13. Corollary.
generic structure
M
If the class
is
unique up to isomorphism.
K
has nite closures then the
2.4. Genericity of countable homogeneous
structures
2.4.0.1. Denition.
hereditary
if
D
consists of (minimal by inclusion) diagrams
0
containing formulas describing, for
Ψ(B)
, how many
are links between elements in these copies.
A generative class
Ψ(B) ∈ D0satisfying
Ψ(B)
-copies are available over
(D0; 6)
Φ(A)
is (
minimal
Φ(A) 6
, and what
Φ(A)
)
136
Chapter
2. GENERIC CONSTRUCTIONS
Note that for the description of
respect to
Φ(A)
it suces, using amalgams (applied as in Remark
2.2.0.17), to provide the following. If
contains all formulas
as well as some formulas
A) ⊂finX(C)
, forΣ-diagrams
∃Y Ψ0(A, Y )
χ∗(a)= ¬∃ZX0(A, Z)
Ψ(B)
-copies and their links with
Φ(A) 6 Ψ(B)
, where
Ψ0(A, B \ A) ⊂finΨ(B)
X(C) ⊃ Φ(A)
then
, where
X0(A, C \
(in some generative
classes) which are inconsistent with some strong extension of
a)
in
D
and this fact is witnessed by
0
2.4.0.2.
geneous(saturated)structure
Theorem.
Every at most countable rst order homo-
M
some hereditary generative class
is a
(D0; 6)(with thed-covering prop-
χ∗(
(D0; 6)
.
-generic structure for
erty).
Proof.
required class is the hereditary class
all copies of complete diagrams
a ∈ A}
(with
Let
M
be a countable homogeneous structure. The
(D0; 6)
Φ(A) = {ϕ(
respect toM) for every nite set
, where
a) |
M |= ϕ(
A ⊆ M
D
consists of
0
, and6is
a)
the inclusion relation.
Indeed, checking thed-amalgamation property we consider com-
plete diagrams
Y
b)]
[tpY(
c)
tp(
[tpZ(
∩ [tpZ(c)]
B
.
Then we take
Z
c)]
o
ver
C
[tpX(
[tpX(
Z
C
[tp
a)]
X
a)]
,
[tpY(b)]
A
,
a, b, c
bc)]
(
Y Z
X
.
Other properties of hereditary generative
A
Y
,
[tpZ(c)]
B
are
tuples in
Y
Z
as the amalgam of
BC
Z
,
C
where
[tpX(
M,tp(a)= tp(
b)]
[tpY(
classes obviously hold.
If
M
is a saturated structure, then the class
thed-covering property, since the theory
Th(M)
(D0; 6)
is small.
possesses
Φ(A)
Φ(A)
where
a)]
Y
B
¤
X
=
A
b) ∩
and
,
Note that complete diagrams, which are uses for the proof of
Theorem 2.4.0.2, can be replaced by subdiagrams isolating these
complete diagrams and containing their restrictions to the set of
quantier-free formulas.
By virtue of the fact that each complete countable theory has
an at most countable homogeneous model, Theorem 2.4.0.2 yields
the following:
2.4.0.3. Corollary.
Every complete countable theory is generic.
2.4.
GENERICITY OF HOMOGENEOUS STRUCTURES
137
As shown in Corollary 2.3.0.12, each generic structure is count-
able and homogeneous. By Theorem 2.4.0.2 and Corollary 2.4.0.3,
having a countable homogeneous structure
erative class for
Th(M)
. J. T. Baldwin suggested a modication
M
we can dene a gen-
of the denition of generative class (excluding the necessity of bi-
jective substitutions of elements into diagrams
quirement for the countability of language and of set of
Φ(A)
and the re-
[Φ(A)]
A
X
so that each structure can be considered as generic.
We know that the property ofTto be small implies that there
exists a countable saturated model ofT. By Theorem 2.4.0.2,
therefore, the theoryTis
(D0; 6)
with thed-covering property.
Conversely, ifTis a
tive class
(D0; 6)
with thed-covering property, then the count-
ability of a number of types
d
-covering property imply that the set
(D0; 6)
(D0; 6)
[Φ(A)]
-generic for some generative class
-generic theory for some genera-
A
, where
X
S(T)
Φ(A) ∈ D0, and the
of types forT, is count-
able, that is,Tis small.
We thus arrive at
)
2.4.0.4. Theorem.
For any complete countable theoryT, the fol-
lowing conditions are equivalent:
(1)Tis small;
(2)Tis
(D0; 6)
-generic for some generative class
(D0; 6)
with
thed-covering property.
2.4.0.5. Denition.
if there exists a diagram
A generative class
(D0; 6)
Φ(A) ∈ D0containing some complete
is
complete
theory of the languageΣ.
The property of
formulas occurring in diagrams of
(D0; 6)
being complete implies that the set of
D
generate a
0
(D0; 6)
-generic
theory. At the same time, the condition of being hereditary
for
(D0; 6)
cannot guarantee generation of a complete the-
ory. For example, in the generic construction of an innite
linearly ordered set via a minimal hereditary class
the formula
formulas occurring in types
∀x, y ((x ≤ y) ∨ (y ≤ x))
Φ(X)
, where
is not deduced from a set of
Φ(A) ∈ D0for someA.
(D0; 6)
,
138
Chapter
2. GENERIC CONSTRUCTIONS
In the proof of Theorem 2.4.0.2 (on representability of any
countable homogeneous structure as a generic one), use was made
of complete generative classes. At the same time, generative classes,
in solving various problems, are dened for constructing an
a priori
unknown theory. For a required theory to possess requisite prop-
erties, therefore, it is preferable that our generative classes involve
diagrams containing a minimum of relevant information.
2.4.0.6. Denition.
Let
(D0; 6)
ative classes of languagesΣand
We say that the class
write
D0E D
Φ0(A0) ∈ D
0
, if for any diagram
0
0
such that
0
0
(D
; 60)
0
dominates
Φ(A) ⊆ Φ0(A0)
and
Σ0, respectively, with
Φ(A) ∈ D0there is a diagram
, and the condition of there
0
(D
; 60)
0
the class
be gener-
Σ ⊆ Σ0.
(D0; 6)
, and
being some systems, which are extensions overA, together with
available information on interrelations of elements in these exten-
sions written in the diagram
Φ(A)
, implies that the same extensions
exist overA, and that similar information is available on interrela-
tions of elements in those extensions written in the diagram
Φ0(A0)
Obviously, the relationEis a preorder on the class of generative
classes.
It is easy to see that if a structure
beddable in a structure
class
Φ0(B)
0
(D
; 60)
, which is equal to the closure of a set of diagrams
0
for all possible nite sets
M0¹ Σ
, then the (minimal) hereditary
B ∈ M0with respect to bijective
substitutions of constants on which the inclusion relation
ned, dominates the (minimal) hereditary class
equal to the closure of a set of diagrams
sets
A ∈ M
with respect to bijective substitutions of constants on
M
is isomorphically em-
(D0; 6)
Φ(A)
for all possible nite
0
6
is de-
, which is
which the inclusion relation6is dened.
At the same time, the condition
the
(D0; 6)
restriction of the
-generic structure is isomorphically embeddable in the
0
(D
; 60)
-generic structure to the languageΣ.
0
D0E D
0
implies that
0
Thus, we have
.
2.4.0.7. Theorem.
Let
M
structures of languagesΣand
ditions are equivalent:
and
M0be countable homogeneous
Σ0, respectively. The following con-
2.4.
GENERICITY OF HOMOGENEOUS STRUCTURES
(1)
the structure
ture
M0¹ Σ
(2)
;
there are generative classes
Mis(D0; 6)
-generic,
M
is isomorphically embeddable in the struc-
(D0; 6)
M0is
(D
0
; 60)
-generic, and
0
and
(D
0
; 60)
0
D0E D
139
such that
0
.
0
2.4.0.8. Denition.
erative classes
write
D0∼ D
(D0; 6)
0
.
0
If
D0E D
and
(D
0
; 60)
0
0
0
and
0
D
E D
0
are
domination-equivalent
we say that gen-
0
and
It is easy to see that∼is an equivalence relation.
In view of Theorem 2.4.0.7. the following corollary holds.
2.4.0.9. Corollary.
Let
M
and
M0be countable homogeneous
structures of a languageΣ. The following conditions are equivalent:
(1)
and
the structures
(2)
there are domination-equivalent generative classes
0
(D
; 60)
0
such that
M
and
M0are isomorphic;
Mis(D0; 6)
-generic and
M0is
(D0; 6)
0
(D
; 60)
0
generic;
(3)
the structures
erative class
Proof.
(2) ⇒ (1)
that
M
(D0; 6)
(1) ⇒ (3)
. Having the hypothesis we see by Theorem 2.4.0.7.
and
M0are mutually embeddable. Since
countable homogeneous, they are isomorphic.
M
.
and
and
M0are
(3) ⇒ (2)
(D0; 6)
-generic for some gen-
are obvious.
¤
M
and
M0are
Thus, by uniqueness of homogeneous structures realizing same
set of types if generative classes are domination-equivalent then
these classes produce isomorphic generic structures. We have the
converse implication for quantier-free generative classes.
At the same time, there are generative classes, being not∼-
equivalent but forming isomorphic generic structures. For instance,
the structure
axiom
ϕ0, is generated both by the quantier-free generative class
(D0; 6)
subsets ofQ) and by the generative class
diagram contains the axiom
hQ; ≤i
, having a nitely axiomatizable theory with an
(whose diagrams describe≤-links of elements for nite
0
(D
; 60)
, where each
0
ϕ0. Clearly,
D0E D
0
0
and
D
0
6E D0.
0
-
140
Chapter
2. GENERIC CONSTRUCTIONS
2.5. The uniformd-amalgamation property
and saturated generic structures
2.5.1. Denition, theorem, and corollaries
2.5.1.1. Denition.
Let
(D0; 6)
be a self-sucient class satisfy-
ing the following conditions:
(a) for any diagram
(
A)
formula
sure
χ
Φ(A)
deducible from
describing
Φ
;
moreover,
Φ(A)
cardinality of the set
(b)
for any self-sucient diagrams
Φ(A) 6 Ψ(B)
andYare
resp
ectively), there exists a formula
Φ(X)
and
∀X ((χ
,
and for any formula
disjoint sets of variables, bijective with sets
is such that the following formula holds true in
(X) ∧ ϕ(X)) →
Φ
Φ(A) ∈ D0, the diagram
(
χ
Φ
and describes an upper bound for the
A
Φ(A)
the self-sucient condition for the clo-
A)
also
contains a formula which is
;
ψ(X, Y )
∃Y (χ
ϕ(X)
(X
Ψ
Φ(A)
and
Ψ(X ∪ Y )
in
which is deducible from
, Y ) ∧ ψ(X, Y ))).
Ψ(B)
A
and
yields
,
where
(here,
B \ A
M
:
X
If the above conditions are satised then we say that the class
(D0; 6)
has the
uniformd-amalgamation property
.
Notice that the concept of uniformd-amalgamation gen-
eralizes
uniform amalgamation
as dened in J. T. Baldwin and
N. Shi [96].
As distinct from the uniform amalgamation property, the uni-
formd-amalgamation property, if satised, is not supposed to in-
volve a function
for the cardinalities of self-sucient closures
f ∈ ωω, representing uniform upper bounds
A
dep
ending only on
f(n)
cardinalitiesnof given setsA.
As an example of a generative class that possesses the uniform
d
-amalgamation property but lacks in the above-mentioned upper
bound, we may take a class of diagrams corresponding to nite
acyclic undigraphs with unbounded degrees of all elements. In fact,
taking elements
anand
bn, that are connected by shortest paths of
lengthn, we have unbounded cardinalities of self-sucient closures,
that formed by adding of elements of shortest paths:
n + 1
.
{an,
|
bn}| =
a
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