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Classification of countable models of complete theories. Р.1. Monograph in two parts

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1.2.

INESSENTIAL COMBINATIONS AND COLORINGS

Col

(a) = µ},µ < λ

is a combination of i. e., with a structure

Comb(M, hM,

Col

A coloring Col of structure over a set

U ⊆ ω

distinct elements and having length together

with colors of elements of

the denition of

Col of structure

IECU(M, hM,

Coli).

For any model Col0:

M0→ λ ∪ {∞}

1) Col0(a) = µifM0|=

2) Col0(a) = ∞ifM06|=

Below, models shall denote a restriction of model

Any expansion predicates Colµ, any colored theory is a theory of some colored structure where

M |= T

A coloring Col of structureMis said to be

U ⊆ ω

if for any model

. Clearly, a colored structure

with a

hM,

Col

coloring of its universe

i = hM;

Colµi

µ<λ

hM,

i).

if for any tuple

is said to be

a ∈ M

a) ∈ U

isolates

innerly inessential

consisting of pairwise

the type of

hM,

Col

(a)

IECU, an inner inessentiality of coloring

overUis equivalent to the equality

M0|= Th(hM,

Col

hM,

a coloring

is dened naturally by the following rules:

Colµ(a)

M0will be denoted by

;

for any

hM0,

µ < λ

hM0,

Col0i

, and by

to language

T0of a theoryTby pairwise inconsistent unary

µ < λ

, is said to be a

colored theory

. Clearly,

hM,

inessential over a set

hM0,

Col0i

of colored theory

Th(hM,

a corresponding coloring Col0is innerly inessential overU.

As above, considering the inessentiality over simply about the

A coloring Col of structure

inessential

, i. e.,

if this coloring is innerly inessential over an innite set

hM,

A coloring Col of structure if there is an innite set of colored theory

inessentiality

Col

i = AIEC(M,hM,

U ⊆ ω

Th(hM,

is said to be

Col

i).

is said to be

such that for any model

Coli), a corresponding coloring Col0is

U = ω

innerly almost

almost)inessential

hM0,

innerly inessential overU.

hM,

Col

ainM

Col

M0we

Σ(M)

Coli,

Coli),

we say

Col0i

Col

i =

The following example shows that an inner inessentiality of col- oring of structure does not imply an inessentiality of the coloring, although by Corollary 1.2.1.11 for a weakly saturated structure such implication holds.

Chapter

1. CHARACTERIZATION OF EHRENFEUCHTNESS

1.2.2.2. Example.

| n ∈ ω},i ∈ {0, 1, 2}

f(c

) = c

2 2n

f(c

2 2n

Let

be a structure consisting of constants

, and expanded by substitutionfacting

) = c

f(c

) = c

2 2n+1

f(c

2 2n+1

) = c

n ∈ ω

Now let all these constants and the symbolfform the language of

. Consider a coloring Col:

i,n ∈ ω,i ∈ {0, 1, 2}

inessential, because a type some set of formulas

. Clearly, any coloring of

{(xj≈ c

Col0of weakly saturated model

M → {0, 1, 2}

any tuple

) | 1 ≤ j ≤ l(

hM0,

Col0i |= Th(hM,

, dened by Col

a ∈ M

x)}

isolated by

But a coloring

Coli)is not

) =

is innerly

inessential.

Indeed, consider elements

Col2(ak) ∧ ¬(ak≈ c

n ∈ ω

hM0,

Col

k = 0, 1

(a0) = tp

. Clearly,

hM0,

Col

ak∈ M0such that

) ∧ ∃xk(

Colk(xk) ∧ (f(xk) ≈ ak)),

(a1)

, but

(a0) 6= tp

hM0,

(a0) = tp

(a1),

Col

(a1)

and

i. e.,

hM0,

The type

(a0, a1)

Col0i 6= IEC(M0, hM0,

also witnesses this inequality.

Col0i).

Note that for any colored structure ory

∆

Col

Th(hM0,

is a closure of

Col0i)

is totally transcendental and

{(x ≈ y)} ∪ {

Colµ(x) | µ < λ}

hM0,

Col0i

∆

-based, where

Col

with respect to

, the the-

substitutions of variables.

The following assertions hold in view of Theorem 1.2.1.12, Corollary 1.2.1.13, and Theorem 1.2.1.15.

1.2.2.3. Theorem.

theoryTover a set

(1) Col

(2)

is an inessential coloring overU;

Th(hM,

1.2.2.4. Corollary.

Let

U ⊆ ω

Coli)is

Let

Col

be a coloring of model

Mof∆

. The following conditions are equivalent:

(∆ ∪ ∆

Col

)

-based overU.

Col

be a coloring of model

Mof∆

theoryT. The following conditions are equivalent:

(1) Col

(2)

is an(almost)inessential coloring;

Th(hM,

Coli)is an(almost)(∆ ∪ ∆

)

-based theory.

Col

-based

1.2.

INESSENTIAL COMBINATIONS AND COLORINGS

1.2.2.5. Theorem.IfCol

structureM, then

Th(M)isλ

-stable(small).

Th(hM,

is an almost inessential coloring of a

Coli)isλ-stable(small)if and only if

1.2.3. Ordered colorings

1.2.3.1. Denition.

be a formula ofT. A coloring nite cardinality) is said to beϕ-ordered

Let

be a model of a theoryTand

Col:M → λ∪{∞}

(whereλis an in-

if the following conditions

ϕ(x, y)

hold:

(a) for any

Colµ(a) ∧

(b) if

Colµ(c) ∧

Recall that a theoryTis said to be

-type over the empty set. A coloring Col of structure

inessential,n ∈ ω \ {0}

model

hM0,

Clearly, any inessential coloring isn-inessential for any

, and a coloring Col isn-inessential if and only if the theory

Th(hM, Th(hM,

Coli)is an inessential combination of theories

Coli)over

Note that if

coloring of a model complete1-types of

µ ∈ λ∪{∞} µ ∈ λ,p∞(x)

{¬

Colµ(x) | µ < λ}

In the Ehrenfeucht example of the theory

able models, the model expansion of a transitive theory by constants

µ ≤ ν < λ

Colν(b) ∧ ϕ(a,b)

µ < ν < λ

Colν(d) ∧ ϕ(d,c)

Col0i |= Th(hM,

U = {1, . . . , n}

Col:M → λ∪{∞}

, where

, there exist elements

;

then there are no elements

transitiveifT

is said to ben-

, if

(M0)n⊆ IECT

Coli).

is a surjective1-inessential

of a transitive theoryT, then the set of

Th(hM,

pµ(x)

Coli)over∅consists of types

is a type isolated by the formula Colµ(x)

a, b ∈ M

c, d ∈ M

has a unique

hM0,

Col

is a (unique) nonprincipal type isolated by the set

of formulas.

T3with three count-

ck,

k ∈ ω

, can be interpretable as an inessential col-

such that

for any

n ≥

Th(M)

pµ(x)

Th(hQ; <i)

oring Col specied by the following conditions:



a < c0,

a = ck,

ck< a < c

k+1

Col

(a) =





2k + 1 2k + 2

and

, ,

Chapter

1. CHARACTERIZATION OF EHRENFEUCHTNESS

It is easy to note that Col isϕ-ordered, where Besides, the relation type

p∞is non-symmetric, which the formulaϕis a witness for.

In Ehrenfeucht examples of structures

hQ; <, P0, . . . , P

on the set of realizations of powerful

∞

Tn,

n ≥ 4

, constant expansions of the

can also be conceived of as color

n−3

ϕ(x, y)  x < y

models with inessential ordered colorings.

Now we show that if a theory is obtained by means of an1-

inessential ordered coloring of a transitive theory and has a unique nonprincipal complete1-type then that coloring has the cardinality

λ = ω

1.2.3.2. Proposition.Ifϕ(x, y)

oryTand ordered coloring of a model

Proof.

is a formula of a transitive the-

Col:M → λ ∪ {∞}

MofT

is a surjective

, then

λ = ω

As it was noticed above, the1-inessentiality of coloring implies that there exists a unique nonprincipal complete1-type of the theory

Now assume that

Th(hM,

Coli).

λ > ω

. Consider the following sets of formu-

las:

q0(x) = {∃y, z(Coln(y) ∧ Colω(z) ∧ ϕ(y, x) ∧ ϕ(x, z)) | n ∈ ω},

q1(x) = {∃y(Colµ(y) ∧ ϕ(y, x)) | µ < λ}.

By compactness each

Colisϕ

-ordered, the set

time, since the coloring

i = 0,1

theory

, can be extended to a complete nonprincipal1-type of the

Th(hM,

Coli). Thus we get a contradiction.

qi,

i = 0, 1

q0(x) ∪ q1(x)

Col

is surjective andϕ-ordered, each

, is consistent. Since the coloring

is inconsistent. At the same

qi,

Consider sucient conditions for aϕ-ordered1-inessential col-

oring to imply a non-symmetry of relation

1.2.3.3. Proposition.

Let

ϕ(x, y)

be a principal(i. e., isolating

witnessed byϕ.

∞

a complete type and consistent)formula of a transitive theory

and

Col

be the

MofT

ϕ(a, b)

such that

imply

(a, b) ∈ IECT

realizationaof type

hM0,

Col0i |= Th(hM,

hM0,

p∞(x)

the following conditions hold:

-ordered coloring of a model

Coli)and

. Then for any(i. e., some

Col

hM0,

Col0i |=

)

1.2.

INESSENTIAL COMBINATIONS AND COLORINGS

(1)if|= ϕ(a, b) (2)if|= ϕ(a, b)

Proof.

By Proposition 1.2.3.2, without loss of generality,

then

|= p∞(b)

andasemi-isolatesb;

thenbdoes not semi-isolatea.

λ =

(1) Consider the following set of formulas:

¯ ¯ ¯ ¯

q(x) = {¬Colm(x) | m < ω } ∪(¬∃yÃϕ(x, y) ∧

is a nite set of natural numbers).

Coln(y)

n∈w

By the item (b) of the denition ofϕ-ordered coloring, that set is locally consistent (it suces to consider a formula

with a nite set

¬∃yÃϕ(x, y) ∧

w = {i1, . . . , ik}

Coln(y)

n∈w

, a nite set of formulas

¬Colj(x),j = j1, . . . , jm, and to take, for a realization, an ele-

ment of a color compactness the set

-inessential, the theory

-type

p∞(x)

plies the validity of the inclusion that, by the denition of cannot have realizationsbwith some condition So

|= ϕ(a, b)

k < ω

, greater than all

q(x)

is consistent. Since the coloring

Th(hM, Coli)

i1, . . . , ik, j1, . . . , jm). By

has a unique nonprincipal

, and the consistency of that type with the set

q(x)

, the formula

q(x) ⊂ p∞(x)

. Finally we notice

ϕ(a, y)

, where

Col(b) = n,n < ω

implies

|= p∞(b),ϕ(a, y) ` p∞(y)

, and thusasemi-

Col

q(x)

im-

|= p∞(a)

isolatesb.

(2) At rst we show that the formula

ϕ(x, y)

is not

p∞-preserving

with respect to the rst coordinate. To do so, we consider, for an arbitrary the structure

m < ω

hM,

, the following set of formulas in the language of

Coli:

, .

rm(y)  {¬Coln(y) | n < ω} ∪ {∃x(Colm(x) ∧ ϕ(x,y))}.

Since the set

rm(y)

is consistent with the type

p∞(y)

by the item

(a) of the denition ofϕ-ordering, and theϕ-ordered coloring

Col

Chapter

1. CHARACTERIZATION OF EHRENFEUCHTNESS

is1-inessential, we obtain an inclusion that for any realizationbof and for any element

m < ω,N |= ϕ(am, b) ∧ Colm(am)

aminN. Hence,

p∞(y)

ϕ(x, y)

is not

rm(y) ⊂ p∞(y)

in a modelNof

. It means

Th(hM,

holds for some

p∞-preserving with respect

Col

to the rst coordinate.

Assume now to the contrary, that semi-isolatesa. The condition that the formula preserving with respect to the rst coordinate implies that

|= p∞(a),|= ϕ(a, b)

ϕ(x, y)

is not

, and

p∞-

ϕ(x, b)

cannot witness thatbsemi-isolatesa. On the other hand, by the assumption, there is a formula

ψ(x, b) ` p∞(x) ψ(x, y)}

is consistent. By compactness and since

principal type,

. In this instance the set

p∞(x) ∪ p∞(y) ∪ {ϕ(x, y) ∧ ¬ψ(x, y)}

ψ(x, y)

such that

|= ψ(a, b)

and

p∞(x)∪ p∞(y) ∪ {ϕ(x, y)∧

p∞(x)

is a non-

is likewise.

Hence the set

{¬

Colm(x) ∧ ¬

Colm(y) | m < ω} ∪ {ϕ(x, y )}

does not semi-isolate a complete type. The latter conicts with the fact that relation

(a, b) ∈ IECT

|= ϕ(a, b)

and

ϕ(x, y)

|= p∞(a)

is a principal formula inT, and with the

Col

for any

hM0,

imply thatbdoes not semi-isolatea.

(a, b)

with

|= ϕ(a, b)

. Thus,

Note that the conclusion of Proposition 1.2.3.3 is true if we assume that

ϕ(x, y)

is a disjunction of principal formulas.

Proposition 1.2.3.3, being a sucient condition for non-sym- metry of semi-isolation, will be several times used for further con- structions. Below, in this Section, we introduce an example, real- izing properties that described in this Proposition.

Recall several notions in Graph Theory.

1.2.3.4. Denition.

A structure

Γ = hX ; Qi

with a (non-sym- metric, symmetric) binary relationQis said to be a spectively a

undirected graph

more, the setXis said to be a set of a set of of vertices in a graphΓ, such that

n − 1

, is said to be a

also said to be a

is the

directed graph

or, in abbreviated form, a

or, in abbreviated form, an

vertices

arcs

od graphΓ. Any nonempty sequence

route

(a0, an)

lengthofS

, or a

-route

or a

path

(a0, an)

undigraph

and the relationQis

S = (a0, . . . , an)

Γ |= Q(ai, a

i+1

, inΓ. Here, the routeSis

-path

, and the number

graph

digraph

(re-

, an

). Further-

),i = 0,. . . ,

1.2.

INESSENTIAL COMBINATIONS AND COLORINGS

By the denition, any path of length0has unique vertex. A

cycle

length. If digraph does not have cycles it is said to be graph

a, b ∈ X (a, a)

-path such that there are no arcs with

in a digraphΓis an arbitrary

hX; Qi

is said to be

are connected by a

(a0, . . . , an)

of nonzero length in a undirected graphΓ,

connected

(a, b)

(ai, a

, aj)

for

j+1

ai6= a

, is said to be a

i+1

(a, a)

-route of nonzero

acyclic

if any two distinct vertices

-path in graph

)

that are repeated or coincide

i+1

hX; Q ∪Q−1i

cycleinΓ

. A undigraph

. A

. Any

ΓisacyclicΓdoes not have cycles.

Recall that any maximal, with respect to inclusion, connected

subgraph of graph

Γ = hX; Qi

is said to be a

connected component

ofΓ. Each connected componentCofΓis uniquely dened by every its element

a ∈ C

and is denoted by

C(a, Γ)

, or by

C(a, Q)

the universeXis given.

For a graph

relations

Qn,

Q−n (Qn)−1,

Γ = hX; Qi

, we dene inductively the following

n ∈ Z:Q0 idX,

n ∈ ω

Q1 Q,Q

n+1

 Qn◦ Q

The following example, constructed on the base of the free di- rected pseudoplane, found independently by A. Pillay [345] and by the author [393], shows:

(1) the non-symmetry of relation

-stable theories,

SIpis realized in the class of

(2) distinct elementary chains over a same type may gener- ate non-isomorphic limit models forming continuum many pairwise non-isomorphic limit models.

Notice that any formula witnessing that

SIpis non-symmetric

denes a binary relation and, hence, a graph, whose vertices are colored by neighbourhoods ofp. Thus, constructing an example with non-symmetric tain a2n-ary relationQandn-ary relation

n = 1

there

and such that

is a pair

y the formula

SIp(in particular, an example below) we ob-

Pi,

i ∈ ω

a, b) ∈ Q

(

a, y)

x) | i ∈ ω}

{Pi(

andbdo

realizations ofp, for which

isolates

the type

es not semi-isolate

, reduced to

and

isolates

These cir- cumstances explain the language of the following example as well as languages of many further examples.

Chapter

1. CHARACTERIZATION OF EHRENFEUCHTNESS

1.2.3.5. Example.

graph

M0= hM0; Qi

Consider a countable connected acyclic di-

with acyclic undigraph

hM0; Q ∪ Q−1i

such that every element has innitely many images and innitely many preimages4. The structure

plane

Extend the language by new binary predicates

M0is said to be a

free directed pseudo-

Q0and

Q1, form-

ing a partition of the predicateQwith the following condition: for any element nitely many preimages with respect to a1-inessentialQ-ordered coloring Col:

a ∈ M0there exist innitely many images and in-

Q0and to M0→ ω ∪ {∞}

Q1. Dene then

of the re-

sulting structure so that every element of colornhas the following:

(a) innitely many images of colorµrelative to

for any

µ ≥ n

(including∞);

(b) innitely many preimages of color

Q1for any

The theory

m ≤ n

T0 Th(hhM0; Q, Q0, Q1i,

Coli)is axiomatized by

Q0and to

relative to

Q0and

the following axioms:

(1)

∃≥ωxColn(x),n ∈ ω

(2)

¬∃x (Colm(x) ∧ Coln(x)),m 6= n

(3)

∀x, y (Q(x, y) ↔ (Q0(x, y) ∨ Q1(x, y)))

;

(4)

¬∃x, y(Q0(x, y) ∧ Q1(x, y))

(5)

∀x∃≥ωyQi(x, y),i = 0,1

(6)

∀x∃≥ωyQi(y, x),i = 0,1

(7)

∀x, y(Q(x, y) → ¬Q(y, x))

(8)

∀x¡Colm(x) → ∃≥ωy (Qi(x, y) ∧ Coln(y))¢,

n < ω

m < ω

by the author [393] for a realization of non-symmetric relation of semi-isolation in the class of stable theories.

;

(9)

∀x¡Colm(x) → ∃≥ωy (Qi(y, x) ∧ Coln(y))¢,

;

(10)

¬∃x, y (Colm(x) ∧ Coln(y) ∧ Q(y, x)),m < n < ω

The

structure

M0has constructed independently by A. Pillay [345] and

; ; ;

;

i ∈ {0, 1},m ≤

i ∈ {0, 1},n ≤

;

1.2.

INESSENTIAL COMBINATIONS AND COLORINGS

(11)

ÃÃ

∀x, z, y0, . . . , y

Q(x, z) ∧ (y0≈ x) ∧ (yn≈ z) ∧

n−1

→

Note that the theory

((yi≈ x) ∧ (y

i=0

T0is∆-based, where∆is the least closed,

i+1

n−1

i=0

≈ z))

Q±1(yi, y

i+1

)!→

relative to substitutions of variables, set of formulas with at most two free variables such that this set contains formulas

n ∈ ω,(x ≈ y)

then

∃z (ϕ(x, z) ∧ Q

i, δ2∈ {0, 1};Q Coln(z),Col

and satises the following condition: if

(z, y) ∧ Col

(x, y) = Qi(x, y),Q

(z) = ¬Coln(z)

(y)) ∈ ∆

−1

(x, y) = Qi(y, x),Col

, where

Coln(x)

ϕ(x, y) ∈ ∆

δ1∈ {−1, 1}

(z) =

Indeed, by Lemma 1.2.1.7 it suces to show that for each tuple

a model of

a)is∆

tp(

(i)

Col(a)

-based.

is connected withaby an automorphism, i. e.,

isolated by set of formulas

n ∈ ω

;

(ii) if

con

then, up to a permutation,

a2suc C

tp(

h that

a2do

and

a1) ∪ tp(a2) ∪ Φ

that elementsbin

(Q ∪ Q−1)

a2)

tp(

-arcs; since by induction hypothesis the types

are∆-based,

(iii) if all elements of

T0, with pairwise distinct coordinates, the type

It is true by induction on length

a)ofa

a singletonathen any elementbsuch that

Col(b) =

tp(a)

Col

(x)

with

|= Col

(a),δn∈ {0, 1}

tains elements in at least two connected components

a = a1ˆa2for

a1consists

of elements in a connected component

es not contain elements inC;

whereΦis the set of formulas in∆\saying"

a1andcin

we get that

a2are

elong to a connected component

not linked by a sequence of

a)is∆

tp(

nonempty tuples

tp(

-based

too;

isolated by

tp(

a1)

and

a1,

then, by the acyclicity of the graph, there is a verticesbandcin

suc

h thatcseparatesbfrom sequentially incident arcs linking the separability property ofband

set

Ψ(x,

formulas in∆describing distances (i. e., lengths of

a1 a \

{b}

a1withbcon

a1b

yccan be expressed by a

, i. e., all tuples of

tain the vertexc;

;

Chapter

1. CHARACTERIZATION OF EHRENFEUCHTNESS

shortest if

|= Ψ(b0,

mapping

(Q ∪ Q−1)

a1)

and

b0to

b00;

-paths) betweenband elements in

thus,

Ψ(b00,

tp(b/

a1)

then

there is an

isolated by

Lemma 1.2.1.5 and the induction hypothesis for

a)is∆

tp(

The∆-basedness for a model countable set

-based. of

T0implies that

T0isω-stabile. Indeed,

M |= T0there are countably many1-types over a

A ⊂ M

, since there are countably many possibilities

Ψ(x,

a1;

moreover,

a1-automorphism

a1)

and using

a1)

tp(

e get that

for distance distributions from elements ofAto elements inM, intermediate with respect toAor separated by an element inA.

AsTisω-stable there exists a prime model

ization of type

p∞(x)

. By Proposition 1.2.3.3, the relation

is non-symmetric and it is witnessed by the formulas

Q1(x, y)

limit models over ductively elementary chains take for of

as where model

We argue to show that there exist

p∞. For this goal to be met, we construct in-

α¹n)n∈ω

p∞. If models

is a prime model over a realization

α¹n

α¹n+1

n∈ω

The model

an arbitrary prime model over a realization

α¹0

, . . . , M

α¹0

α¹n

we take a prime model over a realization

α¹n

≺ M

α¹n

α¹n+1

Mα.

and

|= Q

Mαis dened (as a union of elementary chain

2ωpairwise non-isomorphic

α ∈ 2ω, over

are already constructed, and

α¹n+1

α(n)

of models, each of which is prime over unique realization of sequence by predicate symbol of the elements, over which the models in

Sequencesαandβin

k, m ∈ ω

models

Sαof arcs on realizations of

. In this case endpoints of arcs consist

α(n)

2ωare said to be

such that

Mαand

α(k + n) = β(m + n)

Mβare isomorphic if and only ifαandβare

p∞, wheren-th arc is labelled

Cαare prime.

for all

of type

α¹n

, a

)

α¹n

equivalent

n ∈ ω

over a real-

∞

Q0(x, y)

and

p∞. We

p∞, then

α¹n+1

p∞,

. Denote the

p∞) by

if there exist

. Note that

α¹0

∞

equivalent.

Indeed, having the equivalence (i. e., when there exist such that phism of models (on the set of realizations of sequence of arcs with the labels

α(k + n) = β(m + n)

Mαand

Mβ, transforming the sequence of arcs

p∞) with the labels

for all

β(m+n)

n ∈ ω

), a partial isomor-

, can be extended to an

α(k+n)

k, m ∈ ω

on the

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