Добавил:
ivanov666
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Classification of countable models of complete theories. Р.1. Monograph in two parts
.pdf
2.8.
ON VARIETIES OF GENERATIVE CLASSES
161
2.8.0.3. Theorem.
Let
M
and
M0be countable homogeneous
algebras of languageΣ. The following conditions are equivalent:
(1) M0is isomorphic to a quotient-algebra ofM;
(2)
there are generative classes
Mis(D0; 6)
quotient-class of
Proof.
complete diagrams
[Φ0(A0)]
A
B
generic,
quotient-algebra ofM,
(2) ⇒ (1).
ing diagram
quotient-algebra of
bras
M
algebra ofM.
2.8.0.4. Denition.
diagrams
-generic,
(1) ⇒ (2).
0
) for some
0
M0is
(D
(D0; 6)
M0is
.
Let
Φ(A)(Φ0(A0)
B ⊆ M(B0⊆ M0),
0
; ⊆)
-generic and, since
0
(D
Since for each diagram
Φ0(A0) ∈ D
and
M0we observe that
0
0
A
, constructing step-by-step generic alge-
Φ(A)
¤
A diagram
Φi(Ai),i < λ,λ ∈ ω ∪ {ω}(Φ(A)
is a Cartesian product of
product of algebras
Let
(D0; 6)
languageΣ. The class
A
Φi(Ai)
and
(Di0; 6i),i < λ
(D0; 6)
(Di0; 6i),i < λ:(D0; 6)
(D0; 6)
0
(D
; 60)
-generic and
0
D
(respectively,
0
) such that
i < λ
0
; 60)
0
is the quotient-class of
Φ(A) ∈ D0and correspond-
, the algebra
A
Φ0(A0)
M0is isomorphic to a quotient-
Φ(A)
Aiand an algebra
:
A
Φ(A)
=
is a
Q
A
i<λ
, be generative classes of
is a
Q
(Di0; 6i)
i<λ
Cartesian product
if the following conditions
and
D
M |= [Φ(A)]
M0is isomorphic to a
Φi(Ai)
0
(D
; 60)
0
0
) be the class of all
0
. Now
such that
0
(D
; 60)
0
A
(
M0|=
B
Mis(D0; ⊆)
(D0; 6)
is isomorphic to a
Cartesian product
Q
A
Φi(Ai)
i<λ
is a Cartesian
Φ(A)
), if
.
of classes
is a
hold:
1) any diagram
Φi(Ai) ∈ Di0,
2) for any diagrams
Φ(A) ∈ D0that is a Cartesian product of
3) for any diagrams
Ψ(B)
is equivalent to
Φ(A) ∈ D0is a Cartesian product of diagrams
i < λ
;
Φi(Ai) ∈ Di0,
i < λ
, there is a diagram
Φi(Ai) ∈ Di0,
Φ(A), Ψ(B) ∈ D0, the condition
Φi(Ai) 6iΨi(Bi),i < λ
.
i < λ
Φ(A) 6
;
-
.
of
A
2.8.0.5. Theorem.
Let
M
and
Mi,
i < λ
, be countable ho-
mogeneous algebras of languageΣ. The following conditions are
equivalent:
162
Chapter
2. GENERIC CONSTRUCTIONS
(1) M
(2)
such that
and
(D0; 6)
Proof.
is isomorphic to the Cartesian product of
there are generative classes
Mis(D0; 6)
is a Cartesian product of
(1) ⇒ (2).
of all complete diagrams
A
(
Mi|= [Φi(Ai)]
generic,
Miare
Cartesian product of
i < λ
.
(2) ⇒ (1).
ing diagrams
i
) for some
B
i
(Di0; ⊆)
Mi,
Since for each diagram
Φi(Ai) ∈ Di0, the algebra
the Cartesian product of
generic algebras
Q
phic to
i<λ
Mi.
M
¤
and
(D0; 6)
-generic,
Miis
(Di0; 6i)
(Di0; 6i),i < λ
Let
D
(respectively,
0
Φ(A)(Φi(Ai)
) such that
B ⊆ M(Bi⊆ Mi). Now
-generic and, since
i < λ,D
0
is a Cartesian product of
0
M
Φ(A) ∈ D0and correspond-
A
A
Mi,
Φi(Ai)
i < λ
,
i < λ
, constructing step-by-step
, we observe that
Mi,
i < λ
and
(Di0; 6i),i < λ
-generic,
i < λ
.
D
) be the class
i0
M |= [Φ(A)]
Mis(D0; ⊆)
is isomorphic to the
is isomorphic to
Φ(A)
M
is isomor-
D
;
,
,
A
B
-
,
i0
Chapter
3
ALGEBRAS OF DISTRIBUTIONS
FOR BINARY SEMI-ISOLATING
FORMULAS
OF A COMPLETE THEORY
In this Chapter, we consider a general approach to the descrip-
tion of binary links between realizations of1-types in terms of labels
of pairwise non-equivalent semi-isolating formulas. This approach
is naturally interpretable in the class of relation partial algebras
[27, 35].
3.1. Preliminary notions, notations, and prop-
erties
3.1.0.1. Denition
M |= T
all
there is
each such a formula
{(a, b) | M |= p(a) ∧ ϕ(a, b)}.If(a, b) ∈ R
a
(p, ϕ, q)
is also
a
(q, p)
be a
. Consider types
(p, q)
-preserving
a ∈ M
-arc
. If
principal
If
ϕ(x, y)
-preserving then the set
(p, ϕ, q)
.
is a
-edge
[51, 87, 413]. LetTbe a complete theory,
p(x), q(y) ∈ S(∅)
formulas
such that
ϕ(x, y)
ϕ(a, y)
(p ↔ q)
. If the
ϕ(x, y)ofT
|= p(a)
, we dene a binary relation
is principal (overa), the
-formula
(p, ϕ, q)
and
, i. e., it is both a
[a, b] {(a, b), (b, a)}
-edge
, realized inM, and
, i. e., formulas for which
ϕ(a, y) ` q(y )
, then
p,ϕ,q
[a, b]
consists of principal
. Now, for
(a, b)
(p, ϕ, q)
R
is called
-arc
(a, b)
(p, q)
- and
is said to
164
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
(p, ϕ, q)
[a, b]
spectively if we speak of xed or some formula
a
formula, then
3.1.0.2. Denition.
PF(p, q)
in
solutions for
Notice that each
- and
is a
(p, ϕ, q)
(p, ϕ, q)
Let
PE(p, q)
PF(p, q)
Clearly,
(q, ϕ−1, p)
principal
-arcs and
-arc such that the pair
(a, b)
-arcs, where
(p, ϕ, q)
(p, ϕ, q)
is called
For types
ϕ−1(x, y)
-edge.
-edges are called
(b, a)
is not an arc for any
irreversible
p(x), q(y) ∈ S(∅)
.
denotes
arcs
and
ϕ(x, y)
, we denote by
ϕ(y, x)
edges
. If
(a, b)
the set
{ϕ(x, y) | ϕ(a,y)
ϕ(a, y) ` q(y),
be the set of all pairs
is a principal formula,
where
|= p(a)}.
(ϕ(x, y), ψ(x, y))
of formulas
such that for any (some) realizationaofpthe sets of
ϕ(a, y)
PE(p, q)
and
ψ(a, y)
coincide.
is an equivalence relation on the set
PE(p, q)
-class
E
corresponds to either a prin-
PF(p, q)
, then
re-
is
(q, p)
cipal edge or to an irreversible principal arc connecting realiza-
tions ofpandqby some (any) formula inE. Thus the quo-
tient
PF(p, q)/PE(p, q)
PFS(p, q)
and
PFN(p, q)
classes corresponding to principal edges and
PE(p, q)
noted by
-classes corresponding to irreversible principal arcs.
The sets
PF(p, p),PE(p, p),PFS(p, p)
PF(p),PE(p),PFS(p)
LetTbe a complete theory,
of cardinality
positive elements
for any
u ∈ U−and
denoted by
are called
Let
≥ |S(T )|
u+∈ U+, and zero0. As usual, we write
U≤0and
labels
.
ν(p, q):PF(p, q)/PE(p, q) → U
functions,p(x), q(y) ∈ S(∅)
correspond to classes in
is represented as a disjoint union of sets
, where
, consisting of
u > 0
for any
U+∪ {0}
PFS(p, q)
, and
, and
U = U
PFN(p)
−
˙
∪ {0}˙∪ U+be an alphabet
negative elements
u ∈ U+.1The set
is denoted by
be an injective
consists of
PFN(p, q)
PE(p, q)
consists of
PFN(p, p)
respectively.
u−∈ U−,
U−∪ {0}
U≥0. Elements of
labelling
, for which negative elements
PFN(p, q)/PE(p, q)
and non-negative
are de-
u < 0
is
U
-
.
-
1
IfUis
integers.
at most countable, we suppose thatUis a subset of the setZof
3.1.
PRELIMINARY NOTIONS, NOTATIONS, AND PROPERTIES
165
elements correspond to classes in
is dened only for
ν(p) ν(p, p)
for
p 6= q
(where, as usual, we denote by
functionf) and
p = q
and is represented by the formula
. We additionally suppose that
ρ
ν(p,q)
∩ ρ
ν(p0,q0)
PFS(p, q)/PE(p, q)
ρfthe image of the
= ∅ifp 6= q
ρ
∩ ρ
ν(p)
and
(p, q) 6= (p0, q0)
such that
(x ≈ y)
= {0}
ν(q)
Labelling functions with the properties above as well families of
these functions are said to be
regular
. Below we shall consider only
regular labelling functions and their regular families.
We denote by
u ∈ ρ
. If a typepis xed and
ν(p,q)
is denoted by
Note that if
θ
p,u,q
θu(x, y)
θ
p,u,q
(x, y)
.
(x, y)
a formula in
p = q
and
θ
(x, y)
q,v,p
PF(p, q)
then a formula
with the label
θ
p,u,q
are formulas witness-
(x, y)
ing that for realizationsaandbofpandqrespectively the pairs
(a, b)
θ
(non-negative) labelvcorresponds uniquely to the
u
are denoted by
and
q,v,p
(b, a)
(y, x)
are principal arcs then the formula
witnesses that
[a, b]
is a principal edge. Moreover the
and vice versa. The labelsuandvare
For types
v−1and
p1, p2, . . . , p
u−1respectively.
k+1
∈ S1(∅)
and sets
θ
(x, y) ∧
p,u,q
invertible
reciprocally inverse
X1, X2, . . . , Xk⊆ U
label
and
of labels we denote by
P (p1, X1, p2, X2, . . . , pk, Xk, p
k+1
)
0
,
.
the set of all labels
satisfying, for realizationsaof
u ∈ U
corresponding to formulas
p1and some
the following condition:
θ
p1,u,p
(a, y) ` θ
k+1
p1,u1,p2,u2,...,pk,uk,p
where
θ
∃x2, x3, . . . , xk(θ
. . . ∧ θ
p
k−1,uk−1,pk
Thus the Boolean
p1,u1,p2,u2,...,pk,uk,p
(x
k−1
(x, x2) ∧ θ
2
, xk) ∧ θ
p1,u1,p
P(U)ofU
(x, y)
k+1
is the universe of an
distributions of binary isolating formulas
P (p1, ·, p2, ·, . . . , pk, ·, p
where
to any family
p1, . . . , p
∈ S1(∅)
k+1
R ⊆ S1(∅)
. This algebra has a natural restriction
.
θ
p1,u,p
k+1
u1∈ X1, . . . , uk∈ Xk,
(a, y),
k+1
p2,u2,p
pk,uk,p
(x2, x3) ∧ .. .
3
(xk, y)).
k+1
algebra of
withk-ary operations
),
k+1
(x, y)
166
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
Clearly, replacing the set of labels bijectively we get an iso-
morphic algebra. In particular, there is a
canonical algebra
labels are presented by elements
[
PF(p, q)/PE(p, q).
p,q
, where
Nevertheless, we shall use an abstract setUof labels reecting their
signs and clarifying algebraic properties for operations on
Note that if some set
Xiis disjoint from
ρ
ν(pi,p
, in particular,
)
i+1
P(U)
2
.
if it is empty then
P (p1, X1, p2, X2, . . . , pk, Xk, p
Note also that if
Xi6⊆ ρ
ν(pi,p
for someithen
)
i+1
P (p1, X1, p2, X2, . . . , pk, Xk, p
= P(p1, X1∩ρ
ν(p1,p2)
, p2, X2∩ρ
ν(p1,p2)
, . . . , pk, Xk∩ρ
In view of the previous equality, it is enough to assume
ρ
ν(pi,p
,
i = 1,. . . , k
)
i+1
, for the values
P (p1, X1, p2, X2, . . . , pk, Xk, p
If each set
use
uiinstead of
Xiis a singleton consisting of an element
Xiin
P (p1, X1, p2, X2, . . . , pk, Xk, p
P (p1, u1, p2, u2, . . . , pk, uk, p
2
Considering
P (p1, X1, p2, X2, . . . , pk, Xk, p
and same formulas can be used for distinct tuples of types. We can assume
that for
(p1, p
separated by some
∃yθ
p1,u,p
0
p
,u0,p
1
(x, y) ` ψ
k+1
and
∃xθ
time for innite families of types this procedure may fail. For these algebras,
we put the dierence into consideration and essentially use labels instead sets
of formulas.
formulas instead of labels, the value
)
depends on the choice of formulas
k+1
0
(y)
1
, p
and
0
k+1
),θ
) and
∃xθ
p1,u,p
0
p
1
0
1
,u0,p
(x)
ψ
(x, y)
k+1
(with
(y) ∈ p
k+1
0
(x, y) ` ¬ψ
k+1
) 6= (p
k+1
ψ1(x) ∈ p1(x) \ p
0
(x, y) ` ¬ψ1(x)
k+1
k+1
k+1
k+1
k+1
∃yθ
) = ∅.
) =
).
k+1
).
and
p1,u,p
(y) \ p
k+1
k+1
ν(pk,p
k+1
uithen we
)
and write
k+1
0
0
θ
p
,u0,p
1
k+1
(x, y) ` ψ1(x)
k+1
0
k+1
(y)
). At the same
, p
)
(x, y)
(y)
k+1
Xi⊆
are
(with
).
θ
3.1.
PRELIMINARY NOTIONS, NOTATIONS, AND PROPERTIES
By the denition the following equality holds:
167
P (p1, X1, p2, X2, . . . , pk, Xk, p
= ∪{P(p1, u1, p2, u2, . . . , pk, uk, p
Hence the specication of
duced to the specications of
also that
Clearly, if
P (p, X, q) = X
ui= 0
then
P (p1, X1, p2, X2, . . . , pk, Xk, p
P (p1, u1, p2, u2, . . . , pk, uk, p
for any
pi= p
i+1
P (p1, u1, p2, u2, . . . , pi, 0, p
and the following conditions hold:
P (p1, 0, p1) = {0},
P (p1, u1, p2, u2, . . . , pi, 0, p
i+1
= P(p1, u1, p2, u2, . . . , pi, u
If all types
and
Pp(u1, u2, . . . , uk)
piequal to a typepthen we write
as well as
bu1, u2, . . . , ukc
instead of
P (p1, X1, p2, X2, . . . , pk, Xk, p
) =
k+1
) | u1∈ X1, . . . , uk∈ Xk}.
k+1
X ⊆ ρ
ν(p,q)
.
for nonempty sets
, . . . , pk, uk, p
, u
i+1
i+1
i+1
, p
i+2
, p
, . . . , pk, uk, p
i+2
, . . . , pk, uk, p
k+1
Pp(X1, X2, . . . , Xk)
bX1, X2, . . . , Xkcpand
p
)
k+1
)
k+1
k+1
k+1
k+1
)
)
) =
).
is re-
. Note
and
P (p1, u1, p2, u2, . . . , pk, uk, p
respectively. We omit the index
case, we write
For a family
θ
u1,u2,...,u
R ⊂ S(T )
(x, y)
k
instead of
of1-types we denote by
modelM) the set
{(a, b) | tp(a), tp(b) ∈ R
and by
SIR(inM) the set
{(a, b) | tp(a), tp(b) ∈ R
)
k+1
·pif the typepis xed. In this
θ
p,u1,p,u2,...,p,uk,p
(x, y)
.
IR(in the
andaisolates
andasemi-isolates
b}
b}.
168
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
Clearly,
the relations
IR⊆ SIRand, for any set of realizations of types inR,
IRand
SIRare reexive. As shown in Section 1.1, the
relation of semi-isolation on the set of tuples in an arbitrary model
is transitive and, in particular, any relation
SIRis transitive.
By Lemma 1.1.3.7, we have
3.1.0.3. Lemma.If(a, b) ∈ IRand
3.1.0.4. Proposition.
ρ
ν(p,q)
∪ ρ
ν(q,p)
⊆ U≥0.
(2)Ifp, q ∈ R,p
type then
Proof.
ρ
ν(p,q)
= ∅
(1) If
(1)Ifp, q ∈ R
is a principal type andqis a nonprincipal
and
ρ
ν(q,p)
ρ
contains a label
ν(p,q)
realizationsaandbofpandqrespectively such that
and
(b, a) /∈ IR. So by Lemma 3.1.0.3,
p(x)
contains a principal formula
(b, a) ∈ SIR. The contradiction implies that
larly we obtain
(2) Let
and
(a, b) ∈ IRthat witnessed by a formula
∃x(ϕ(x) ∧ θu(x, y))
ρ
ν(p,q)
= ∅
ρ
ϕ(x)
⊆ U≥0.
ν(q,p)
be a principal formula of
isolates
q(y)
. By the same reason,
(b, a) ∈ SIRthen
(b, a) ∈ IR.
are principal types then
⊆ U−.
u < 0
then there are
(a, b) ∈ I
(b, a) /∈ SIR. But since
ϕ(x)
, this formula witnesses that
ρ
p(x)
θu(x, y)
⊆ U≥0. Simi-
ν(p,q)
. If
|= p(a),|= q(b)
, the formula
. Sinceqis not isolated we obtain
ρ
ν(q,p)
⊆ U−.
¤
R
,
3.1.0.5. Corollary.Ifp(x)
3.1.0.6. Proposition.
is a principal type then
Let
p1, p2, . . . , p
following assertions hold.
(1)Ifui∈ ρ
ν(pi,p
,
i = 1,. . . , k
)
i+1
P (p1, u1, p2, u2, . . . , pk, uk, p
(2)Ifui∈ ρ
ν(pi,p
,
i = 1, . . . , k
)
i+1
negative then
P (p1, u1, p2, u2, . . . , pk, uk, p
(3)Ifui∈ ρ
ν(pi,p
,
i = 1, . . . , k
)
i+1
negative, then all elements of the set
X P(p1, u1, p2, u2, . . . , pk, uk, p
be types in
k+1
, and some
k+1
uiis negative then
) ⊆ U−.
, and all elements
) ⊆ U≥0.
k+1
, and all elements
)
k+1
ρ
⊆ U≥0.
ν(p)
S1(∅)
. The
uiare not
uiare non-
3.2.
EXAMPLES
169
are invertible and the set
X−1 {v−1| v ∈ X}
set
P (p
Proof.
(1) Letvbe a label in
Consider realizations
|= θ
pi,ui,p
For the family
(ai, a
ui< 0
SIR. If
(a
i+1
) ∈ IR,
i+1
then
v ≥ 0
, a
k+1
(a
), (a
(ai, a
i+1
R = {p1, p2, . . . , p
i = 1, . . . , k
, ai) /∈ IRand then, by Lemma 3.1.0.3,
i+1
then
, a1), (a1, ai) ∈ SIRwe get
k+1
−1
, u
k+1
(a
, pk, u
k
aiof
pisuch that
), i = 1, . . . , k, |= θ
i+1
, a1) ∈ IRand, by transitivity of
k+1
impossible. Since the element
is taken arbitrarily the set
P (p1, u1, p2, u2, . . . , pk, uk, p
of negative elements.
(2) Take again elements
ui≥ 0
relation
of
so
then
SIR, the element
(a1, a
v ≥ 0
) ∈ IR, by Lemma 3.1.0.3, we have
k+1
. Since the element
(a
i+1
, ai) ∈ IR,
a
k+1
v ∈ P (p1, u1, p2, u2, . . . , pk, uk, p
coincides with the
−1
, . . . , p2, u
k−1
−1
1
, p1).
P (p1, u1, p2, u2, . . . , pk, uk, p
k+1
(a1, a
(a1, a
k+1
k+1
, and so
p1,v,p
}
we have
(ai, aj) ∈ SIRfor
(a
(a
, ai) ∈ SIRthat is
i+1
v ∈ P (p1, u1, p2, u2, . . . , pk, uk, p
k+1
v, a1, a2, . . . , a
i = 1, . . . , k
. By transitivity of the
semi-isolates the element
as for (1). If
k+1
(a
, a1) ∈ IRand
k+1
)
k+1
a1. In view
k+1
).
k+1
) ∈ IR,
i ≤ j
, ai) /∈
i+1
SIRand
)
consists
. If
k+1
).
)
is taken arbitrarily the set
P (p1, u1, p2, u2, . . . , pk, uk, p
k+1
)
consists
of non-negative elements.
(3) follows immediately from (2).
3.1.0.7. Corollary.
Restrictions ofUto the sets
¤
U≤0and
≥0
U
form subalgebras of the algebra of distributions of binary isolating
formulas. Each element of the restriction to
U≥0has a unique
inverse element. The operation of inversion is coordinated with
the operations of the algebra.
3.2. Examples
Consider some examples for distributions of labels of binary
isolating formulas on sets of realizations of types
countable theoriesT.
p(x) ∈ S(∅)
for
170
I. If
Chapter
|ρ
ν(p)
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
| = 1
then
(x ≈ y)
is the unique principal formula up
to equivalence. It is possible only in the following cases:
(1)Tis small (i. e., with countable
S(∅)
) and satises some of
the following condition:
(a)
p(x)
is a principal type with a unique realization;
(b)
p(x)
y)} ∪ p(y)
then
ϕ(a, y) 6` p(y)
is a nonprincipal type such that if a set
is consistent, where
ϕ(x, y)
is a formula ofT,
;
{ϕ(a, y)∧¬(a ≈
|= p(a)
(2)Tis a theory with continuum many types and for any for-
mula
ϕ(x, y)ofT
¬(a ≈ y)} ∪ p(y)
isolating formulas
and for a realizationaof
is consistent and
ψ(a, y)
such that
ϕ(a, y) ` p(y)
ψ(a, y) ` ϕ(a, y) ∧ ¬(a ≈ y)
p(x)
if the set
{ϕ(a, y)∧
then there are no
.
The case 1,a is represented by a type being realized by a con-
stant; the cases 1,b and 2 are represented by theories of unary pred-
icates with nonprincipal types
p(x)
and having countably many and
continuum many types respectively.
,
II. Let
is linked with a unique realizationbofpfor which
moreover,
on two-element equivalence classes consisting of
principal type of a small theory then a
ρ
= {0, 1}
ν(p)
|= θ1(b, a)
. Then
1−1= 1
and any realizationaof
|= θ1(a, b)
and,
. Then the set of all realizations ofpsplits
θ1-edges. Ifpis a
θ1-edge is unique, and if
is nonprincipal, then the number of these edges can vary from1to
innity depending on a model of a theory.
More generally, having a nontrivial denable equivalence rela-
tionE(with at least two-element inE-classes) on the set of all
realizations of a1-typepwith
E(x, y) ∧ ¬(x ≈ y)
, we get the
ρ
= {0, 1}
ν(p)
{p}
-restriction for the algebra
, where
θ1(x, y) =
of distributions of binary isolating formulas with the condition
Pp(1, 1) = {0, 1}
III. Let
lary 3.1.0.5 the type
witnesses that
∃z(θ−1(x, z)∧θ−1(z, y))
By assumption the formula
θ−1(a, y)
. It means that, on the set of all realizations ofp, the re-
lation described by the formula
.
ρ
= {−1, 0}
ν(p)
p(x)
be a set for a small theoryT. By Corol-
is nonprincipal and the formula
SIpis non-symmetric. The formula
is also witnessing that
θ
(a, y)
−1,−1
is equivalent to the formula
θ−1(x, y) ∨ (x ≈ y)
θ−1(x, y)
θ
−1,−1
(x, y)
SIpis non-symmetric.
is an innite
partial order. This partial order is dense since if an element
p
p
a
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]