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Файл:Classification of countable models of complete theories. Р.1. Monograph in two parts
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S. V. SUDOPLATOV
CLASSIFICATION
OF COUNTABLE MODELS
OF COMPLETE THEORIES
Рart 1
Monograph
NOVOSIBIRSK
2018
УДК 510.67
С892
Reviewers:
Member Corresponding of National Academy of Sciences at Republic
of Kazakhstan, Professor B. S. Baizhanov, D.Sc. (Phys. & Math.),
Professor E. A. Palyutin, D.Sc. (Phys. & Math.),
Professor A. G. Pinus, D.Sc. (Phys. & Math.)
Sudoplatov S. V.
С892 Classification of countable models of complete theories: monograph in two
parts / S. V. Sudoplatov. – Novosibirsk : NSTU Publisher, 2018.
ISBN 978-5-7782-3526-7
Рart 1. – 326 p.
ISBN 978-5-7782-3527-4
The book is the first part of the monograph “Classification of countable models
of complete theories” consisting of two parts. In the monograph, a classification of
countable models of complete theories with respect to two basic characteristics
(Rudin–Keisler preorders and distribution functions for numbers of limit models) is
presented and applied to the most important classes of countable theories such as the
class of Ehrenfeucht theories (i. e., complete first-order theories with finitely many
but more than one pairwise non-isomorphic countable models), the class of small
theories (i. e., complete first-order theories with countably many types), and the
class of countable first-order theories with continuum many types. For realizations
of basic characteristics of countable complete theories, syntactic generic
constructions, generalizing the Jonsson–Fraïssé construction and the Hrushovski
construction, are presented. Using these constructions a solution of the Goncharov–
Millar problem (on the existence of Ehrenfeucht theories with countable models
which are not almost homogeneous) is described. Modifying the Hrushovski–
Herwig generic construction, a solution of the Lachlan problem on the existence of
stable Ehrenfeucht theories is shown. In the first part, a characterization of
Ehrenfeuchtness, properties of Ehrenfeucht theories, generic constructions, and
algebras for distributions of binary semi-isolating formulas of a complete theory are
considered.
The book is intended for specialists interested in Mathematical Logic.
УДК 510.67
ISBN 978-5-7782-3527-4 (Part 1)
ISBN 978-5-7782-3526-7
© Novosibirsk State Technical
© Sudoplatov S. V., 2018
University, 2018
Con
tents
Preface
Introduction and historical survey
Chapter 1.
Chapter 2.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Characterization of Ehrenfeuchtness. Prop-
erties of Ehrenfeucht theories
§
1.1. Syntactic characterization of the class of complete
theories with nitely many countable models . . 24
§
1.2. Inessential combinations and colorings of structures 53
§
1.3. Type reducibility, powerful types, and the strict
order property . . . . . . . . . . . . . . . . . . . 74
§
1.4. Powerful digraphs . . . . . . . . . . . . . . . . . 93
§
1.5. The Tsuboi and Kim theorems . . . . . . . . . . 105
Generic constructions
§
2.1. Semantic generic constructions . . . . . . . . . . 113
§
2.2. Syntactic generic constructions . . . . . . . . . . 114
§
2.3. Self-sucient classes . . . . . . . . . . . . . . . . 129
. . . . . . . . . . . . . 11
. . . . . . . . 24
. . . . . . . . . . . . . . 113
§
2.4. Genericity of countable homogeneous structures . 135
§
2.5. The uniformd-amalgamation property and satu-
rated generic structures . . . . . . . . . . . . . . 140
§
2.6. On the nite closure property in fusions of gener-
ative classes . . . . . . . . . . . . . . . . . . . . . 147
§
2.7. On generating elements in generic algebras . . . 155
§
2.8. On varieties of generative classes . . . . . . . . . 159
4
Chapter
§
§
§
§
§
§
§
§
3.
Algebras of distributions for binary semi-
isolating formulas of a complete theory
. . 163
3.1. Preliminary notions, notations, and properties . 163
3.2. Examples . . . . . . . . . . . . . . . . . . . . . . 169
3.3. Algebra of distributions for binary isolating for-
mulas on a set of realizations of a type . . . . . . 174
3.4. Characterization for transitivity of the relation
Deterministic, almost deterministic
I
-groupoids
ν(p)
Ip.
and elements . . . . . . . . . . . . . . . . . . . . 180
3.5. Graph and monoid compositions . . . . . . . . . 187
3.6.I-groupoids . . . . . . . . . . . . . . . . . . . . . 190
3.7. Groupoids of binary isolating formulas on sets of
realizations of types of special theories . . . . . . 196
3.8. Partial groupoid of binary isolating formulas on
a set of realizations for a family of1-types of a
complete theory . . . . . . . . . . . . . . . . . . 199
§
3.9.
IR-structures . . . . . . . . . . . . . . . . . . . . 203
§
3.10. Notions, notations, and properties . . . . . . . . 206
§
3.11. Preordered algebras of distributions of binary semi-
isolating formulas . . . . . . . . . . . . . . . . . . 212
§
3.12. Ranks and degrees of semi-isolation . . . . . . . 215
§
3.13. Monoid of distributions of binary semi-isolating
formulas on a set of realizations of a type . . . . 220
§
3.14.α-deterministic and almostα-deterministic
SI
ν(p)
-
monoids . . . . . . . . . . . . . . . . . . . . . . . 222
§
3.15.
POSTC
-monoids . . . . . . . . . . . . . . . . . . 228
§
3.16.
Partial
for a family of1-types of a complete theory . . . 232
§
3.17.
POSTCR-structures . . . . . . . . . . . . . . . . 237
§
3.18. Algebras of distributions of binary semi-isolating
formulas for families of isolated types and for count-
ably categorical theories . . . . . . . . . . . . . . 241
§
3.19. Forcing of innity and algebras of distributions of
binary semi-isolating formulas for strongly mini-
mal theories . . . . . . . . . . . . . . . . . . . . . 243
§
3.20. Absorbing structures . . . . . . . . . . . . . . . . 248
§
3.21. Structures of distributions of isolating formulas
as derivative structures: for acyclic graphs . . . . 252
POSTC
-monoid on a set of realizations
5
References
Index of names
Index of terms
Index of symbols
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
. . . . . . . . . . . . . . . . . . . . . . . . . 306
. . . . . . . . . . . . . . . . . . . . . . . . . . 311
. . . . . . . . . . . . . . . . . . . . . . . . 320
PREF
theories is one of the basic problems of modern model theory.
Model Theory, formed in an independent area in 1950th years,
is on a joint of Mathematical Logic and Algebra. Syntactic objects
(
theories
objects (algebraic systems, or structures, describing interrelations
of elements of real objects) are its subjects, as well as classica-
tions of syntactic objects by properties of semantic objects and
vice versa. Essentially distinct (non-isomorphic) realizations of
these theories by algebraic systems (their models) can correspond
to complete theories (i. e., theories without consistent information
in a xed language that can be added but not added). The num-
ber of that realizations may vary in distinct innite cardinalities
(i. e., with distinct innite number of elements) of algebraic sys-
tems. Thus, there are spectrum functions, mapping the number
of non-isomorphic models of given theory to cardinalities of that
models, and the problem of descriptions of all possible spectrum
functions for the class of all theories, and also for various essential
subclasses of this class.
(uncountable) cardinalities in the class of all theories. Here the
basic achievements are connected with works by S. Shelah [47] and
nally represented in the work by B. Hart, E. Hrushovski, and
M. S. Laskowski [196].
turned out much more dicult. Firstly, it is not known till now on
an existence of theories with uncountable and non-maximal num-
ber of countable models (the Vaught Problem). Secondly, con-
structed by A. Ehrenfeucht (see [465]) initial examples of theories
ACE
The problem of classication for countable models of complete
representing descriptions of real objects) and semantic
It is surprising that the Spectrum Problem is solved for large
For the countable (minimal innite) cardinality, the situation
PREF
ACE
with nitely many, but more than one, countable models (now such
theories, in his honour, are called
Ehrenfeucht
) had been in essence
unique for a long time: all modications had been reduced to su-
perstructures on innite dense linearly ordered sets. With this
circumstance, the Lachlan problem on the existence of essentially
other (i. e., not having innite linear orders) Ehrenfeucht theories
has been arised. In a brief formulation, the
Lachlan Problem
is:
to dene, whether there exists a stable Ehrenfeucht theory.
This problem has been partially solved by A. H. Lachlan [278]
who published in 1973 a proof of the absence of Ehrenfeucht theo-
ries in the class of superstable theories, that is an important sub-
class of the class of stable theories. It was supposed for a long
time, that the statement is true for stable theories, and it
was referred in the literature, alongside with the Lachlan
Problem, to the
Lachlan Conjecture
(see, for example, [59, p.
202]). The Lachlan Conjecture is proved to be true for many sub-
classes of the class of stable theories in the works by D. Lascar [283],
S. Shelah [47], A. Pillay [337, 341, 344, 346], T. G. Mustan [316],
J. Sae [370], A. Tsuboi [461], E. Hrushovski [222], A. A. Vikent'ev
[58], B. Kim [257], P. Tanovic [448, 449]. At the same time, struc-
tural properties of a counterexample, assuming that it exists, has
been accumulated. The following works are connected with that
accumulation: by M. G. Peretyat'kin [335, 336], M. Benda [125],
R. Woodrow [60, 475, 476], A. Pillay [338, 339], B. Omarov [324],
A. Tsuboi [460], S. S. Goncharov, M. Pourmahdian [187], B. Her-
wig [206] and by the author. A solution of the problem, namely
a proof of the existence of a stable Ehrenfeucht theory, became
possible only after an occurrence of subtle construction created
by E. Hrushovski [221] in 1988 and applied for solutions of many
model-theoretic problems. Now this known construction is called
the generic Hrushovski construction. It allows \to collect" required
structures, formed via classes of nite objects, using amalgams.
Another important component is the theory of group polygono-
metries created by the author [54, 397, 398] and generalizing clas-
sical trigonometries. The class of group polygonometries is a con-
venient and geometrically clear object that has allowed to real-
ize many structural properties of stable Ehrenfeucht theories. At
the same time, now, when the general mechanism of constructions
of Ehrenfeucht theories became clear, the explicit description of
7
8
PREF
ACE
implicitly presented polygonometrical apparatus is superuous in
the construction. Therefore, the Polygonometrical Theory and its
applications are not represented in the book.
For the construction of stable Ehrenfeucht theories, we have
involved a subtle modication of Hrushovski construction, oered
by B. Herwig [206] for a realization of a basic structural property |
innite weight. At the same time, this modication in an original
form has been insucient, since the Hrushovski{Herwig construc-
tion is semantic and does not take into consideration a possibility
of occurrence of external connections with respect to given nite
objects, being \bricks" of the generic construction.
For the elimination of this lack, the theory of syntactic generic
constructions [407] has been developed by the author. Syntactic
constructions are based on types (not on nite objects), i. e., on de-
scriptions (possible to be external) of nite objects, that then allow
to generate models for required theories step-by-step.
Using the aforesaid tools, we construct a wide family of sta-
ble Ehrenfeucht theories and to solve a series of related problems
including the Goncharov{Millar problem on the existence of Ehren-
feucht theories with countable models which are not almost homo-
geneous. Having a rich class of examples of theories, we developed
a classication of countable models of complete theories described
in the book.
My initial work passed during my study in Novosibirsk State
University, where the rst class specialists in Mathematical Logic
and Algebra worked and continue to work. An occurrence of the
Siberian School of Algebra and Logic, to which I regard me, be-
came possible after the foundation of the Institute of Mathematics
in Academgorodok, Novosibirsk in 1957 and of arrival to Novosi-
birsk the founder of the School, Academician Anatoliy Ivanovich
Mal'tsev. Now already more than forty years this School is headed
by Academician Yuriy Leonidovich Ershov. To the statement the
Problem and to successes in its solution, I am obliged in many re-
spects to my scientic advisor, the Head of the Laboratory of Alge-
braic Systems, Professor Evgeniy Andreevich Palyutin. I had a lot
useful and fruitful discussions with Director of Sobolev Institute
of Mathematics, Academician Sergey Savost'yanovich Goncharov,
with Professor of Chair of Algebra and Mathematical Logic of
Novosibirsk State Technical University Aleksandr Georgievich
PREF
ACE
Pinus, with participants of the seminar \Model Theory" in IM
SB RAS, Academician Asan Dabsovich Taimanov, prof. Vadim
Grigor'evich Puzarenko, doc. Elena Viktorovna Ovchinnikova, doc.
Mikhail Andreevich Rusaleev, doc. Aleksandr Nikolaevich Ryaskin,
doc. Aleksandr Aleksandrovich Vikent'ev, doc. Dmitriy Yur'ev-
ich Vlasov, doc. Roman Andreevich Popkov, the students Ilya
Vladimirovich Shulepov, Kristina Andreevna Baikalova, Dmitry
Yur'evich Emel'yanov, Yiannis Kiouvrekis with lecturers of Chair
of Algebra and Mathematical Logic at NSTU. During all my sci-
entic activity, I approved new results before publications at the
seminar \Model Theory", supervised by Academician Yuriy Leo-
nidovich Ershov and Professor Evgeniy Andreevich Palyutin.
I had useful direct and correspondence dialogues with many
model-theoretic specialists from Russia, France, Kazakhstan, USA,
Great Britain, Israel, Japan, Germany, Poland, Serbia, Czechia,
Greece: Professors Pavel Evgen'evich Alaev, Uri Andrews, Erzhan
Rahmetallaevich Baisalov, Bektur Sembiuly Baizhanov, John Bald-
win, Makhsut Iskanderovich Bekenov, Elisabeth Bouscaren, Oleg
Vil'gel'movich Belegradek, Enrique Casanovas, Ivan Chajda, Ca-
meron Hill, Ehud Hrushovski, Koichiro Ikeda, Alexander Ana-
tol'evich Ivanov, Nazif Garifullinovich Khisamiev, Byunghan Kim,
Julia Knight, Robin Knight, Beibut Shaiykovich Kulpeshov, Daniel
Lascar, Steen Lempp, Terry Millar, Andrey Sergeevich Morozov,
Tulendy Garifovich Mustan, Ludomir Newelski, Tursynbek Ak-
tasovich Nurmagambetov, Baibolat Omarov, Inessa Ivanovna
Pavlyuk, Anand Pillay, Bruno Poizat, Vladimir Nikanorovich Re-
meslennikov, Lynn Scow, Sergei Stepanovich Starchenko, Petros
Stefaneas, Charles Steinhorn, Alena Andreevna Stepanova, Pre-
drag Tanovic, Akito Tsuboi, Jamalbek Aliaskarovich Tus-
supov, Engeniy Vital'evich Vassiliev, Viktor Valerievich Ver-
bovskiy, Frank Wagner, Alex Wilkie, Robert Woodrow, Myrzakhan
Myrzakhmetovich Yerimbetov, Aibat Rafkhatovich Yeshkeyev,
Martin Ziegler, Boris Iosifovich Zilber,.
It has turned out, after my postgraduate study in NSU, since
1990, I work in NSTU already 27 years, and 25 years of them at
Chair of Algebra and Mathematical Logic, founded in 1992, which
since 1992 till 2006 was headed by Professor Aleksandr Georgievich
Pinus, rallied the amicable and fruitful group. Then it was headed
by Professors Konstantin Nikolaevich Ponomaryov and Evgeniy
9
10
PREF
ACE
Iosifovich Timoshenko. Nowadays Chair is headed by me. The
former Rectors of NSTU, Professors Anatoliy Sergeevich Vostrikov
and Nikolai Vasil'evich Pustovoi, the rector of NSTU, Professor
Anatoliy Andreevich Bataev, the former Vice-Rector responsible
for scientic researches, Professor Vladimir Ivanovich Denisov, and
nowadays Vice-Rector responsible for scientic researches, Profes-
sor Aleksey Gennad'evich Vostretsov promoted the creation and
the work of the Chair (which with such name is rather an excep-
tion than a rule in technical institutes). Former Dean of Applied
Mathematics and Computer Science Faculty of NSTU, Professor
Boris Yurievich Lemeshko and nowadays Dean, Professor Vladimir
Semyonovich Timofeev supported the Chair. Docent Vladimir
Mikhailovich Zybarev played an important role for the forming
and the adaptation of principal educational courses of Chair.
The benevolent scientic atmosphere in NSTU, the teaching of
courses of Algebra, Discrete Mathematics and Mathematical Logic,
and an opportunity of editions of tutorials for the disciplines pro-
mote the successful scientic work.
Since 2005, I am Senior (since July 2009, Leading) Researcher
of the Laboratory of Algebraic Systems in Sobolev Institute
of Mathematics of the Siberian Branch of the Russian Academy
of Science. The nal completion of the basic investigations up to
articles has occurred here. I am a docent of Novosibirsk State
University since April 2013.
The work was supported by RFBR, projects 93-011-1520, 96-
01-01675, 99-01-00571, 02-01-00258, 05-01-00411, 09-01-00336, 12-
01-00460, 17-01-00531, by the Council for Grants (under RF Pres-
ident) and State Aid for Fundamental Science Schools via projects
NSh-344.2008.1, NSh-3669.2010.1, NSh-6848.2016.1, by grants of
NSTU, and by Committee of Science in Education and Science
Ministry of the Republic of Kazakhstan (Grants No. 0830/GF4,
AP05132546).
I am grateful to all colleagues mentioned above, as well as the
management of the organizations, where the work, stated in the
book, has become possible to realize.
Sergey Sudoplatov
Novosibirsk, 2018
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