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1.2.
INESSENTIAL COMBINATIONS AND COLORINGS
71
isomorphism of that models, because any nite initial segments
of elementary chains of prime models over
added (they are situated as elementary submodels in
for coinciding sequencesαandβ, an isomorphism between
and
Mβcan be constructed step-by-step (in view of acyclicity of
p∞can be removed or
M
), and
p
∞
M
structure, \oors" of elements are dened uniquely with respect to
another one, besides that we take into consideration shortest path
connections and colors of elements, including intermediate).
If the sequencesαandβare non-equivalent, there are no partial
isomorphisms of
(on the set of realizations of
Mαand
Mβ, transforming the sequence of arcs
p∞) with the labels
Q
α(k+n)
on the
sequence of arcs with the same labels. All the more, there are no
isomorphisms between
Since every equivalence class is countable, there are
lence classes. Choosing one model in each class yields
non-isomorphic limit models over
Mαand
Mβ.
p∞.
2ωequiva-
2ωpairwise
¤
1.2.4. Criterion of non-symmetric semi-isolation
α
Finally in this Section, we introduce a criterion for non-sym-
metry of semi-isolation (for a countable theoryT), connecting re-
alizations of two types
p, q ∈ S(T)
. This criterion is based on
coloring of neighbourhoods of types. The terminology, that used
for this criterion, is not far from the system of notions, that used
for colorings. Proposition 1.2.3.3 is integrated in this terminology.
If
p, q ∈ S(A)
, we denote by
s
SI
(in a modelM) the relation
p,q
of semi-isolation (overA), connecting realizations of typespandq:
SI
Note
s
p,q
∪{(
= {(
b, a) |
a, b) |
M |= p(
M |= p(
a) ∧ q(b)
a) ∧ q(b)
andasemi-isolates
andbsemi-isolates
that if the typespandqare principal, then
SI
a}.
s
p,q
b}∪
is sym-
metric, and if one of that types is principal and the other is non-
principal, then
s
SI
is non-symmetric. So below, in this Section,
p,q
we shall assume thatpandqare nonprincipal.
72
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
1.2.4.1. Denition.
S(A)
model
q(
and
o
ver at most countable setAand realized in a countable
M
of a countable theoryT. Let
y)
b
e isolating sets for these types consisting of formulas
0
y),n ∈ ω
θ
(
n
,
respectively, such that the following conditions
Consider nonprincipal types
x) ⊂ p(x)
Θ(
x),q(y) ∈
p(
and
Θ0(y) ⊂
θn(
x)
are satised:
1)
|= ∀xθ0(
2)
|= ∀
3)
|= ∀
These
x) ∧∀y
x(θ
y(θ
(x) → θn(x)) ∧∃x(θn(x) ∧
n+1
0
(y) → θ
n+1
isolating sets of formulas exist since the set of formu-
0
y)
;
θ
(
0
0
(y)) ∧∃y(θ
n
0
(y) ∧
n
¬θ
¬θ
n+1
0
n+1
(
(
x))
y))
;
.
las with parameters inAis countable. Indeed, we enumerate all
formulas belonging, for instance, to the type
F
or any
ψ0= (
n ∈ ω
x ≈ x)
, we denote by
.
Now we remove from the sequence of formulas
ψnthe formula
x):ϕn,
p(
V
i<n
n ∈ ω
ϕiassuming
ψnall formulas equivalent to some their predecessors and obtain
the sequence
If
p = q
,
The formula
|= θ
θ
n
a
0
n
0
(resp
(
formula
a tuple
(
M
b) ∧
x))
(θn(
we assume that
θnis called ann-neighbourhood
is called ann-neighbourhood of typeq. We say that
ectively
¬θ
realizations of typespandq. Here we assume that
n ∈ ω
.
0
n+1
n∈ω
(
.
b)has
b)
).
θn= θ
0
,
n
n ∈ ω
colornif
We put the
.
of typep, and the
M |= θn(
a) ∧
innite color
n < ∞
¬θ
∞
a)
(
n+1
for the
for any
.
x)
1.2.4.2. Proposition.
in
S(A)
T
by tuples
p
arameters inAsatisfying the condition
x, y)
ϕ(
r
ealized in a countable model
a
andbr
witnesses
that
and only if for any
anwith
tuple
M
satises
Pr
oof.
is
semi-isolated over
M
|= θn(
0
y)
.
θ
(
0
n
Suppose that the formula
For any nonprincipal types
espectively and for any formula
b
is
semi-isolated over
n0∈ ω
a
, there exists
an)
,
any realization of formula
with
respect toA. By Lemmas 1.1.1.7 and
M
of a countable theory
M |= ϕ(
n ∈ ω
x, y)
ϕ(
a, b)
a
with
such that for any
witness3es
and
p(
x, y)
ϕ(
,
the formula
respect toAif
ϕ(
1.1.1.8, it is equivalent to the statement that for any formula
an, y)
that
0
θ
n
q(y)
with
in
y)
(
0
b
,
1.2.
INESSENTIAL COMBINATIONS AND COLORINGS
x)
there exists a formula
θn(
suc
h that
73
M |= ∀
This
means that for any tuple
an
y realization of
1.2.4.3. Corollary.
S(A)
,
realized in a countable model
a
by tuples
with
parameters in
andbr
x, y ((θn(x) ∧ ϕ(x,y)) → θ
anof
an, y)inM
ϕ(
has
For any nonprincipal types
espectively, as well as for any formula
A
satisfying the condition
following conditions are equivalent:
x, y)
1)
the formula
with
respect toAbut cannot witness that
with
respect toA;
2)
the following conditions are satised:
(a)
for any
of
color
≥ n
with
in
M
satises
(b)
ther
e exists an
tuples
|= θ0(
anand
an) ∧ θ
0
n
n0∈ ω
0
θ
n
0
b
n
(b
0
n
ϕ(
M |= θn(
y)
;
(
0
of
0
)
,
such that
witnesses
, there exists
an)
,
n ∈ ω
such that for any
nite colors
M |= ϕ(
any realization of formula
color
a color
that
n ∈ ω
< n
0
(y)) .
0
n
≥ n
with
M |= θn(
≥ n0.
¤
x)
and
p(
M
of a countable theory
M |= ϕ(
b
is
semi-isolated over
a
is
semi-isolated over
such that for any tuple
n0∈ ω
and
≥ n0respectively, with
0
an, b
).
n
an)
q(y)
in
x, y)
ϕ(
a, b)
,
the
a
an, y)
ϕ(
there are
,
T
a
b
n
Pr
oof.
By Proposition 1.2.4.2, the condition that the formula
x, y)
ϕ(
witnesses
that
b
is
semi-isolated over
a
with
respect toAis
equivalent to the property (a).
x, y)
Now we assume that the formula
a
is
semi-isolated over
b
with
ϕ(
respect toA. Then, by Lemmas
do
es not witness that
1.1.1.7 and 1.1.1.8, it means that there exists a formula
0
y)
that for any formula
M |= ∃
This
means that there exists an
there are tuples
such that
anand
M |= ϕ(
an, b
θ
x, y (θ
the
(
0
n
0
b
n
0
),
n
following holds:
0
(y) ∧ ϕ(x, y) ∧
0
n
n ∈ ω
of
nite colors
i.
e., the property (b) holds.
¬θn(
such that for any
< n
and
x)) .
≥ n0respectively
θn(
¤
x)
suc
n0∈ ω
h
74
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
1.3. Type reducibility, powerful types,
and the strict order property
In this Section, we dene concepts ofp-type (with a generaliza-
tion) and of reducibility of a theory over a type and prove, on the
one hand, that for a theoryTwithout the strict order property and
with a nonprincipal powerful typep, there is a non-p-principalp-
type andTis not reducible overp. On the other hand, we show an
example ofω-stable theory, having a non-p-principalp-type such
that this type is realized in models
1.3.1. Almostω-categorical theories and type reducibil-
ity
Mp.
The notion of
(n, p)
-type was introduced by S. V. Sudoplatov
[395]. Before it was used implicitly in R. E. Woordow [475] and
A. Tsuboi [461]. The following notion generalizes this denition.
1.3.1.1. Denition
x1),
p1(
A type
q(
x1,
. . . , pn(
q(
. . . ,
denoted by
any1-types
types
q(x1, . . . , xn) ∈ S
xn)
x1,
. . . ,
xn) ⊇
S
p1,...,p
p1(x1), . . . , pn(xn) ∈ S(T )
(K. Ikeda, A. Pillay, A. Tsuboi [229]). Let
b
e types in
xn) ∈ S(T )
n
S
pi(xi)
.
i=1
(T )
. A theoryTis
n
S(T)
is
with disjoint free variables.
said to be a
The set of all
(p1, . . . , pn)
(p1, . . . , pn)
-types ofTis
almostω-categorical
there are only nitely many
(T )
p1,...,p
.
n
-type
if for
It is shown in [229] that ifTis an almostω-categorical theory
with
I(T, ω) = 3
1.3.1.2. Denition.Ifp1(x)= . . . = pn(x)= p(
q(
x1,
. . . ,
type
types ofTis denoted by
then a dense linear ordering is interpretable inT.
x)
,
a
(p1, . . . , pn)
xn)
is
said to be a
S
n,p
(n, p)
(T )
and elements of
-type
. The set of all
(n, p)
[
Sp(T )
n∈ω\{0}
S
(T )
n,p
if
-
-
arep-types
.
1.3.
TYPE REDUCIBILITY
75
` q(
y)inS
q(
y)
.
ϕ(
If
q(
A type
yi,
pi= pi(yi),i =
there is a formula
y)}
{ϕ(
said to bep-principal
(T )
p1,...,p
n
1, . . . , n
y) ∈ q( y)
y)
is
a
.
,
where
y
is
, is said to be
suc
h that
(p, . . . , p)
-principalp-type, that type is
a concatenation of tuples
(p1, . . . , pn)
yi) | i =
∪{pi(
-principal
1, . . . , n} ∪
The following lemma is obvious.
1.3.1.3. Lemma.
fol
lowing conditions are equivalent:
(1)
the set of
is
nite;
(2)
any
(p1, . . . , pn)
For any types
(p1, . . . , pn)
-type is
-types with free variables in
x1),
p1(
(p1, . . . , pn)
. . . , pn(
-principal.
xn) ∈ S(∅)
x1,
(
. . . ,
the
xn)
By Lemma 1.3.1.3, a theoryTis almostω-categorical if and
only if for any1-types
type is
(p1, . . . , pn)
mits a natural generalization for uncomplete types
xn)
pn(
.
Without
loss of generality of the results, we will consider below
p1(x1), . . . , pn(xn) ∈ S(T )
any
(p1, . . . , pn)
-principal. Notice also that Lemma 1.3.1.3 ad-
x1),
p1(
. . . ,
in this Section, except Proposition 1.3.2.1, for simplicity thatpis
a type in
S1(∅)
.
if
-
1.3.1.4. Denition.
Let
M
be a countable saturated model
of a theoryThaving a predicate language. Consider, induced by
M
, a substructure
T
with the universe
R(p(M)) = R(M) ∩ (p(M))
p(M) = hp(M); Σ(T )i
p(M) = {a ∈ M | |= p(a)}
µ(R)
,
R ∈ Σ(T )
of language
Σ(T )
and relations
. Denote the theory
of
Th(p(M))byTp.
A theoryTis said to be
reduced over a typepifTand
Tpadmits
the quantier elimination.
1.3.1.5. Proposition.
a typep, then, in a model
If a small theory
T
is reduced over
Mp, any non-p-principalp-type is omit-
ted.
Proof.
there exists a bijection
q(M ),q ∈ Sp(T )
Note that by the quantier elimination ofTand
·p:
Sp(T ) → S(Tp)
such that
qp(p(M)) =
, and the restriction of this bijection on the set
Tp,
ofp-principal types implements a one-to-one correspondence with
76
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
the set of principal types of
M0, that exists by the smallness ofT. Assume that, in
non-p-principal type
quantier-free formula
y)inS
q(
ψ(x,
T ` ∃x (ϕ(x) ∧ ∃yψ(x,
for
any
ϕ(x) ∈ p(x)
It
hence follows that
and quantier-free formulas
Tp` ∃x (∃yψ(x,
y)
where
χ(
is
a quantier-free formula of type
Tp. Denote a prime model of
(T )
is
n,p
y)ofT
realized. Then there is a
suc
h that
y) ∧∀y (ψ(x, y) → χ(y)))
χ(
y) ∧∀y (ψ(x, y) → χ(y))),
y)
.
qp(
Tpby
Mp, a
y) ∈ q(y )
Since
Tpis
a transitive theory admitting the quantier elimination, there is an
element
It
a ∈ M0such that
M0|= ∃yψ(a,
means that the type
y) ∧∀y (ψ(a, y) → qp(y)).
y) ∈ Sn(Tp)
qp(
is
realized in
M0. But
q
is a nonprincipal type, since the correspondingp-typeqis non-
p
-principal. Consequently, a nonprincipal type is realized in the
prime model
Fix a theoryTand consider its
to a complete theory
of
T }
T0` Rϕ(
of
T0to the complete theory of language
formula of
M0, a contradiction.
such that
y) ↔ ϕ(y)
T }
is denoted by
Rϕis a
,
T0of language
y)
l(
y)
where
ϕ(
is
T∗. Thus, we get an operation
¤
Morleyzation
Σ(T ) ∪ {Rϕ| ϕ
-ary
, i. e., an expansion
is a formula
predicate symbol with
a formula ofT. A restriction
{Rϕ| ϕ
·∗:
is a
T →
T∗. It's easy to see the existence of a one-to-one correspondence
·∗:
S(T) → S(T∗)
T∗` Rϕ(
the type
y) → ψ(y)
y) ∈ S(T )
q(
simpleness
for which a complete type
for
some formula
.
For this correspondence, theλ-stability, the
y) ∈ q}
ϕ(
(see [59]) and the smallness of theories, and also the
y) {ψ(y) |
q∗(
corresp
onds to
properties of isolation and of powerfulness for types are preserved.
So considering questions on the existence of nonprincipal powerful
types in the classes of theories above, it suces to take theories of
form
T∗.
Then Lemma 1.3.1.3 and Proposition 1.3.1.5 imply
.
p
1.3.1.6. Corollary.If|S
reduced over the type
model
Mp.
p∗, then some
(T )| = ω
n,p
and the small theory
(n, p)
-type is omitted in the
T∗is
1.3.
TYPE REDUCIBILITY
1.3.2. Strict order property and realizability of types
77
Recall that a theoryThas the
exists a formula
x, y)ofT
ϕ(
and
strict order property
ai,
i ∈ ω
tuples
,
if there
such that the
following equivalence holds:
ai, y) → ϕ(aj, y) ⇔ i ≤ j
` ϕ(
.
Note that theories with formula-denable innite linear orders
have the strict order property. In particular, the Ehrenfeucht ex-
amples (see Example 1.1.1.3) have this property because they are
almostω-categorical.
The following proposition, which is implicitly contained
in R. E. Woodrow [475], claries that the described situation is
impossible for the theories without the strict order property.
x)
1.3.2.1. Proposition.
of theory
then
|S
2,p
T
and
(T )| = ω
T
. Moreover, for any model
If
does not have the strict order property,
p(
is
a nonprincipal powerful type
MofT
realizing the
typep, there are innitely manyp-preserving formulas which are
pairwise non-equivalent on the set of realizations ofpinM.
Proof.
symmetry of relation
We set
z(ϕn(x, z) ∧ ϕ(z,y)),n ∈ ω \
∃
p
-preserving
Consider a formula
SIp(such a formula exists by Lemma 1.1.1.12).
x, y) (x ≈ y),ϕ1(x, y) ϕ(x, y),ϕ
ϕ0(
(see Lemma 1.1.1.8 and the proof of Lemma 1.1.1.9),
x, y)ofT
ϕ(
{0}
. Since all formulas
it suces to show, that in a structure
countable saturated model of
n+1
R
(a,p(M∗)) \ R
ϕ
n
(a,p(M∗)) 6= ∅
ϕ
T∗) for any
hold for every
witnessing
p(M∗)
a ∈ p(M )
n+1
ϕn(
(where
,
n ∈ ω
the non-
(x, y)
x, y)
are
M∗is a
inequalities
. As-
sume on the contrary that for somen, an inclusion
n+1
R
(a,p(M∗)) ⊆ R
ϕ
is true. Consider a formula
assumption, for any
a, b ∈ p(M )
x, y)
ψ(
,
satisfying
n
(a,p(M∗))
ϕ
n
W
i=0
ϕi(x, y)
|= ϕ(
a, b)
.
Then, by
and
(b, a) 6∈
78
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
SIp, we get
b
that
then
semi-isolates
|= ∃
y (ψ(a, y)∧
b, y) → ψ(a, y)
` ψ(
all its realizations,
¬ψ(
same typep, there exists a sequence
p
such that
i
< j < ω
. This facts contradict the absence of the strict order
ci, y) → ψ(cj, y)
` ψ(
b, y))
.
Since the formula
|= ψ(
.
Since the tuples
cn)
(
and
|= ∃y(ψ(cj, y)∧
a, a)
n∈ω
a
of
b, y)
ψ(
and
witnesses
(b, a) 6∈ SIp,
andbrealize
realizations of
ci, y))
¬ψ(
the
property inT.
Now we write another proof for the equality
|S
2,p
(T )| = ω
. At
the same time it shows that there are innitely manyp-preserving
formulas that are pairwise non-equivalent on the set of realizations
ofp. This proof is based on the notion of quasi-neighbourhood and
suggested by B. S. Baizhanov.
SinceTdoes not have the strict order property, then for any
realization
a
-denable
M
.
Indeed, otherwise, by Lemma 1.1.1.12, some set
QV
p,M
aofp
,
i. e., is not a set of solutions for a formula
a)isb
(
in
a model
-denable
M |= T
, the set
QV
by means of the formula
p,M
ϕ(
QV
b, y)
(
ϕ(
a)
p,M
is
a, y)
b) ⊂
(
.
This
not
in
contradicts the condition thatTdoes not have the strict order
a
property. The
-denabilit
y of
QV
istence of greatest, by inclusion, set
preserving.
Then there are innitely manyp-preserving formulas
p,M
(
ψ(
a)
is
equivalent to the ex-
a, M)
,
where
x, y)isp
ψ(
which are pairwise non-equivalent on the set of realizations ofp.
Indeed, if there were only nitely many such formulas, we could
take their disjunction to obtain ap-preserving formula producing
the greatest set.
¤
,
-
In view of Corollary 1.3.1.6 and Proposition 1.3.2.1, we have
1.3.2.2. Theorem.
erty,
p(x)
is a nonprincipal powerful type ofT, then
reduced over
p∗.
IfTa theory without the strict order prop-
T∗is not
Lemma 1.1.1.2 and Proposition 1.3.2.1 imply
1.3.2.3. Proposition.
Any almostω-categorical Ehrenfeucht the-
ory has the strict order property.
The following proposition shows that a realizability on non-
1.3.
TYPE REDUCIBILITY
p
-principalp-types in a model
relation
SIp.
79
Mpimplies the non-symmetry of
1.3.2.4. Proposition.
in a model
element
to
Ipand
Proof.
M(a)
, whereais a realization ofp, then for every
biof a realization
(bi, a)
does not belong to
Letabe a realization of typep, and
mula isolating a non-p-principalp-type
element
biof a realization
elementa. Consider a formula
semi-isolatesa. Then the type
∪{p(yi) | yi∈
since thep-type
y}
∪ {∃x (ϕ(x,
y)
q(
If a non-p-principalp-typeqis realized
bofqinM(a)
,
the pair
(a, bi)
SIp.
ϕ(a,
y)
.
b
of
y) ∧ ψ(yi,
is
notp-principal.
q(y)
q(
in
ψ(bi, x)
y)
q(
x))}
¤
Assume that some
M(a)
semi-isolates
witnessing that
is
isolated by set
. This is impossible
belongs
y)
a
for-
the
b
1.3.3. Example
1.3.3.1. Example.
ory
T1, having a typepsuch that some non-p-principalp-type is
realizable in an elementary submodel
The languageΣwill consist of unary predicate symbols
n ∈ ω
, binary predicate symbols
We are going to construct anω-stable the-
Mpof a model
M |= T1.
5
Coln,
Q, R1, R2, and a 3-ary predicate
symbolS.
The predicateQdenes on the universeMa free directed pseu-
doplane, as in Example 1.2.3.5, with a
transitive
(that is connecting
any two elements) automorphism group, with innitely many con-
nected components
coloring
Col
, corresponding to symbols
C(a, Q)
and with an1-inessentialQ-ordered
Coln,
n ∈ ω
, satisfying the
following conditions:
i
5
This
example has a long history. In 1987, the author proposed an easy
\proof" at negative solving Lachlan problem based on the following arguments.
Since prime models omit all nonprincipal types, prime models over a type
omit all non-p-principalp-types. As every stable Ehrenfeucht theories should
contain a powerful typepand this type has a non-p-principalp-type, we get a
contradiction. Example refuted these arguments. Western specialists saw this
example in the talk by the author at Mal'tsev Conference 1989 in Novosibirsk.
Then it was exposed syntactically and published in [395]. Here the exposition
is both semantic and syntactic.
p
80
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
(1) innitely many images of colorµrelative toQfor any
µ ≥ n
(including∞);
(2) innitely many preimages of colormrelative toQfor any
m ≤ n
ors, for which
of solutions of every formula
unique image
such that
.
The predicate
|= ∃x (Q(x, a) ∧ Q(x, b))
c1and a unique preimage
|= Q(a, b)
The predicate
R1connects only elementsaandbof same col-
hold, and denes, in a set
Q(a, y)
, a
successor function
with a
c26= c1for each element
, and without cycles.
R2denes on
Mafree
directed graph with a
transitive automorphism group, with innitely many connected
components
Col
, corresponding to symbols
C(a, R2)
and with an1-inessential
Coln,
n ∈ ω
, satisfying the following
R2-ordered coloring
conditions:
(1) innitely many images of colorµrelative to
µ ≥ n
(including∞);
(2) innitely many preimages of colormrelative to
m ≤ n
to
;
(3) every two distinct preimages of every element, with respect
S
n
R
, lie in distinctQ-components, i. e., connected components
2
n∈ω
R2for any
R2for any
with respect toQ;
(4) any twoQ-components are connected by at most two
R2-
arcs, and having two that arcs the arcs have a common initial
vertex;
(5)Q-components form a free directed pseudoplane with re-
spect to
is exactly one element
R2and lying in the connected component
b
andcsatisfy
Col(a) = n
length of shortest
(b, c) ∈ R
innitely many common preimages ofbandcwith respect to
R2-arcs;
(6) for every imagebof an elementawith respect to
c 6= b
, being an image ofawith respect to
|= ∃x (Q(x, b) ∧ Q(x,c))
then, in the graph with the relation
(b, c)
-path is not less thann;
(7) for any elementsb,cwith
−n
n
1
∪ R
, and for any color
1
C(b, Q)
, have same color, and if
|= ∃x (Q(x, b) ∧ Q(x, c))
m ≤ min{n,Col(b)}
; these elements
R1∪ R
R2, there
−1
, the
1
and
, there are
R2,
having the colorm.
b
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