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INTR
ODUCTION AND HISTORICAL SURVEY
21
properties of
groupoids
P
to the class of partial groupoids being joins of
ν(p)
P
. In nal Section 3.9, we produce a list of properties
ν(p)
characterizing the class of partial groupoids corresponding to al-
gebras of distributions for binary isolating formulas on a family of
types. In sections 3.10{3.17, results on isolating formulas are gener-
alized for semi-isolating formulas. In Sections 3.18{3.20, properties
of special structures, related to categorical, strongly minimal, and
Ehrenfeucht theories, are investigated. In Section 3.21 we propose
a description for deterministic algebras of isolating formulas for
theories of acyclic graphs.
The present book is written on a base of the book \The Lachlan
problem" [51] and includes all its main results.
Originally it was planned to write one part, which included
would contain both the present part and the second part, the
material which develops the presentation of the monograph \The
Lachlan problem". As the text has turned out to be volume and
branching enough, it became clear that it is desirable to divide
the book in two parts. This was done at the suggestion of the
Publishing House of NSTU.
Outside the monograph there are results on some main direc-
tions of studying derived objects in the model theory:
•
Objects related to classications of models of a the-
ory: B. Sh. Kulpeshov, S. V. Sudoplatov [271, 272, 273, 274];
S. V. Sudoplatov, P. Tanovic [426]; S. V. Sudoplatov [444];
•
Generic constructions and generic limits: Y. Kiouvrekis,
P. Stefaneas, S. V. Sudoplatov [259, 441]; S. V. Sudoplatov [427,
428, 429, 442, 443, 445];
•
Algebras of distributions of binary isolating formulas:
K. A. Baikalova, D. Yu. Emel'yanov, B. Sh. Kulpeshov, E. A. Pa-
lyutin, S. V. Sudoplatov [79]; D. Yu. Emel'yanov, B. Sh. Kulpeshov,
S. V. Sudoplatov [160, 162]; D. Yu. Emel'yanov, S. V. Sudoplatov
[161]; B. Sh. Kulpeshov, S. V. Sudoplatov [270]; E. V. Ovchin-
nikova, S. V. Sudoplatov [328];
•
Hypergraphs of models of a theory: B. Sh. Kulpeshov,
S. V. Sudoplatov [275, 276, 277]; S. V. Sudoplatov [433];
•
Operators on families of theories and their topological prop-
erties: In. I. Pavlyuk, S. V. Sudoplatov [333]; S. V. Sudoplatov
[430, 431, 432, 434, 435, 436].
22
INTR
ODUCTION AND HISTORICAL SURVEY
Considered in the book, classes of limit models, applied to nat-
ural classes of theories such as theories of graphs with uniform -
nite separability (in particular, theories of unars, of polygons over a
group, and of ultraat graphs), of ordered and non-ordered abelian
groups, as well as of locally free algebras, are investigated in works
by K. A. Baikalova [74]{[78]. She showed that for small theories of
structures with one unary functions, of graphs with uniform nite
separability, for small theories of locally free algebras, and for a
series of theories of abelian groups there are only0,1,ω, or
limit models. Besides there are
2ωprime over tuples and
2
2ωlimit
models for theories of locally free algebras with continuum many
types. R. A. Popkov [352, 353, 355, 356] obtained a classication
of countable models for theories of unary predicates, investigated
algebras of distributions of binary isolating formulas for the the-
ory of additive group of integers, and found the numbers of special
countable models of this theory.
Considered in the book concrete theories are constructed using
the following ways:
(1) taking a theory of given structure;
(2) explicitly axiomatizing required theory;
(3) applying semantic or syntactic approach (described in Chap-
ter 2), where a generative class is explicitly dened, an amalgama-
tion is shown and desirable properties based on generic construc-
tion are proven;
(4) applying an operator approach, based on (3) and allowing
to get required structure and theory varying arguments (Section
7.6).
ω
Below, we use without specications:
•
model-theoretical notations and terminology in Handbook of
Mathematical Logic [23] and in the book by S. V. Sudoplatov,
E. V. Ovchinnikova [53] (see also J. T. Baldwin [2, 3]; Yu. L. Er-
shov [13]; Yu. L. Ershov, S. S. Goncharov [14]; Yu. L. Ershov,
E. A. Palyutin [15]; C. C. Chang, H. J. Keisler [8]; W. Hodges
[28]; D. Lascar [32]; A. Pillay [39]; B. P. Poizat [43]; G. E. Sacks
[45]; S. Shelah [47]; K. Tent, M. Ziegler; F. O. Wagner [59]);
INTR
ODUCTION AND HISTORICAL SURVEY
•
Graph Theory terminology in the book by S. V. Sudoplatov,
23
E. V. Ovchinnikova [52] (see also F. Harary [24]; [33]; O. Ore [37],
A. A. Zykov [61]);).
•
the terminology of General Algebra in the books [19, 20]
(see also E. S. Lyapin [34]; A. H. Cliord, G. B. Preston [10, 11]).
As usual,is an equals-by-denition symbol and¤is an end-
of-proof mark.
Chapter
1
CHARACTERIZATION
OF EHRENFEUCHTNESS.
PROPERTIES OF EHRENFEUCHT
THEORIES
1.1. Syntactic characterization of the class
of complete theories with nitely many
countable models
In this Section, a syntactic characterization is furnished for the
class of elementary complete theories with nitely many countable
models, which is an analog of known theorem by C. Ryll-Nard-
zewski [368], that the countable categoricity of a theory is equiv-
alent to a niteness of number ofn-types of the theory for every
naturalnand xed set of free variables. Establishing charac-
terization is based on classifying the theories by Rudin{Keisler
preorders and distribution functions of a number of limit
models over types.
Throughout this book we will denote innite structures (i. e.,
models of elementary theories) by
and we use corresponding Latin letters
universes. The type of a tuple
denoted by
A = ∅
and by
ofTover the empty set is denoted by
tpM(a/A)
, we write
Sn(∅)
the set ofn-types of a theoryT. The set of all types
tp(
or by
a)
instead
M, N , . . .
ainM
tp(a/A)
of
tp(
, possibly with indexes;
M, N, . . .
o
ver a set
if the structure is given. If
a/∅)
.
S(T)
and by
to denote their
A ⊆ M
We denote by
S(∅)
.
will be
Sn(T )
1.1.
SYNTACTIC CHARACTERIZATION OF THEORIES
a
andbare
If
aˆ b
b
a, b)
y
(
a
b
the
y
A
F
or a theoryT, we denote by
union ofAwith the set of coordinates of the tuple
tuples of elements we shall often denote the tuple
andbyab
.
IfAis a set then we denote by
I(T, λ)
non-isomorphic models ofTin a powerλ.
25
a
A ∪
and
a
.
the number of pairwise
1.1.0.1. Denition
if
1 < I(T, ω) < ω.
(T. Millar [306]). A theoryTis
Ehrenfeucht
Unless specied otherwise, we deal with complete theories. Ad-
ditionally in this Section, all considered theories are countable.
1.1.1. Powerful types. Relations of semi-isolation.
(p, q)
-Preserving formulas. Quasi-neighbourhoods
1.1.1.1. Denition
to be
powerful
in a theoryTif every model
realizes every type
Since for any type
(M. Benda [125]). A type
q ∈ S(T )
, that is,
p ∈ S(T )
ofT, realizingp, and the model
MofT
M |= S(T )
there exists a countable model
M
realizes exactly countably many
x) ∈ S(T )
p(
realizingpalso
.
types, the availability of a powerful type implies thatTis
that is, the set
and its realization
i.
e., a model ofTcontaining
a)
M(
q
is
elementarily embeddable to any model realizing the type
. Since all prime models over realizations ofqare isomorphic, we
denote these models by
orq-prime
The condition that
type in
in
S(T)
S(T)ofω
S(T)
is countable. Hence for any type
a
,
there exists a
a
Mq. Models
prime model
with
M(a) |= q(a)
Mqare called
.
x)
is
p(
is realized in
a powerful type means that every
Mp, that is,
-categorical theoryTis powerful.
Mp|= S(T )
M(
and
almost prime
. Every type
is
said
small
q ∈ S(T )
a)
overa,
such that
M
,
1.1.1.2. Lemma
(M. Benda [125]).
has a powerful type.
Proof.
there
omits
x, y)
p1(
Assume on the contrary that for any type
exists a type
y)
.
rq(
a
type in
Denote by
y) ∈ S(∅)
rq(
S(∅)
x)
p0(
containing the type
suc
an
arbitrary type in
we construct by induction a sequence
Every Ehrenfeucht theory
x) ∈ S(∅)
q(
h that a model
x) ∪ r
p0(
pn∈ S(∅),n ∈ ω
Mq(ofT)
S(∅)
p
0
and by
(y)
.
, such that
Now
T
,
26
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
pn⊂ p
pn⊂ pmfor
Then
contradiction.
As an illustration, we consider the following
amples
Peretyat'kin example
and
n+1
M
p
n
n < m
M
6' M
p
n
p
m
¤
[465] of theories
omits
p
n+1
and hence
for
n 6= m
Tn,
n ∈ ω
[335] of theory
. By the construction, we have that
M
omits
p
n
. Thus,
pmif and only if
I(T, ω) ≥ ω
n < m
and we get a
Ehrenfeucht ex-
, with
I(Tn, ω) = n ≥ 3
, and
T0with three countable mod-
els.
1.1.1.3. Example.
formed from the structure
k ∈ ω
, such that
lim
k→∞
Let
Tnbe the theory of a structure
hQ; <i
ck= ∞
by adding constants
, and unary predicates
ck,
ck< c
P0, . . . , P
Mn,
k+1
n−3
which form a partition of the setQof rationals, with
|= ∀x, y ((x < y) → ∃z((x < z) ∧(z < y)∧Pi(z))), i = 0,. . . , n−3.
The theory
Tnhas exactlynpairwise non-isomorphic countable
models:
(a) a prime model
(b) prime models
S1(∅)
, isolated by sets of formulas
i = 0, . . . , n − 3(lim
(c) a saturated model
k→∞
Mn(
M
lim
ck= ∞
k→∞
n
over realizations of powerful types
i
);
{ck< x | k ∈ ω} ∪ {Pi(x)}
ck∈ Pi);
Mn(the
limit
lim
ckis irrational).
k→∞
pi(x) ∈
.
,
,
1.1.1.4. Example.
Let
M = hM; ≤i
be a lower semilattice with-
out least and greatest elements such that:
(a) for each pair of incomparable elements, their join does not
exist;
(b) for each pair of distinct comparable elements, there is an
element between them;
(c) for each elementathere exist innitely many pairwise in-
comparable elements greater thana, whose inmum is equal
toa.
Expand the structure
cn< c
n+1
,
n ∈ ω
. The theory
M
by constants
T0of this expansion has exactly
cn,
n ∈ ω
, such that
three countable models: the prime model; the saturated model;
the prime model over the realization of the powerful type
isolated by the set
{cn< x | n ∈ ω}
of formulas.
p∞(x)
,
1.1.
SYNTACTIC CHARACTERIZATION OF THEORIES
27
Recall that`means proof from no hypotheses deducing
` ϕ
for a formulaϕof a vocabularyΣ. If deducingϕ, hypotheses in
a setΦof formulas can be used, we write
Ψ
of formulas,
Φ ` Ψ
means that each formula inΨis forced by
Φ ` ϕ
. For setsΦand
some nite set of formulas inΦ.
1.1.1.5. Denition(A. Pillay [339, 341]). Let
a
a theoryT,
semi-isolates
a, y) ∈ tp(b/Aa)
ϕ(
we say that the formula
b
that
is
Similarly, a tuple
formula
is
a principal (i. e., isolating) formula over
ϕ(
with names for
(with
r
parameters inA)
espect toA.
x, y)
If
ϕ(
toA, we also say that
a
over
b
;
∅
with
a
(semi-)isolatesbo
If
and if a formula
then we say that
andbtuples
for
b
which
the
tuple
ϕ(
semi-isolated over
a
isolates
a, y) ∈ tp(b/Aa)
a
.
A
In this case we say that the formula
witnesses
that
ϕ(
respect toA.
ver∅, we simply say that
a, y)
ϕ(
a, y)
ϕ(
inM,Aa subset ofM. The tuple
o
ver the setAif there exists a formula
a, y) ` tp(b/
ϕ(
x, y)
a
for
(with
with
a
which
parameters inA)
respect toA.
b
tuple
o
a, y) ` tp(b/
ϕ(
A
b
witnesses that
b
is
(semi-)isolated over
a, y)
witnesses
witnesses
witnesses
is
that
a
that
a(semi-)isolatesb.
that
M
be a model of
A)
holds. In this case
witnesses
verAif there exists a
A)
and
a
,
that is, in a language
isolated over
a
with
bis(
semi-)isolate
a, y)
ϕ(
x, y)
ϕ(
a
with
respect
a(semi-)isolates
(semi-)isolatesbo
ver
a
d
a
Note
that if
a, y)
and
ϕ(
b = b1b2thena(semi-)isolates
means of the formulas
(semi-)isolatesbo
y2ϕ(a, y1, y2)
∃
verAby means of a formula
b1and
and
∃y1ϕ(a, y1, y2)
b2o
verAby
resp
ec-
tively.
The following notion proposed by B. S. Baizhanov general-
izes the notion ofp-stability, that introduced in [84] (see also
B. S. Baizhanov, B. Sh. Kulpeshov [86]).
x)
and
1.1.1.6. Denition.
Let
p p(
be incomplete) types over a set
x, y)
T
. A formula
(p, q)
-semi-isolating
ϕ(
for any realization
with
parameters in
, a
(p → q)
aofp,ϕ(a, y) ` q(y)
A ⊆ M
-formula
q q(y)
in a model
Ais(p, q)
, or a
holds.
b
e some (may
M
(q ← p)
A formula
of a theory
-preserving
-formula
x, y)
ϕ(
,
if
28
is a
(p ↔ q)
(p ← q)
p
-preserving
1.1.1.7. Lemma.
(p, q)
ters inA)
p
arameters inA)
Chapter
-formula. If
1. CHARACTERIZATION OF EHRENFEUCHTNESS
x, y)
-formulaifϕ(
p = q
or a
(p → p)
A formula
then a
-formula
is
(p, q)
.
ϕ(
-preserving if and only if for any formula
satisfying
such that
y) ` θ(y)
q(
ther
x) ` θ0(x)
p(
both a
(p → q)
-formula and a
-preserving formula is called
x, y)
with
parameters in
y)(with
θ(
e exists a formula
parame-
x)(with
θ0(
and
A
is
M
Pr
oof.
Suppose that a formula
Consider an arbitrary formula
that
parameters
y) ` θ(y )
q(
inA) such that
M
b
y compactness we obtain a realization
y (ϕ(a, y) ∧
∃
¬θ(
vation for the formula
No
w, suppose that for any formula
there
exists a formula
M
but
the formula
a
realization
formula
y) ` θ(y)
q(
M |= ∃
assumption.
ofp,
a, y)
ϕ(
.
Then for any formula
x, y (θ0(x) ∧ ϕ(x,y) ∧
¤
x, y ((θ0(x) ∧ ϕ(x,y)) → θ(y))) .
|= ∀
y)
(with
θ(
.
Assuming that there are no formulas
x) ` θ0(x)
p(
x, y ((θ0(x) ∧ ϕ(x,y)) → θ(y))) ,
|= ∀
y)) .
It
contradicts the condition of
x, y)
x)
.
suc
h that
is
not
(p, q)
holds.
ϕ(
θ0(
x, y ((θ0(x) ∧ ϕ(x,y)) → θ(y)))
|= ∀
x, y)
ϕ(
ϕ(a, y) 6` q(y)
is
consistent with some formula
θ0(
y)) ,
¬θ(
x, y)is(p,
ϕ(
q)
-preserving.
parameters inA) such
x)
(with
θ0(
and
aofp
y)
θ(
x) ` θ0(x)
p(
suc
suc
h that
and
h that
(p, q)
y) ` θ(y)
q(
M |=
-preser-
-preserving, i. e., for some
This means that the
y)
¬θ(
x)
with
p(x) ` θ0(x)
whic
h is impossible by the
suc
w
h that
e have
,
The following statement is obvious.
1.1.1.8. Lemma.
b
if
and only if
1.1.1.9.
Lemma
x, y)is(tp(a), tp(b))
ϕ(
A formula
ϕ(
(A. Pillay [341]).
witnesses
-pr
eserving and
The relation of semi-isolation
that
a
semi-isolates
M |= ϕ(
a, b)
a, y)
with respect toAforms a preorder(i. e., a reexive and transitive
relation)on the set of tuples in the modelM.
.
1.1.
SYNTACTIC CHARACTERIZATION OF THEORIES
29
Proof.
Notice that any tuple
itself with respect to
n
V
(
(xi≈ yi)
i=1
is a
(tp(
A
by means of the formula
a) ↔ tp(a))
a = ha1,
. . . , ani
semi-isolates
V
i=1
-form
ula), i. e., the relation of
n
semi-isolation is reexive.
a, b
SI
,
b
M
p
Now, suppose that for tuples
(with
a
isolated over
preserving
resp
parameters inA) witnesses that
b, z)
formula
ψ(
(with
b
.
The formula
parameters inA) witnesses that
y (ϕ(x, y) ∧ ψ(y,z))is(tp(a), tp(c))
∃
and, moreover, it witnesses that
ect toA. Thus, the relation of semi-isolation is transitive.
If
p ∈ S(T )
and
M |= T
then
c
,
and
is
semi-isolated over
a formula
a
semi-isolatescwith
denotes the relation of
semi-isolation (over∅) on the set of all realizations ofp:
M
SI
p
Similarly
a, b) |
{(
M |= p(
, we denote by
a) ∧ p(b)
M
I
the relation of isolation (over∅)
p
andasemi-isolates
on the set of all realizations ofp:
M
I
p
{(
a, b) |
M |= p(
a) ∧ p(b)
andaisolates
(ai≈ yi)
a, y)
ϕ(
a
and
c
is
semi-
¤
b}.
b}.
-
Belo
model
w we omit the top index
M
is xed or it is a monster-model, i. e., a model containing
M
in notations
SI
M
p
and
M
I
if the
p
all considered models of a theory as elementary submodels.
Repeating arguments on the preorder of semi-isolation for the
set of realizations of typep, we obtain that the relation
preorder. The preorder
SIpis called the
preorder of semi-isolation
SIpis also a
on the set of realizations of typep.
At the same time, contrasting to the semi-isolation, it is easy
to construct an example of a theory with a non-transitive relation
Ip,1i. e., generally speaking, the isolation can be not preserved
under two-step transitions by the relation of isolation.
1
Belo
w we shall consider these examples several times.
30
Chapter
1. CHARACTERIZATION OF EHRENFEUCHTNESS
1.1.1.10. Lemma
powerful type having a realization in a model
relation
SIpon the set of realization ofpin
Moreover, there exist realizations
b/a)
type
tp(
Pr
oof.
x, y) p(x)∪p(y)∪{¬ϕ(x, y) | ϕ(x, y)
q(
is
At rst we consider the set
(A. Pillay [341]).Ifp ∈ S(T )
a
andbofpin
b
principal and
do
es not semi-isolate
is a nonprincipal
M
ofT, then the
M
is non-symmetric.
M
such
that the
a
.
is
ap-preserving formula
and show that it is consistent. Since any disjunction ofp-preserving
formulas isp-preserving as well, by compactness it suces to prove
that any formula
x, y)
ϕ(
is
ap-preserving formula, and
θ(
y) ∧
of that formula follows from
non-principality ofpimplies the existence of a tuple
b)
and
that
M 6|= p(
θ(
y) ∧
Since
a, y)inM
¬ϕ(
the set
plete type
x, y)
a)
is
realized by some pair
for
some realization
r(
M(
semi-isolate
semi-isolates
x, y) ∈ S(T )
r(
a
,
c
M
.
x, y)
q(
since otherwise, by transitivity of semi-isolation,
in
spite of denition ofr.
a, y)
¬ϕ(
is
consistent, where
a, y) ` p(y)
ϕ(
M |= p(
and
a)
.
The consistency
the fact that the
y) ∈ p(y)
θ(
b ∈ M
b)
|= θ(
is
.
That tuple
consistent, it can be extended to a com-
.
As the typepis powerful, the type
b, c)
in
(
aofp
.
Then
b
realizes
the model
a, b) ∈ Ipbutbcannot
(
the formula
Mp, which is
¤
suc
}
,
h
b
Thus, the availability of a nonprincipal powerful type
sumes
any (some) realization
the existence of a formula
aofp
(1)
(2)
a, y) ` p(y)
ϕ(
x, a) 6` p(x)
ϕ(
;
,
and, moreover, there exists a tuple
,
realizes typepand is such that
b
isolate
called a
.
x, y)
Ev
ery formula
ϕ(
,
satisfying the conditions 1 and 2, is
formula witnessing that the relation
The formula
(x < y)
witnesses that the relations
x, y),l(x)
ϕ(
= l(
the following conditions hold:
b, a)
|= ϕ(
andado
SIpis non-symmetric
symmetric in the Ehrenfeucht examples.
p(
y)
,
such that for
es not semi-
SI
p
i
x)
pre-
b
whic
are non-
h
.
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