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Серия «Математика»
2017. Т. 22. С. 106-117
УДК 510.67
MSG 03С15, 03С30, 03С50
DOI https://doi.org/10.26516/1997-7670.2017.22.106
Generations of generative classes *
S. V. Sudoplatov
Sobolev Institute of Mathematics, Novosibirsk State Technical University, Novosibirsk State University, Institute of Mathematics and Mathematical Modeling
Abstract. We study generating sets of diagrams for generative classes. The generative classes appeared solving a series of model-theoretic problems. They are divided into semantic and syntactic ones. The fists ones are witnessed by well-known Fraïssé constructions and Hrushovski constructions. Syntactic generative classes and syntactic generic constructions were introduced by the author. They allow to consider any ui-homogeneous structure as a generic limit of diagrams over finite sets. Therefore any elementary theory is represented by some their generic models. Moreover, an information written by diagrams is realized in these models.
We consider generic constructions both in general case and with some natural restrictions, in particular, with the self-sufficiency property. We study the dominating relation and domination-equivalence for generative classes. These relations allow to characterize the finiteness of generic structure reducing the construction of generic structures to maximal diagrams. We also have that a generic structure is finite if and only if given generative class is finitely generated, i.e., all diagrams of this class are reduced to copying of some finite set of diagrams.
It is shown that a generative class without maximal diagrams is countably generated, i.e., reduced to some at most countable set of diagrams if and only if there is a countable generic structure. And the uncountable generation is equivalent to the absence of generic structures or to the existence only uncountable generative structures.
Keywords: generative class, generic structure, generation of generative class.
* The research is partially supported by Russian Foundation for Basic Researches (Grant No. 17-01-00531), by the Grants Council (under RF President) for State Aid of Leading Scientific Schools (grant NSh-6848.2016.1), and by Committee of Science in Education and Science Ministry of the Republic of Kazakhstan (Grant No. 0830/GF4).
1. Introduction
The notion of generative class was introduced in [9] and used in [10; 11] solving a series of complicated model-theoretic problems. This notion produces syntactic generic constructions which naturally generalize semantic ones including well-known Frai'sse constructions [2-4] and Hrushovski constructions [1; 5-7].
In the present paper we continue to study structural properties of generative classes and related objects [8; 12-14]. In Section 2, we consider a series of notions and results on generative class including the notions of domination and domination-equivalence for generative classes. In Section 3, we introduce the notions of finite, countable, and uncountable generations for generative classes and characterize these syntactic properties in terms of generic structures being semantic objects.
2. Preliminaries
We consider collections of sentence and formulas in first order logic over a language £. Thus, as usual, h means proof from no hypotheses deducing h ^ for a formula ^ of language £, which may contain function symbols and constants. If deducing hypotheses in a set $ of formulas can be used, we write $ h Usually £ will be fixed in context and not mentioned explicitly.
Below we write X, Y, Z,... for finite sets of variables, and denote by A, B, C,... finite sets of elements, as well as finite sets in structures, or else the structures with finite universes themselves.
In diagrams, A, B, C,... denote finite sets of constant symbols disjoint from the constant symbols in £ and £(A) is the vocabulary with the constants from A adjoined. $(A), ^(B), X(C) stand for £-diagrams (of sets A, B, C), that is, consistent sets of £(A)-, £(B)-, £(C)-sentences, respectively.
Below we assume that for any considered diagram $(A), if a^a2 are distinct elements in A then —(ai œ a2) € $(A). This means that if c is a constant symbol in £, then there is at most one element a € A such that (a œ c) € $(A).
If $(A) is a diagram and B is a set, we denote by $(A)|B the set |^>(a) € $(A) | a € B}. Similarly, for a language £, we denote by $(A)|s the restriction of $(A) to the set of formulas in the language £.
Definition 1. [8-14]. We denote by [$(A)]B the diagram $(B) obtained by replacing a subset A' C A by a set B' C B of constants disjoint from £ and with |A'| = |Bwhere A\ A' = B\B'. Similarly we call the consistent set of formulas denoted by [$(A)]X the type $(X) if it is the result of a
bijective substitution into of variables of X for the constants in A. In this case, we say that ®(B) is a copy of and a representative of We also denote the diagram by
Remark 1. If the vocabulary contains functional symbols then diagrams containing equalities and inequalities of terms can generate both finite and infinite structures. The same effect is observed for purely predicate vocabularies if it is written in that the model for should be
infinite. For instance, diagrams containing axioms for finitely axiomatizable theories have this property.
By the definition, for any diagram each constant symbol in £
appears in some formula of Thus, can be considered as
K), where K is the set of constant symbols in £.
We now give conditions on a partial ordering of a collection of diagrams which suffice for it to determine a structure. We modify some of the conditions for structures by d to signify they are conditions on diagrams not structures.
Definition 2. [8-14]. Let £ be a vocabulary. We say that (Do; (or Do) is generic, or generative, if Do is a class of £-diagrams of finite sets so that Do is partially ordered by a binary relation ^ such that ^ is preserved by bijective substitutions, i. e., if ^ ^(B), and A' C B' such that = and = fy(B') are defined, then
are in D0 and ^ [^(B)]^.1 Furthermore:
(i) if € Do then for any quantifier free formula <p(x) and any tuple a £ A either <p(a) € or ~«p(a) €
(ii) if $ < tf then $ C \[/;2
(iii) if $ ^ X, ^ € D0, and $ C V C X, then $ ^
(iv) some diagram $o(0) is the least element of the system (Do; and Do \ {^0(0)} is nonempty;
(v) (the d-amalgamation property) for any diagrams ^(B), X(C) € Do, if there exist injections /0: A —>• B and go: A —>• C with [<S>(A)]f^A) ^ and [<S>(A)}j^A) ^ X(C), then there are a diagram Q(D) € Do and injections f\: B —> D and g\: C —> D for which M^MB) < P^&c) < @(D) and /o°/i= 9oogv, the diagram Q(D) is called the amalgam of ^(B) and X(C) over the diagram and witnessed by the four maps (fo,9o, /1, <71);
1 Note that Do is closed under bijective substitutions since < is preserved by bijective substitutions and < is reflexive.
2 Note that < ^(B) implies A C B, since if a € A then (a ¡=s a) £ so <&(A) < ^(B) implies C i'(B) and we have (a ¡=s a) £ ^(B), whence a€ B.
(vi) (the local realizability property) if € Do and b 3x ip(x), then there are a diagram \Jj(B) e Do, ^ ^(B), and an element b £ B for which B) I- ip(b)]
(vii) (the d-uniqueness property) for any diagrams ^(B) £ Do if A C B and the set $(A) U B) is consistent then $(A) = {<p(b) £ ^(B) \ be A}.
A diagram $ is called a strong subdiagram of a diagram \Jj if $ ^ \Jj.
A diagram is said to be (strongly) embeddable in a diagram \Jj(B) if there is an injection f: A ^ B such that [<£>(A)]f{A) C ty(B) ([<£>(A)]f{A) ^
B)). The injection /, in this instance, is called a (strong) embedding of diagram in diagram \Jj(B) and is denoted by /: —>■ \Jj(B). A
diagram is said to be (strongly) embeddable in a structure M if is (strongly) embeddable in some diagram \Jj(B), where A4 |= ^(B). The corresponding embedding /: —>• \Jj(B), in this case, is called a (strong)
embedding of diagram in structure M and is denoted by /: —>•
M.
Let Do be a class of diagrams, Po be a class of structures of some language, and M be a structure in Po- The class Do is cofinal in the structure A4 if for each finite set A C M, there are a finite set B, A C B C M, and a diagram <&(B) £ D0 such that M \= §(B). The class D0 is co final in Po if Do is cofinal in every structure of Po- We denote by K(Do) the class of all structures M with the condition that Do is cofinal in A4, and by P a subclass of K(Do) such that each diagram $ € Do is true in some structure in P.
Now we extend the relation ^ from the generative class (Do;^) to a class of subsets of structures in the class K(Do).
Let M be a structure in K(Do), A and B be finite sets in M with A C B. We call A a strong subset of the set B (in the structure A4), and write A ^ B, if there exist diagrams $(^1), ^(U) £ Do, for which < and M |= ^(B).
A finite set A is called a strong subset of a set Mo C M (in the structure M), where A C M0, if A ^ B for any finite set B such that A C B C M0 and C ¡P(B) for some diagrams ^(A),^(B) £ D0 with M |= V(B). If A is a strong subset of Mo then, as above, we write A ^ Mo. If A ^ M in M then we refer to A as a self-sufficient set (in A4).
Notice that, by the d-uniqueness property, the diagrams and ^(B) specified in the definition of strong subsets are defined uniquely. A diagram £ Do, corresponding to a self-sufficient set A in M, is said to be a self-sufficient diagram (in M).
Definition 3. [8-14]. A class (Do;^) possesses the joint embedding property (JEP) if for any diagrams ^(B) £ Do, there is a diagram
X(C) € D0 such that and ^>(B) are strongly embeddable in X(C).
Clearly, every generative class has JEP since JEP means the (¿-amalgamation property over the empty set.
Definition 4. [8-14]. A structure Л4 £ P has finite closures with respect to the class (Do; or is finitely generated over E, if any finite set А С M is contained in some finite self-sufficient set in Л4, i. е., there is a finite set В with ACBCMand Ф (B) € D0 such that M |= Ф (B) and Ф (В) ^ X(C) for any X(C) € D0 with M |= X(C) and Ф(5) С X(C). A class P has finite closures with respect to the class (Do; or is finitely generated over E, if each structure in P has finite closures (with respect to (Do;
Clearly, an at most countable structure Л4 has finite closures with respect to (Do;^) if and only if M = (J Ai for some self-sufficient sets
i£ui
Ai with Ai ^ Ai+1, i € oj.
Note that the finite closure property is defined modulo E and does not correlate with the cardinalities of algebraic closures. For instance, if E contains infinitely many constant symbols then acl(A) is always infinite whereas a finite set A can or can not be extended to a self-sufficient set.
Besides, for the finite closures of sets A we consider finite self-sufficient extensions Б in a given structure Л4 with respect to (Do; only and В can be both a universe of a substructure of Л4 or not. Moreover, it is permitted that corresponding diagrams Ф(-В) can have only finite, finite and infinite, or only infinite models.
Thus, for instance, a finitely axiomatizable theory without finite models and with a generative class (Do; C), containing diagrams for all finite sets and with axioms in diagrams, has identical finite closures whereas each diagram in Do has only infinite models.
Definition 5. [8-14]. A structure M € K(D0) is (D0; -generic, or a generic limit for the class (Do; and denoted by glim (Do; if it satisfies the following conditions:
(a) Л4 has finite closures with respect to Do;
(b) if А С M is a finite set, Ф(А), Ф(B) € D0, M |= Ф(A) and Ф(A) ^ Ф (B), then there exists a set В' ^ M such that A (IB1 and M |= Ф (B').
Clearly, uncountable (Do; ^)-generic structures can be non-isomorphic. Indeed, for instance, all infinite structures in the empty language are generic for a given generative class although these structures are non-isomorphic for distinct cardinalities. But, as the following theorem shows, they are isomorphic for at most countable cases.
Theorem 1. [11]. For any generative class (Do;^) with at most count-ably many diagrams whose copies form Do, there exists at most countable (Do; generic structure, unique up to isomorphism.
Theorem 2. [8; 12; 14]. Every uj-homogeneous structure M. is (Do;^)-generic for some generative class (Do;
Thus any first-order theory has a generic model and therefore can be represented by it.
Definition 6. [8-14]. A generative class (Do;^) is self-sufficient if the following axiom of self-sufficiency holds:
(viii) if Do, i> ^ and X C then $ n X ^ X.
Note that in the proof of Theorem 2 the required generative class (Do; is self-sufficient.
Theorem 3. [8-11; 14]. Let (D0;^) be a self-sufficient class, M be at most countable (Do; generic structure, and K be the class of all models ofT = Th(.M) which has finite closures. Then the generic structure M. is homogeneous.
Thus, since any w-homogeneous structure can be considered as generic with respect to a generic class with complete diagrams, a countable structure A4 is homogeneous if and only if it is generic for an appropriate self-sufficient generative class (Do;
Definition 7. [9-11]. Let (D0;^) and (Dq;^') be generative classes of languages E and E', respectively, with E C E'. We say that the class (Dq; dominates the class (Do; and write Do<!Dq, if for any diagram € D0 there is a diagram ^'(A') € Dq such that C ^'(A'), and the condition of there being some systems, which are extensions over A, together with available information on interrelations of elements in these extensions written in the diagram implies that the same extensions
exist over A, and that similar information is available on interrelations of elements in those extensions written in the diagram ^'(.A').
If Do < Dq and Dq < Do we say that generative classes (Do; and (Dq; are domination-equivalent and write Do ~ Dq.
Theorem 4. [9-11]. Let M and M' be countable homogeneous structures of languages E and E'; respectively. The following conditions are equivalent:
(1) the structure M. is isomorphically embeddable in the structure M' \
(2) there are generative classes (Do;^) and (Dq;^') such that M is (Do; generic, M' is (Dq; -generic, and Do < Dq.
Since mutually embeddable countable homogeneous structures are isomorphic, Theorem 4 implies the following corollary.
Corollary 1. Let M and M' be countable homogeneous structures of a language E. The following conditions are equivalent:
(1) the structures A4 and M' are isomorphic;
(2) there are domination-equivalent generative classes (Do;^) and (Dq; such that A4 is (Do; generic and Ai' is (Dq; -generic;
(3) the structures A4 and Ai' are (Do; ^-generic for some generative class (Do;
Proof (1) (3) and (3) (2) are obvious.
(2) (1). Having the hypothesis we see by Theorem 4 that M and Ai' are mutually embeddable. Since Ai and Ai' are countable homogeneous, they are isomorphic. □
3. Finite, countable and uncountable generations of generative
classes
Definition 8. [13]. Let (Do;^) be a generative class in a language E, Ф(А) € Do- The diagram Ф(А) is called structural if it satisfies the following modification of the local realizability property: if Ф(А) b 3x<p(x) then there is a constant term t(a), a € A, such that Ф(А) b Lp(t(a)).
Note that there exist generative classes containing non-structural diagrams. Indeed, consider a finitely axiomatizable, by an axiom ipo, complete theory T of relational language and without finite models (for instance, consider the theory of dense linear order without endpoints). Now, take a generative class (Do; for T such that all diagrams in Do contain ipo. Obviously, Do does not contain structural diagrams since for any Ф(А) € Do, every model Ai |= Ф(А) is infinite, being a model of T, whereas constant terms t(a), a € A, can have values only in the finite set A.
Theorem 5. [13, Theorem 4.1]. For any diagram Ф(А) € D0, А ф 0, the following conditions are equivalent:
(1) Ф(А) is structural;
(2) there exists a structure Ai consisting of some constant terms in the language SUA and such that Ai |= Ф(А).
Proof. (1) => (2). Let Ф(A) be structural. Denote by N the set of all constant terms t(a), a € A, in the language E U A. For constant terms ii and ¿2, we put t\ ~ ¿2 if and only if Ф(А) b (t\ pa ¿2)- Clearly, ~ is an equivalence relation and there is a canonical structure Л/"/~ having the universe and satisfying the quantifier-free part of Ф(А). By (vi') and induction on length of formulas in Ф(А) we get М/ ~ |= Ф(-А). Taking representatives for each ~-class in N/ ~ we form a required structure Ai isomorphic to W/<~.
(2) => (1). If there exists a required structure Ai having the universe M consisting of some constant terms (one representative for each ~-class)
in the language S U A and such that A4 |= then, by completeness of
the first-order calculus, for any formula <p(x) with h 3x<p(x), there is a term t(a) € M such that h <p(t(a)). Thus, is structural. □
The structure W/<~ in the proof of Theorem 5 is called & (A)-canonical, or simply canonical and denoted by The structure Ai, in the proof,
is a representation of
By Theorem 5, any structural diagram for A / 0, defines the al-
gebra with the universe Nbeing the restriction of A/"/~ to the functional sublanguage and finitely generated by A (relative constant symbols in E). At the same time, for quantifier-free diagrams, this condition is sufficient:
Corollary 2. [13, Corollary 4.2]. Any quantifier-free diagram € Do is structural.
Definition 9. [13]. A diagram € D0 is called self-structural if
A / 0 and satisfies the following: if h 3x <p(x) then there is an element a € A such that h <p(a).
Theorem 6. [13, Theorem 25] For a generative class (Do;^) with a language having a finite set C of pairwise distinct constants, the following conditions are equivalent:
(1) the (Do; ^-generic structure is finite;
(2) (Do;^) has maximal diagrams;
(3) (Do; is domination-equivalent to a minimal generative class consisting of a diagram $0(0) and of copies of a self-structural diagram
(4) the (Do; generic structure is isomorphic, for a quantifier-free diagram to a representation, with the universe A\JC, of § (A)-canonical structure.
Remark 2. In Theorem 6 the existence of finite C is implied by each of the conditions (1), (2), (3).
Definition 10. A generative class (Do; is called A-generated, where A is a cardinality, if Do contains a set Z of diagrams such that each diagram in Do is a copy of some diagram in Z. The generative class (Do; is called finitely generated if it is n-generated for some new. The generative class (Do; is called countably generated if it is uj-generated. If (Do; is not countably generated it is called uncountably generated.
The following theorem extends the list of criteria for the existence of finite generic structures in Theorem 6:
Theorem 7. For a generative class (Do;^) the following conditions are equivalent:
(1) (Do;^) is finitely generated;
(2) each diagram in (Do; is extensible till a maximal one;
(3) (Do; has a maximal diagram.
Proof. (1) => (2). If (Do; is finitely generated by diagrams
then their amalgam is a copy of some and it is maximal. Thus any
diagram $ being in the list ..., $ra is extensible till a maximal one.
(2) (3) is obvious.
(3) (1). If (Do;^) has a maximal diagram then, since Do is closed under amalgams, is a copy of amalgam of with arbitrary diagram ^(A) € Do- Therefore, each diagram in Do is a copy of a restriction ^(A)^ of to some B ç A. As A is finite there are finitely many these restrictions. Thus, (Do; is finitely generated. □
Theorem 8. For any generative class (Do; the following conditions are equivalent:
(1) there is a countable (Do; generic structure (and the language has at most countable set C of pairwise distinct constants);
(2) (Do; is countably generated and does not have maximal diagrams.
Proof. (1) (2) holds by the definition of countable generic structure and Theorem 6 with Remark 2. In particular, the language has at most countable set of pairwise distinct constants.
(2) (1). If (Do; is countably generated and does not have maximal diagrams then we construct step-by-step a countable generic structure M as in the proof of Theorem 1 producing at most countably many pairwise distinct constants. □
Theorem 8 immediately implies the following characterization for the uncountably generated generative classes (Do;
Corollary 3. For any generative class (Do;^) the following conditions are equivalent:
(1) (Do;^) is uncountably generated;
(2) the setC of pairwise distinct constants in the language is uncountable and/or all (Do; generic structures are uncountable or they do not exist.
Corollary 3 with the following theorem allows to divide (in terms of meeting of contradictions for the cardinalities of definable sets) the uncountable generation of (Do; into two cases: with or without generic structures.
Theorem 9. [8]. For any generative class (Do;^) the following conditions are equivalent:
(1) there exists a (Do; ^-generic structure;
(2) there are no type-definable sets X constructed with respect to (Do; such that these X meet contradictions for their cardinality;
(3) there are no definable sets X constructed with respect to (Do; such that these X meet contradictions for their cardinality.
In conclusion we note that by Theorems above generative classes can, on syntactic level, control existence of finite, countable, or uncountable generic structures, and their absence as well.
References
1. Baldwin J.T., Shi N. Stable generic structures Ann. Pure and Appl. Logic, 1996, vol. 79, no 1, pp. 1-35. https://doi.org/10.1016/0168-0072(95)00027-5
2. Fraïsse R. Sur certaines relations qui generalisent l'ordre des nombres rationnels C.R. Acad. Sci. Paris, 1953, vol. 237, pp. 540-542.
3. Fraïsse R. Sur l'extension aux relations de quelques proprietes des ordres. Annales Scientifiques de l'Ecole Normale Supérieure. Troisième Série, 1954, vol. 71, pp. 363-388.
4. Hodges W. Model theory. Cambridge, Cambridge University Press, 1993. https://doi.org/10.1017/CB09780511551574
5. Hrushovski E. Strongly minimal expansions of algebraically closed fields. Israel J. Math., 1992, vol. 79, no 2-3, pp. 129-151. https://doi.org/10.1007/BF02808211
6. Hrushovski E. Extending partial isomorphisms of graphs. Combinatorica, 1992, vol. 12, no 4. pp. 204-218. https://doi.org/10.1007/BF01305233
7. Hrushovski E. A new strongly minimal set. Ann. Pure and Appl. Logic, 1993, vol. 62, no 2. pp. 147-166. ttps://doi.org/10.1016/0168-0072(93)90171-9
8. Kiouvrekis Y., Stefaneas P., Sudoplatov S. V. Definable sets in generic structures and their cardinalities. Siberian Advances in Mathematics, 2018. (to appear)
9. Sudoplatov S. V. Syntactic approach to constructions of generic models. Algebra and Logic. 2007, vol. 46, no 2, pp. 134-146. https://doi.org/10.1007/s10469-007-0012-4
10. Sudoplatov S.V. The Lachlan Problem. Novosibirsk: NSTU, 2009. (in Russian)
11. Sudoplatov S. V. Classification of Countable Models of Complete Theories. Novosibirsk, NSTU, 2014. (in Russian)
12. Sudoplatov S.V. Generative and pre-generative classes Proceedings of the 10th Panhellenic Logic Symposium, June 11-15, 2015, Samos, Greece. University of Aegean, University of Crete, and University of Athens, 2015, pp. 30-34.
13. Sudoplatov S.V. Generative classes generated by sets of diagrams.Algebra and, Model Theory 10. Collection of papers. Edited by A.G. Pinus, K.N. Ponomaryov, S.V. Sudoplatov, E.I. Timoshenko. Novosibirsk, NSTU, 2015, pp. 163-174.
14. Sudoplatov S.V., Kiouvrekis Y., Stefaneas P. Generic constructions and generic limits. Algebraic Modeling of Topological and Computational Structures and Applications. Springer Proc. in Math. & Statist., 219. S. Lambropoulou (ed.) et al. Berlin, Springer Int. Publ., 2017.
Sudoplatov Sergey Vladimirovich, Doctor of Sciences (Physics and Mathematics), Associate Professor, Leading Researcher, Sobolev Institute of Mathematics SB RAS, 4, Academician Koptyug Avenue, Novosibirsk, 630090, Russian Federation, tel.: (383)3297586; Head of Chair, Novosibirsk State Technical University, 20, K. Marx Avenue, Novosibirsk, 630073, Russian Federation, tel.: (383)3461166; Docent, Novosibirsk State University, 1, Pirogov st., Novosibirsk, 630090, Russian Federation , tel.: (383)3634020;
Principal Researcher, Institute of Mathematics and Mathematical Modeling, 125, Pushkin St., Almaty, 050010, Kazakhstan, tel.: +7(727)2720046 (e-mail: [email protected])
С. В. Судоплатов
Порождения в генерирующих классах
Аннотация. В работе исследуются порождающие множества диаграмм для генерирующих классов. Сами генерирующие классы возникли при решении ряда теоретико-модельных проблем. Они подразделяются на семантические и синтаксические. К первым относятся широко известные конструкции Фраиссе и Хрушов-ского. Синтаксические генерирующие классы и синтаксические генерические конструкции были введены в работах автора. Они позволяют рассматривать любую ш-однородную структуру в виде генерического предела диаграмм над конечными множествами. Тем самым, любая элементарная теория представляется некоторыми своими генерическими моделями. При этом информация, заданная диаграммами, реализуется в этих моделях.
Мы рассматриваем генерические конструкции как в общем виде, так и при некоторых естественных ограничениях, в частности при выполнении свойства самодостаточности. Исследуется отношение доминирования и эквивалентности по доминированию для генерирующих классов. С помощью этого отношения характеризуется условие конечности генерической структуры, сводящее построение генерической структуры к использованию лишь максимальных диаграмм. Условие конечности генерической структуры также эквивалентно конечной порожденности генерирующего класса, т. е. сведению всех диаграмм данного класса к копированию некоторого конечного множества диаграмм.
Доказано, что счетная порожденность (сведение к некоторому, не более чем счетному множеству диаграмм) генерирующего класса без максимальных диаграмм равносильна существованию счетной генерической структуры, а несчетная порожденность — отсутствию генерических структур или наличию лишь несчетных гене-рических структур.
Ключевые слова: генерирующий класс, генерическая структура, порождение генерирующего класса.
Список литературы
1. Baldwin J. T. Stable generic structures / J. T. Baldwin, Shi N. // Ann. Pure and Appl. Logic. - 1996. - Vol. 79, N 1. - P. 1-35.
2. Fraïssé R. Sur certaines relations qui généralisent l'ordre des nombres rationnels // C.R. Acad. Sei. Paris. - 1953. - Vol. 237. - P. 540-542.
3. Fraïssé R. Sur l'extension aux relations de quelques propriétés des ordres // Annales Scientifiques de l'Ecole Normale Supérieure. Troisième Série. — 1954. -Vol. 71. - P. 363-388.
4. Hodges W. Model theory / W. Hodges. - Cambridge : Cambridge University Press, 1993.
5. Hrushovski E. Strongly minimal expansions of algebraically closed fields // Israel J. Math. - 1992. - Vol. 79, N 2-3. - P. 129-151.
6. Hrushovski E. Extending partial isomorphisms of graphs // Combinatorica. -1992. - Vol. 12, N 4. - P. 204-218.
7. Hrushovski E. A new strongly minimal set // Ann. Pure and Appl. Logic. - 1993. -Vol. 62, N 2. - P. 147-166.
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Судоплатов Сергей Владимирович, доктор физико-математических наук, доцент, ведущий научный сотрудник, Институт математики им. С. JI. Соболева СО РАН, 630090, Россия, Новосибирск, пр. Академика Коптюга, 4, тел.: (383)3297586; заведующий кафедрой алгебры и математической логики, Новосибирский государственный технический университет, 630073, Россия, Новосибирск, пр. К. Маркса, 20, тел. (383)3461166; доцент кафедры алгебры и математической логики, Новосибирский государственный университет, 630090, Россия, Новосибирск, ул. Пирогова, 1, тел. (383)3634020; главный научный сотрудник, Институт математики и математического моделирования МОН РК, 050010, Казахстан, Алматы, ул. Пушкина, 125, тел. +7(727)2720046 (e-mail: sudoplat@math. nsc . ru)
