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1.3.
TYPE REDUCIBILITY
Let
type
p1(x), . . . , pn(x)
q ∈ S(T )
is called
be some 1-types of
(p1, . . . , pn)
realizingq, realizes all types
y
yl∈
internally powerful
some
type
pi(yl)
, if
is contained in
p(x)isp
-powerful
y) ∈ S(T )
r(
-powerful.
LCC1
-theoryT. A
, if any model
suc
h that for each variable
y)
.
r(
A type
p(x)
M |= T
is called
91
,
1.3.4.2. Theorem.
types of
S(T)
LCC1
is powerful.
Proof.
-theoryT, then each(p
Letqbe a
bitrary nonprincipal type
we assume that
eac
h element of the tuple
y = y1ˆ y2,
and each element of the tuple
ψ(x) ¬ϕ1(x) ∧ .. . ∧ ¬ϕn(x)
By the condition the subtype
M
realizingq. It means that there is a principal formula
a ∈
M
, and a tuple
´
³
1
a, b
χ
.
Now sinceTis a
³
formula
is a tuple
θ
a, b1, y
b2∈
M
If
(p1, . . . , pn)
b1∈
´
2
suc
h that
such that
p1(x), . . . , pn(x)
are all nonprincipal1-
, . . . , pn)
1
-powerful type. Consider an ar-
y) ∈ S(T )
r(
.
Without loss of generality
where for any realization
b1realizes
, where
M
, for which
LCC1
θ
|= r
some type of
b2satises
some uniform formula
ϕi(x) ∈ pi(x),i = 1,. . . , n
r1(
y1)ofr
is
realized in any model
|= q(
-theory, then there is a principal
³
a, b1, y
³
b1, b
´
2
´
2
andris
` r³b1, y
-powerful typeq∈
b1ˆ b2of
p1(x), . . . , pn(x)
χ (
a),|= r1(b1)
´
2
.
Then there
realized inM.
Since the typerand the modelM, realizingq, is chosen arbi-
trarily, the typeqis powerful.
¤
Theorem 1.3.4.2 immediately implies
a, y1)
and
r
.
,
|=
1.3.4.3. Corollary.Ifp(x)
theoryTand
p(x)
is an internally powerful type, then
is unique nonprincipal1-type of
p(x)
erful.
The following assertions in this section are proved in the paper
by B. Sh. Kulpeshov and S. V. Sudoplatov [273].
1.3.4.4. Proposition.
if and only if
|S1(T )| ≤ ω
Any almostω-categorical theoryTis small
.
LCC1
is pow-
-
92
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
Proof.
|S1(T )| ≤ ω
each union
pletions in
Clearly, ifTis small then
|S1(T )| ≤ ω
. Assume that
. Since by the denition of almostω-categorical theory
p1(x1) ∪ . . . ∪ pn(xn)of1
S
p1,...,p
(T )
, then we have the following estimate for the
n
-types has nitely many com-
number of complete types of the theoryT:
|S(T)| ≤
i. e.,Tis a small theory.
∞
X
(ωn· ω) =
n=1
¤
∞
X
n=1
n+1
ω
= ω,
Proposition 1.3.4.4 admits the following generalization produc-
ing a characterization of the smallness of a theory.
1.3.4.5. Proposition.
only if
the set
|S1(T )| ≤ ω
S
p1,...,p
Proof
(T )
is at most countable.
n
repeats the proof of Proposition 1.3.4.4.
1.3.4.6. Theorem.
language and with nitely many nonprincipal types
S1(∅)
, and the set
Any countable theoryTis small if and
and for any types
p1(x1), . . . , pn(xn) ∈ S1(T )
¤
LetTbe a countable theory of a predicate
p1,
. . .,pn∈
S
p1,...,p
(T )
is nite. Then the following condi-
n
tions are equivalent:
(1)
the theoryTis almostω-categorical;
(2)
the theoryTis1-locallyω-categorical.
Proof.
(1) ⇒ (2)
. LetTbe an almostω-categorical theory
and assume thatTis not1-locally countably categorical. Then
for some formulas
a
,
whose coordinates realize types
formulas with parameters in
¬ϕ1(x)∧ . . . ∧ ¬ϕn(x),ϕi(x) ∈ pi(x)
Nardzewski Theorem it means that for somemthe set
a)
types in
S(
,
ϕi(x) ∈ pi(x),i = 1, . . . , n
pi, the structureM, dened by
a
,
on some setM, dened by a formula
, is notω-categorical. By Ryll-
realized inM, is innite. Since
, and some tuple
Xofm
p1(x), . . . , pn(x)
are
all nonprincipal1-types ofTthenMrealizes only nitely many1-
types, therefore there are innitely manym-types
inX,
a =
set
containing same1-types
(a1, . . . , ak)
S
r
,...,r
i
i
1
and
|= p
1
,...,p
j
k
(T )
,p
j
m
r
(x1), . . . , r
i
1
(as),s ∈ {1,. . . , k}
j
s
(xm)
i
m
we obtain the innite
, that contradicts the almostω-categoricity
q(x1, . . . , xm,
. Assuming that
a)
ofT.
-
1.4.
POWERFUL DIGRAPHS
93
(2) ⇒ (1)
set
X S
from
p1, . . . , pn. If
Assuming that
pn(x)
M
types
separating
, dened by the formula
r
i
1
Then each type inXcontains the formula
for any tuple
structure, dened by formulas with parameters in
M
, isω-categorical. By Ryll-Nardzewski Theorem each element
in
S
p
,...,p
j
1
condition the set
. LetTbe a
r
,...,r
i
1
m > 0
, . . . , r
j
i
m
a
,
(T )
has nitely many completions inX. Since by the
k
,p
,...,p
i
j
m
j
1
k
m = 0
we nd formulas
r
, . . . , r
i
1
i
m
LCC1
(T )
then by the condition this set is nite.
from
-theory. Consider an arbitrary
, where the types
r
i
1
, . . . , r
ϕ1(x) ∈ p1(x), . . . , ϕn(x) ∈
p1, . . . , pnsuch that on some set
ψ(x) ¬ϕ1(x) ∧ . . . ∧ ¬ϕn(x)
are realized and all types
p1, . . . , pnare omitted.
ψ(x)
. By the condition
whose each coordinate realizes some type
a
,
on the set
S
p1,...,p
(T )
is nite, and so
n
S
p
j
1
,...,p
j
k
(T )
dier
i
m
, all
pi, the
is nite
too, then the setXis also nite. As the setXis chosen arbitrarily,
the theoryTis almostω-categorical.
¤
Proposition 1.3.4.5 and the proof of Theorem 1.3.4.6 implies
the following
1.3.4.7. Proposition.
nonprincipal1-types
if and only if the set
LetTbe a
LCC1
p1(x), . . . , pn(x)
S
p1,...,p
(T )
is at most countable.
n
-theory with nitely many
. Then the theoryTis small
1.4. Powerful digraphs
In this Section, we introduce the concept of a powerful di-
graph and establish its \local" presence in the structure of any
nonprincipal powerful typep. We also show, that, having the
invariance property of a theory, a structure of powerful digraph is
contained in a restriction of saturated structure to a structure of re-
alizations of nonprincipal powerful type, having the global pairwise
intersection property. We describe structures of transitive closures
of saturated powerful digraphs, formed in models of theories with
nonprincipal powerful1-types, when the number of nonprincipal
1
-types is nite. Moreover, we prove, that the structure of pow-
erful digraph, considered in a model of
simple
theory [59], induces
(2, p)
-
94
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
the innite weight. It means, that there are no powerful graphs
in structures of known classes of simple theories (such as super-
simple or nitely based theories), that do not contain Ehrenfeucht
theories.
1.4.1. Cones. Powerful digraphs. Pairwise intersection
properties
1.4.1.1. Denition.
sets of solutions of formulas
for
some
n ∈ ω
algebraic(denable)closure
denoted by
acl(A)
1.4.1.2. Denition.
inΓ. The set
S
Qn(Γ, a)
n∈ω
cones
5Q(a)
) is called a
and
Recall that for any setAofTthe union of
ϕ(x,
(respectively
a),a ∈ A
|= ∃=1x ϕ(x,
suc
ofA. An algebraic closure ofAis
and its denable closure, by
Let
5Q(a)
upper(lower)Q
4Q(a)
Γ = hX; Qi
S
Qn(a, Γ)
n∈ω
by
cones
be a graph,
(respectively,
-coneofa
and denote by
respectively, ifQis xed.
A countable acyclic digraph
Γ = hX; Qi
if the following conditions hold:
(a) the automorphism group ofΓis
transitive
vertices are connected by an automorphism;
(b) the formula
Q(x, y)
is equivalent in the theory
a disjunction of principal formulas;
(c)
(d)
property
acl({a}) ∩ 4Q(a) = {a}
Γ |= ∀x, y ∃z (Q(z, x) ∧ Q(z, y))
).
for each vertex
(the
h that
a)
|= ∃=nxϕ(x,
)
is said to be an
dcl(A)
.
a
a vertex
4Q(a)
. We call theQ-
5(a)
is said to be
and
4(a)
powerful
, that is, any two
Th(Γ)
a ∈ X
;
pairwise intersection
a)
,
to
Clearly, in the classical examples of Ehrenfeucht theories, the
countable graph with the relation
x < y
of dense linear order is
powerful.
The following example, represented in A. I. Mal'tsev [295]
and E. Bouscaren, B. P. Poizat [133], denes a stable theory of
acyclic pairing function
with a structure of powerful digraph.
1.4.
POWERFUL DIGRAPHS
95
1.4.1.3. Example.
M
such that
(f1, f2):M → M × M
are no nonempty sequences
that
f
(. . . f
i
n
(a) . . .) = a
i
1
ing a theory of a locally free algebra, is stable6. Consider the
1
-type
p ∈ S1(T )
f
(. . . f
i
n
(a) . . .) = f
i
1
LetMbe a set with two functions
is a bijection, for which there
i1, . . . , inand elements
. The theory
T Th(hM ; f1, f2i)
a ∈ M
, i. e., the type of elementsasuch that
j
m
(. . . f
(a) . . .) ⇔ i1. . . in= j1. . . jm.
j
1
f1, f2:
M →
such
, be-
free
It's easy to see, that some countable set of realizations ofpwith
the relationQ, dened by means of the formula
f2(x))
, forms a powerful digraph.
(y ≈ f1(x)) ∨ (y ≈
At the same time, as B. Herwig showed [206, Proposition 3],
related constructions in the class of stable theories, being based
on the property that
b ∈ acl({a})
holds for any
(a, b) ∈ Q
, are
impossible in the class of small theories, i. e., that constructions
generate equalities
|S(T)| = 2ω.
¤
As P. Tanovic noticed, a pairing function appears on a set of
realizations of (nonprincipal) type
relation
SIpif
SIpis antisymmetric, there is a
with realizationsaandb, being not linked by
strongly
for some realizations
formulas
since for any element
∃x, y(ϕ(x, y, c) ∧ ϕ(x, y, c0))
to
SIp, and thus
RK
-equivalent top.
Indeed, in view of
ϕ(a, b, z)
and
c = c0.
q ≡RKp
a, b, cofp
ϕ(x, y, c)
c0with
witnesses that
there is a formula
p(x)
with the non-symmetric
SIp, and
ϕ(x, y, z)
with
|= q(a, b)
and
are principal. Then
|= ϕ(a, b, c0)
(c, c0)
(2, p)
-type
q(x, y)
q(x, y)
such that
|= ϕ(a, b, c)
c ∈ dcl(ab)
, the formula
and
(c0, c)
belong
is
, the
In the Peretyat'kin example [335] of theory with three countable
models the powerful digraph also includes the relation<.
Apart from these examples, a rather rich class of pow-
erful digraphs is formed by the acyclic digraphs
hP ; Qi =
hP ; {(p, p0) | p0= pg0on some line}i, corresponding to polygonome-
tries
pm(G, hP, L, ∈i, g0)
6
The
stability of theories of locally free algebras is proved by O. V. Bele-
gradek [123].
on projective planes [54, 397].
96
Let
of
T
l(x)= l(
and
by
Chapter
M
be a model of a theoryT,
over the empty set,
y)
.
Denote by
p
R
(M)
ψ
1. CHARACTERIZATION OF EHRENFEUCHTNESS
x)
b
e a complete type
x)inM
p(
p(M)
p(
x, y)
ψ(
b
e a formula ofT, where
the set of realizations of
, the binary relation
,
a, b) ∈ (p(M))2|
{(
The
following statement shows that the powerful digraphs re-
M |= ψ(
a, b)}.
side \locally" in the structure of each nonprincipal powerful type.
x)
1.4.1.4. Proposition.Ifp(
a theoryTand
formula
l(x)= l(
(1)
ϕ(
y)),
for each
M
is a countable saturated model ofTthen for each
x) ∈ p(x)
,
there exists a formula
satisfying the following conditions:
a ∈ p(M )
a disjunction of principal formulas
that
isolate
|= ϕ(
alizations
semi-isolate
there
a, x) ` p(x)
ψi(
a
;
(2)
for
every
c) ∧ ψ(c, a) ∧ ψ(c, b)
Pr
oof.
By the hypothesis and Lemma 1.1.1.12, there are re-
a
andbof
a
.
is a principal formula
ate all realizations of
a, y) χ
χn(
m
W
i=0
Fix
χi(
x, y)
c
n
some formula
can
,
and
a, b ∈ p(M)
p(x)
Since
M(
p(
(
a, y)
for
ϕ(
be taken as
|= ψi(
.
in
b) ≺
x)inM(b):p(M(b))
n ∈ ω
x) ∈ p(x)
is
a nonprincipal powerful type of
,
the formula
ψi(
a, b)
implies,
,
there exists a tuple
the model
M(
(
a, y)
χ
c
M(
a)
,
for each realization
suc
h that
.
and
show that some formula
x, y)
ψ(
.
Clearly, each of those formulas
x, y)ofT(wher
ψ(
a, x)
ψ(
is
a, x),i ≤ m
b
suc
h that
(
|= χ
c
cn| n ∈ ω}
= {
do
a, c)
that
a)
equivalent to
,
such
es not semi-
c
such
that
b
do
es not
c ∈ M(b)
.
Enumer-
.
Put
satises condition 1. Assuming that none of them satises condi-
tion 2, by Compactness Theorem, we get the consistency of the
set
x, y) p(x) ∪ p(x)∪
r(
!
¯
¯
¯
m ∈ ω).
¯
¯
∪(¬∃
z
ÃÃ
m
_
χi(z,x)!∧
i=0
Ã
m
_
χi(z,y)!∧ ϕ(z)
i=0
e
1.4.
POWERFUL DIGRAPHS
x)
p(
d1and
and
is
powerful, the type
d2:
M(b) |= r(d1, d2)
(
χ
x, y)
d
2
suc
h that
Since
tuples
(
x, y)
χ
d
1
contradicts the consistency of
sistent; hence, for some
x) ∪ p(y) ∪(¬∃z
p(
Putting
x, y)
ψ(
m0, we have the inconsistency of the set
ÃÃ
m
0
_
χi(z,x)!∧
i=0
m
0
W
χi(x, y)
i=0
x, y)
r(
r(
x, y)
is
realized in
.
Thus, there exist formulas
(
|= χ
a, d1) ∧ χ
d
1
.
Therefore,
Ã
m
0
_
χi(z,y)!∧ ϕ(z)
i=0
,
we deduce the claim.
d
r(
M(
(
a, d2)
2
x, y)
¤
b)
b
y some
,
which
is
incon-
!)
97
.
1.4.1.5. Denition.
called the
(LPIP)
(
20) for every
that
|= ψ(
w
e then call it the
with
respect to
local pairwise intersection property
. If for the formula
a, b ∈ p(M)
c, a) ∧ ψ(c, b)
ψ(
Whenever a formula
the digraphp(M); R
Recall that theories
tively are said to be
there are formulas of
constants of language
of
Σ
is a model of
1−i
A theoryTis said to be
(where
the language
l(
x)
= n
{Rψ}
) of
of the theory of the structure of some formula-denable set
(where
ϕ ∈ p,M |= T
The property 2 of Proposition 1.4.1.4 is
and denote it by
x, y)
ψ(
the
stronger property is true:
,
there exists a tuple
c ∈ p(M )
suc
,
global pairwise intersection property
x, y)
and
it will be denoted by
x, y)
ψ(
p
(M)®prepowerful
ψ
T0and
similar
with
T1of languages
if for any models
Ti, dening in
Σ
such that the corresponding structure
1−i
T
.
1−i
(n, p)
-invariant
properties 1 and
.
Mipredicates, functions and
if for each formula
(GPIP)
Σ0and
Mi|= Ti,
Tp, the restriction of the Morleyzation of
for
p(
.
20exist, call
Σ1respec-
i = 0, 1
ψ(
Tpto
is similar to the restriction of the Morleyzation
ϕ(M)
) to the same language.
x)
x)
We will show that the existence of a prepowerful structure on
the set of realizations of a nonprincipal powerful typepof some
(2, p)
-invariant theory implies the existence of a formula-dening
a powerful digraph structure on this set.
h
,
98
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
1.4.1.6. Proposition.Ifp(x)
a
(2, p)
-invariant theory
T
ful digraph then for some formula
ψ(x, y)
, the digraph
Proof.
The
hp(M); R
(2, p)
-invariance ofTimplies that we can choose
to make principal the formulas
ψi(x, y)
pal formulas for each
such that
ψ(a, x) =
a ∈ p(M)
ementabelongs to a prime model
leyzation of
Tpto the language
cipal formulas, corresponding to the formulas
belong to the same model. Consider formulas
{Rψ}
, corresponding to complete formulas of the types
i = 0, . . . , m
. Take as
θ(x, y)
is a nonprincipal powerful type of
andp(M); R
p
(M)i
θ
m
W
is powerful.
p
R
(x, y)
0
ψ
i
ψi(a, x)
i=0
and all
p
(M)®is a prepower-
ψ
θ(x, y)
with
corresponding in
, where
ψi(a, x)
i = 0,. . . , m
T ` θ(x, y) →
are princi-
. Indeed, the el-
M0of the restriction of the Mor-
{Rψ}
. Some realizations
diof prin-
ψi(a, y) i = 0, . . . , m
0
ψ
(x, y)
i
of language
some formula ofTsatisfying the
following conditions:
(1) the restriction of the Morleyzation of the theory of the struc-
ture of some formula-denable set
guage
{Rθ}
is similar to the restriction of the Morleyzation of
to the same language;
(2)
` ((ϕ(x) ∧ ϕ(y)) → (θ(x, y) ↔
(3)
` θ(x, y) → ψ(x, y)
Since the modelMis saturated, the automorphism group of the
digraph
Γ = hp(M); R
Note that for each
b ∈ p(M )
andbdoes not semi-isolatea. Thus, since the relation
.
p
(M)i
θ
a ∈ p(M)
ϕ(M)
m
W
i=0
is transitive.
it follows from
(where
0
ψ
(x, y))
i
ϕ ∈ p
;
|= θ(a, b)
) to the lan-
of semi-isolation is transitive, the digraphΓis acyclic.
The non-symmetry of the relation
nessed by means of the formulaθ, implies the equality
p
4
(b) = {b}
R
θ
exists
dinM
d ∈ (acl({b}) ∩ 4
. Thus, because the relation of semi-isolation is transitive,
for each
b ∈ p(M)
p
(b)) \ {b},
R
θ
the elementbwill semi-isolatea, where
SIpof semi-isolation, wit-
acl({b}) ∩
. Indeed, assuming that there
we deduce thatbsemi-isolates
|= θ(a, b)
and
a ∈ p(M)
this is a contradiction.
x)
The
(GPIP)
for
property with respect to
p(
with
θ(x, y)
respect to
ψ(x, y)
implies the same
, and so, we have the pairwise in-
T
tp(adi)
T
that
to
,
,
p
;
1.4.
POWERFUL DIGRAPHS
99
tersection property forΓ. Therefore,Γis a powerful digraph.
¤
Note that under the assumptions of Proposition 1.4.1.6 the
niteness of number of nonprincipal1-types implies the relation
acl({a}) ∩ 5
for each vertexaof the digraph
Indeed, assume that the type
b ∈ (R
a formula
η(a, b) ∧ ∃=ky η(a, y )
a
p
(M))n(a, Γ),b 6= a
θ
η(a, y)
witnessing the semi-isolation and such that
. Then the original theory contains
for some
and the number of nonprincipal1-types is nite, it follows that
there exists an elementcrealizing a principal type such that
η(c, b) ∧ ∃=ky η(c, y )
. This means that the nonprincipal type
p
(a) = {a}
R
θ
Γ = hp(M); R
tp(b/a)
k ∈ ω
. Sincebdoes not semi-isolate
p
(M)i
θ
.
is algebraic for some
(1.1)
|=
|=
p(x)
is realized in the prime model; this is a contradiction.
If there are innitely many nonprincipal1-types then (1.1) need
not hold. To illustrate that, consider the following example of an
ω
-stable theory with a nonprincipal1-type
symmetric relation of semi-isolation via a formula
acl({a}) = 5Q(a)
1.4.1.7. Example.
sequences
α = hα0,
l(α)= α0+ 2
Let
T0be a theory of language
for each realizationaof
Denote by
α1, . . . , αni
.
Ω
the set of nonempty nite
with
(1)
,
hP
α
p0(x)
αi∈ ω
(2)
Q
i
p0(x)
, having a non-
Q(x, y)
.
for
with
α∈Ω
such that
i ≤ n
and
the following
axioms:
α = α0ˆ m ∈ Ω
(1) if
then
`¡P
then
α0ˆ
(2) if
`
(3)
(x) → P
(m+1)
α1= α
¬∃x (P
0
ˆ
1
(x) ∧ P
α
1
0
the relationQforms the graph of a free (acyclic) unar with
α0ˆ m
and
α
(x)¢∧
α2= α
(x))
2
∃≥ωx¡P
0
ˆ
2
;
α0ˆ m
0
are tuples inΩand
innitely many preimages of each element;
(4)
` ∀x, y ((P
¬P
h0,m+1i
(5) if
(y)))
|= P
of the formula
formula
P
h0,mi
of the formulas
for
h0,mi
Q(x, a)
(x) ∧ ¬P
P
(x) ∧ ¬P
h0,mi
m ∈ ω
(a) ∧ ¬P
h0,m+1i
;
h0,m+1i
(x) ∧ Q(x, y)) → (P
(a)
then the set of realizations
consists of innitely many realizations of the
(x)
, and innitely many realizations
h1,k,m+1i
(x)
h1,k,mi
h0,m+1i
(x) ∧ ¬P
(x) ∧
for each
¬P
k ∈ ω
α0ˆ
(m+1)
α
h0,mi
;
(x)¢;
0
6= α
1
(y) ∧
0
2
100
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
α = k ˆ α0ˆ l ˆ m
(6) if
µ
(k−1) ˆ
k 6= 0
¡
P
α0ˆ l ˆ m
k ˆ
α0ˆ m
and
` ∀x, y
→¡P
(7) if
of realizations of the formula
alizations of the formulas
for each
l ∈ ω
;
(8) a universe of a model of
tions
α ∈ Ω
P
α
The
construction of a saturated model satisfying axioms 1{8,
.
,
enables us to verify the completeness of
(y) ∧
|= P
is
a tuple inΩ,
(x) ∧
¬P
¬P
(k−1) ˆ
(a) ∧
α ˆ m
k ˆ
Q(x, a)
P
(k+1) ˆ
k ≥ 1
, then
k ˆ
α0ˆl ˆ
α0ˆ
¬P
(m+1)
k ˆ
(x) ∧ Q(x,y)¢→
(m+1)
¶
¢
α ˆ
(y)
(m+1)
, m ∈ ω;
(a)
, then the set
consists of innitely many re-
(x) ∧
α ˆ l ˆ m
¬P
(k+1) ˆ
α ˆ l ˆ
(m+1)
T0consists of elements of rela-
T0. Theω-stability of
(x)
T
follows because each formula without parameters is equivalent to
(x)
,
a Boolean combination of formulas of the form
and
∃z (Q
n
1
(x,
z)∧ Q
n
2
(y, z)),n1, n2∈ ω
. Moreover, like Example
P
α
α ∈ Ω
1.2.3.5, the countable number of1-types over each countable set
A
is followed by the countable number of possibilities of distance
distributions from elements ofAto realizations of types, which in
turn is implied by acyclicity of graph with the relationQ.
For the type
p0(x) ∈ S1(∅)
©
P
h0,mi
, isolated by the set
ª
(x) | m ∈ ω
0
,
of formulas, the relation of semi-isolation is not symmetric via the
formula
formula
and of the nonprincipal types
sets©P
Q(x, y)
Q(x, a)
(x) | m ∈ ωª,
h1,k,mi
. For each
a |= p0, the set of realizations of the
is exhausted by the realizations of the type
p
k ∈ ω
(x) ∈ S1(∅)
h1,ki
. Since the relationQforms the
, isolated by the
p
graph of a free unar with innitely many preimages of each element,
acl({a}) = dcl({a}) = 5(a)
M |= T0.
¤
for each element
a
of a model
In connection with the argument above, the problem seems in-
teresting of describing the powerful digraphs that can be expanded
to the structures of powerful1-types both in the case of nitely
many and innitely many nonprincipal 1-types.
0
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