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THE LACHLAN PROBLEM
S. V. SUDOPLATOV
NOVOSIBIRSK2008
UDC 510.6
Sudoplatov S. V. The Lachlan Problem. | Novosibirsk, 2008.
A generic construction realizing basic characteristics of Ehren-feucht theories, i.e. of complete rst order theories with nitelymany but more then one pairwise non-isomorphic countable mod-els, is stated. On the basis of that construction as well as of theHrushovski | Herwig generic construction, a solution of the Lachlanproblem on existence of stable Ehrenfeucht theories is shown.
For persons who interest in the Mathematical Logic.
c© Sudoplatov S.V., 2008
Contents
Preface 5
Introduction and historical review 9
C h a p t e r 1. Characterization of Ehrenfeuchtness. Propertiesof Ehrenfeucht theories 17
§ 1.1. Syntactic characterization of the class of complete theorieswith nitely many countable models . . . . . . . . . . . . 17
§ 1.2. Inessential combinations and colorings of models . . . . . 28§ 1.3. Type reducibility and powerful types . . . . . . . . . . . . 39§ 1.4. Powerful digraphs . . . . . . . . . . . . . . . . . . . . . . 44
C h a p t e r 2. Generic constructions 55§ 2.1. Semantic generic constructions . . . . . . . . . . . . . . . 55§ 2.2. Syntactic generic constructions . . . . . . . . . . . . . . . 56§ 2.3. Self-sucient classes . . . . . . . . . . . . . . . . . . . . . 60§ 2.4. Genericity of countable homogeneous models . . . . . . . 65§ 2.5. The uniform t-amalgamation property and saturated generic
models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67§ 2.6. On the nite closure property in fusions of generic classes 70
C h a p t e r 3. Generic Ehrenfeucht theories 78§ 3.1. Generic theory with a non-symmetric semi-isolation relation 78§ 3.2. Generic theories with nonprincipal powerful types . . . . 86§ 3.3. A theory with three countable models . . . . . . . . . . . 95§ 3.4. Realizations of basic characteristics of complete theories
with nitely many countable models . . . . . . . . . . . . 105§ 3.5. Theories with nite Rudin | Keisler preorders . . . . . . 109§ 3.6. Rudin | Keisler preorders in small theories . . . . . . . . 115§ 3.7. Theories with non-dense structures of powerful digraphs
and theories with powerful types and without powerfuldigraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
C h a p t e r 4. Stable generic Ehrenfeucht theories (a solutionof the Lachlan problem) 121
§ 4.1. Small stable generic graphs with innite weight. Bipartitedigraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
§ 4.2. Small stable generic graphs with innite weight. Digraphswithout furcations . . . . . . . . . . . . . . . . . . . . . . 136
§ 4.3. Small stable generic graphs with innite weight. Powerfuldigraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
§ 4.4. On expansions of powerful digraphs . . . . . . . . . . . . 174
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§ 4.5. Description of features of generic construction of stableEhrenfeucht theories . . . . . . . . . . . . . . . . . . . . . 178
§ 4.6. Stable graph extensions of colored powerful digraphs . . . 180§ 4.7. Stable theories with nonprincipal powerful types . . . . . 183§ 4.8. Stable theories with three countable models . . . . . . . . 189§ 4.9. Realizations of basic characteristics of stable Ehrenfeucht
theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
C h a p t e r 5. Hypergraphs of prime models and distributionsof countable of small theories 199
§ 5.1. Hypergraphs of prime models . . . . . . . . . . . . . . . . 200§ 5.2. HPKB-Hypergraphs and theorem on structure of type . . 202§ 5.3. Graph connections between types . . . . . . . . . . . . . . 206§ 5.4. Limit models . . . . . . . . . . . . . . . . . . . . . . . . . 208§ 5.5. λ-Model hypergraphs . . . . . . . . . . . . . . . . . . . . 211§ 5.6. Distributions of prime and limit models. . . . . . . . . . . 214
References 222
Index 237
PREFACE
The Lachlan problem is one of the basic problems of modernabstract model theory. Model Theory, formed in an independentarea in 1950th years, is on a joint of Mathematical Logic and Al-gebra. Its subjects are syntactic objects (theories representing de-scriptions of real objects) and semantic objects (algebraic systemsre ecting interrelations of elements of real objects), and also classi-cations of syntactic objects by properties of semantic objects andvice versa. Essentially various (non-isomorphic) realizations of thesetheories by algebraic systems (models) are possible at descriptionsof complete theories (i.e., theories without consistent information ina xed language that can be added but not added). The numberof such realizations can be various in dierent innite cardinalities(i.e., with dierent innite number of elements) of algebraic sys-tems. Thus, there are spectrum functions, re ecting the number ofnon-isomorphic models of given theory depending on cardinalitiesof models, and the problem of descriptions of all possible spectrumfunctions for the class of all theories, and also for various essentialsubclasses of this class.
It is surprising that the spectrum problem is solved for large(uncountable) cardinalities in the class of all theories. Here thebasic achievements are connected with works by S. Shelah [22] andnally represented in the work by B. Hart, E. Hrushovski andM. S. Laskowski [74].
For the countable (minimal innite) cardinality the situationappeared much more dicult. Firstly, it is not known till now onan existence of theories with uncountable and non-maximal numberof countable models (the Vaught problem). Secondly, constructed byA. Ehrenfeucht (see [198]) initial examples of theories with a nitelymany, but more than one, countable models (now such theories, in
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his honour, are called Ehrenfeucht) had been in essence unique fora long time: all modications had been reduced to superstructureson innite dense linearly ordered sets. With this circumstance, theLachlan problem on existence of essential other (i.e., not havinginnite linear orders) Ehrenfeucht theories has been arised. In abrief formulation, the Lachlan problem is: to dene, whether thereexists a stable Ehrenfeucht theory.
This problem has been partially solved by A. H. Lachlan [109],published in 1973 a proof of an absence of Ehrenfeucht theories inthe class of superstable theories, that is an important subclass ofthe class of stable theories. It was supposed for a long time, thatthe statement is true for stable theories, and it was referred in theliterature, alongside with the Lachlan Problem, to the Lachlan Con-jecture (see, for example, [27], p. 202). The Lachlan Conjecture isproved to be true for many subclasses of the class of stable theoriesin the works by D. Lascar [111], S. Shelah [22], A. Pillay [139], [143],[145], [146], T. G. Mustan [128], U. Sae [160], A. Tsuboi [197],E. Hrushovski [90], A. A. Vikent'ev [202], B. Kim [105], P. Tanovic[191], [192]. At the same time, structural properties of a coun-terexample, if it exists, has been accumulated. The following worksare connected with this accumulation: by M. G. Peretyat'kin [137],[138], M. Benda [56], R. Woodrow [205], [206], A. Pillay [140], [141],B. Omarov [132], A. Tsuboi [196], S. S. Goncharov, M. Pourmah-dian [70], B. Herwig [81] and by the author. A solution of theproblem, namely a proof of existence of stable Ehrenfeucht theory,became possible only after an occurrence of a nice construction cre-ated by E. Hrushovski [89] in 1988 and applied for solutions of manymodel-theoretic problems. Now this known construction is calledthe generic Hrushovski construction. It allows \to collect" requiredmodels formed via classes of nite objects using amalgams.
Another important component is the theory of group polygono-metries created by the author [172], [173] and generalizing classi-cal trigonometries. The class of group polygonometries is a con-venient and geometrically clear object that has allowed to realizemany structural properties of stable Ehrenfeucht theories. At thesame time, now, when the general mechanism of constructions ofEhrenfeucht theories became clear, the explicit description of im-plicitly presented polygonometrical apparatus is super uous in theconstruction. Therefore, the Polygonometrical Theory and its ap-plications are not represented in the book.
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For the construction of stable Ehrenfeucht theories, we have in-volved a nice modication of Hrushovski construction, oered byB. Herwig [81] for a realization of a basic structural property | theinnite weight. At the same time, this modication in an originalform has been insucient, since the Hrushovski | Herwig construc-tion is semantic and does not take into consideration a possibility ofoccurrence of external connections w.r.t given nite objects, whichtaken as \bricks" of the generic construction.
For the elimination of this lack, the theory of syntactic genericconstructions [180] has been developed by the author. Syntacticconstructions are based on types (not on nite objects), i.e., ondescriptions (possible to be external) of nite objects, that thenallow to generate models of the required theories step-by-step.
Using the aforesaid tools, we construct a series of stable Ehren-feucht theories.
Initial my work passed during my study in Novosibirsk StateUniversity, where the rst class specialists in Mathematical Logicand Algebra worked and continue to work. An occurrence of theSiberian School of Algebra and Logic, to which I regard me, be-came possible after the foundation of the Institute of Mathematicsin Academgorodok, Novosibirsk in 1957 and of arrival to Novosibirskthe founder of the School, Academician Anatoliy Ivanovich Mal'tsev.Now already more thirty years this School is headed by the Direc-tor of Institute of Mathematics, Academician Yuriy Leonidovich Er-shov. To the statement the Problem and to successes in its solution,I am obliged in many respects to my scientic advisor, the Head ofthe Laboratory of Algebraic Systems, Professor Evgeniy AndreevichPalyutin. I had a lot useful and fruitful discussions with the Corre-spondent member of the Russian Academy of Science, the head ofDepartment of Mathematical Logic in IM SB RAS, the Dean of Fac-ulty of Mechanics and Mathematics of Novosibirsk State University,Professor Sergey Savost'yanovich Goncharov, with the Professor ofDepartment of Algebra and Mathematical Logic of Novosibirsk StateTechnical University Aleksandr Georgievich Pinus, with participantsof the seminar \Model Theory" in IM SB RAS, doc. AleksandrNikolaevich Ryaskin, doc. Aleksandr Aleksandrovich Vikent'ev,doc. Dmitriy Yur'evich Vlasov, the post-graduate student MikhailAndreevich Rusaleev, with lecturers of the Department of Algebraand Mathematical Logic in NSTU. During all my scientic activity,I approved new results before publications at the seminar \Model
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Theory", supervised by Academician Yuriy Leonidovich Ershov andProfessor Evgeniy Andreevich Palyutin.
I had useful direct and correspondence dialogues with manymodel-theoretic specialists from France, Kazakhstan, USA, GreatBritain, Israel, Japan, Germany, Poland, Czechia, Serbia.
It has turned out, after my postgraduate study in NSU, since1990, I work already 17 years in NSTU, and 15 years of them atthe Department of Algebra and Mathematical Logic, founded in1992, which since 1992 till 2006 was headed by Professor AleksandrGeorgievich Pinus, rallied the amicable and fruitful collective. Theformer rector of NSTU, Professor Anatoliy Sergeevich Vostrikov andthe rst vice-rector (nowadays the rector) of NSTU, Professor Niko-lay Vasil'evich Pustovoy promoted the creation of the Department(which with such name is rather an exception than a rule in technicalinstitutes). The benevolent scientic atmosphere in NSTU, readingof courses of Algebra, Discrete mathematics and Mathematical logic,and also an opportunity of editions of textbooks for disciplines helpto successful scientic work.
Since 2005 till present, I am a Senior Researcher of the labora-tory of algebraic systems in Sobolev Institute of Mathematics of theSiberian Branch of the Russian Academy of Science, and the nalnishing of the basic results up to articles has occurred here.
The work was supported by RFBR, projects 93-011-1520, 96-01-01675, 99-01-00571, 02-01-00258, 05-01-00411, and by the Councilfor Grants (under RF President) and State Aid of FundamentalScience Schools via project NSh-4787.2006.1.
I am grateful to all colleagues mentioned above, as well as themanagement of the organizations, where the work, stated in thebook, has become possible to carry out.
Sergey Sudoplatov
Novosibirsk, October 2007.
INTRODUCTION AND HISTORICALREVIEW
As it was already mentioned in the preface, one of the main aimof the modern model theory is a solution of spectrum problem, i.e.,the problem of the description of functions I(T, λ) for numbers ofpairwise non-isomorphic models for theories T in cardinalities λ forvarious classes of theories T . An interest to this problem is causedmainly by that the substantial structural theory is required for itssolution.
The problem of description of spectrum functions, and also ofclasses of theories depending on these functions has paid and con-tinues to pay attention of wide group of model-theoretic specialists,forming an extensive area of researches. It is re ected in manyworks, among which we mention the following: books and disser-tations by J. T. Baldwin [1]; O. V. Belegradek [4]; C. C. Chang,H. J. Keisler [6]; Yu. L. Ershov, S. S. Goncharov [8]; Yu. L. Ershov,E. A. Palyutin [9]; Handbook of mathematical logic [3]; W. Hodges[15]; A. Pillay [19]; B. P. Poizat [20]; G. E. Sacks [21]; S. Shelah [22];F. O. Wagner [27]; papers by J. T. Baldwin, A. H. Lachlan [33];M. Benda [56]; S. Buechler [59]; S. S. Goncharov, M. Pourmah-dian [70]; B. Hart, S. S. Starchenko, M. Valeriote [73]; B. Hart,E. Hrushovski, M. S. Laskowski [74]; B. Herwig, J. Loveys, A. Pil-lay, P. Tanovic, F. O. Wagner [80]; B. Herwig [81]; E. Hrushovski[90]; K. Ikeda, A. Pillay, A. Tsuboi [95]; B. Khusainov, A. Nies,R. A. Shore [103]; B. Kim [105]; A. H. Lachlan [109]; D. Lascar [111];J. Loveys, P. Tanovic [116]; L. F. Low, A. Pillay [117]; L. Mayer[120]; T. Millar [121][124]; M. Morley [126], [127]; T. G. Mustan[128]; B. Omarov [132]; E. A. Palyutin [134]; E. A. Palyutin,S. S. Starchenko [135]; M. G. Peretyat'kin [137], [138]; A. Pillay[139][143], [145], [146]; R. Reed [156]; A. N. Ryaskin [158]; C. Ryll-
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Nardzewski [159]; U. Sae [160], [161]; S. Shelah [163]; S. Shelah,L. Harrington, M. Makkai [165]; S. Thomas [195]; A. Tsuboi [196],[197]; P. Tanovic [191]; R. Vaught [198]; R. E. Woodrow [205], [206].
As already mentioned (see S. Shelah [22]; B. Hart, E. Hrushovski,M. S. Laskowski [74]), the spectrum problem is solved as a wholefor countable complete theories in uncountable cardinalities λ.
The problem of the description of number I(T, ω) of pairwisenon-isomorphic countable models of the theory T for the givenclasses of complete theories is not well-studied till present. Here,it should to mention the Vaught Conjecture according to whichthere does not exist a theory T such that ω < I(T, ω) < 2ω.This conjecture has been conrmed for theories of trees (J. Steel[168]), unars (L. Marcus [119]; A. Miller [125]), varieties (B. Hart,S. S. Starchenko, M. Valeriote [73]), for o-minimal theories (L. Mayer[120]), for theories of modules over some rings (V. A. Puninskaya[152], [153]; V. A. Puninskaya, C. Toalori [154]). In the class ofstable theories, the Vaught conjecture has proved for ω-stable the-ories (S. Shelah [163]; S. Shelah, L. Harrington, M. Makkai [165]),for various classes of superstable theories (E. R. Baisalov [30], [31];S. Buechler [59], [60]; L. F. Low, A. Pillay [117]; L. Newelski [129],[130], [131]), and also for 1-based theories with a nonisolated typeover a nite set such that this type is orthogonal to the empty set(P. Tanovic [192]). Attempts of constructions of examples denyingthe Vaught conjecture had taken (see R. W. Knight [107]). However,the problem is still open.
Another interesting conjecture is the Pillay Conjecture. It as-serts that, for any countable theory T , the condition dcl(∅) |= Timplies I(T, ω) ≥ ω. A. Pillay [142] has proven this conjecture forstable theories and has shown (see [139]), that dcl(∅) |= T impliesI(T, ω) ≥ 4. P. Tanovic [193] has proven, that the Pillay Conjectureholds for theories without the strict order property.
C. Ryll-Nardzewski [159], in 1959, has published his well-knowntheorem representing a syntactic criterion of countable categoricityof a theory (i.e., the conditions I(T, ω) = 1), according which thecountable categoricity of a theory is equivalent to a niteness ofnumber of n-types of the theory for every natural n and xed set offree variables. It means that every countable categorical theory isdened by one characteristic, namely, the Ryll-Nardzewski function,that puts in the correspondence to each natural n the number oftypes of n xed free variables.
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Many results are connected with Ehrenfeucht theories, i.e., the-ories having nitely many (> 1) countable models. R. Vaught[198] proved that there are no complete theories having exactlytwo countable models. Using the theory of the dense linear or-der, A. Ehrenfeucht [198] has constructed initial examples of the-ories having equally n countable models for each natural n ≥ 3.The further researches have been connected with constructions ofEhrenfeucht theories possessing various additional properties, withndings and investigations of structural properties of Ehrenfeuchttheories, and also with ndings of classes of complete theories thatdon't contain Ehrenfeucht theories.
M. G. Peretyat'kin [137] has constructed, for every n ≥ 3, a com-plete decidable theory having exactly n countable models such thatthe unique is constructible. B. Omarov [132], M. G. Peretyat'kin[138], T. Millar [121], [124], S. Thomas [195], R. E. Woodrow [206]have constructed examples of Ehrenfeucht theories admitting con-stant expansions up to theories with innitely many countable mod-els, and also of non-Ehrenfeucht theories such that some theirs con-stant expansions are Ehrenfeucht. R. E. Woodrow [205] has shown,that assuming the quantier elimination and restricting the languageon a binary predicate symbol and constant symbols, the count-able complete theories with exactly three countable models are, inessence, the Ehrenfeucht examples. A. Pillay [141] has shown, thatan innite dense partial order is interpretable in any Ehrenfeuchttheory with few links. S. S. Goncharov and M. Pourmahdian [70]have proved that every Ehrenfeucht theory has a nite rank. It isshown in the work by K. Ikeda, A. Pillay, A. Tsuboi [95], that thedense linear order is interpretable in any almost ω-categorical theorywith three countable models. P. Tanovic [194] has shown that theEhrenfeucht example or the Peretyat'kin example is interpretable inany theory with three countable models having an innite set of pair-wise dierent constants. E. R. Baisalov [32] has described possiblenumbers of countable models of o-minimal theories (the class of o-minimal theories includes the classical examples of Ehrenfeucht the-ories). S. Lempp and T. Slaman [115] have shown, that the propertyof Ehrenfeuchtness is Π1
1-complete. W. Calvert, V. S. Harizanov,J. F. Knight, S. Miller [61] have described the complexity of in-dex sets of classical Ehrenfeucht theories. Constructive models ofEhrenfeucht theories are considered in the works by C. J. Ash andT. Millar [29], G. A. Omarova [133], B. Khusainov, A. Nies and
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R. A. Shore [103], A. N. Gavryushkin [69], decidable Ehrenfeuchttheories are in the works by T. Millar [122], [123], R. Reed [156],V. S. Harizanov [13].
The Lachlan Problem, that is solved in the book, is known morethan thirty years. As a direction to the solution of this problem forvarious subclasses of the class of stable theories, an absence of theo-ries T with 1 < I(T, ω) < ω is shown. This absence has been provedfor the class of uncountably categorical theories (J. T. Baldwin,A. H. Lachlan [33]), for superstable theories (A. H. Lachlan [109],D. Lascar [111], S. Shelah [163], U. Sae [160], A. Pillay [143]), fortheories with nonprincipal superstable types (T. G. Mustan [128]),for stable theories, in which dcl(∅) are models (A. Pillay [142]),for normal theories (A. Pillay [143]), for weakly normal (1-based)theories (A. Pillay [145], [146]), for theories admitting nite cod-ings (E. Hrushovski [90]), for unions of pseudo-superstable theories(A. Tsuboi [197]), for theories without dense forking chains (B. Her-wig, J. Loveys, A. Pillay, P. Tanovic, F. O. Wagner [80]). A. Tsuboi[196] has proved that any Ehrenfeucht theory, being a countableunion of ω-categorical theories, is unstable. A. A. Vikent'ev [202]has shown a heredity of non-Ehrenfeuchtness for extensions of non-Ehrenfeucht formula restrictions. P. Tanovic [191] has shown thatany stable theory, interpreting an innite set in pairwise dierentconstants, is non-Ehrenfeucht. He also has proved [193], that ifa theory T is Ehrenfeucht then the set dcl(∅) is nite or the the-ory T has the strict order property.
A development of Theory of simple theories (see. F. O. Wagner[27]; Z. Chatzidakis, A. Pillay [63]; B. Kim, A. Pillay [104]; B. Kim[105]; M. Pourmahdian [151]; S. Shelah [164]), alongside with theLachlan problem for stable theories, has generated a similar problemfor simple theories: the Lachlan problem for simple theories. B. Kim[105] has generalized the Lachlan theorem (see A. H. Lachlan [109])on superstable theories and has shown, that the class of supersimpletheories doesn't contain Ehrenfeucht theories.
So-called powerful types, that always are represented in Ehren-feucht theories (see M. Benda [56]), play an important role for thending of number of countable models. In essence, the proof of ab-sence of Ehrenfeucht theories in aforesaid classes is reduced to theassertion that these classes don't contain theories with nonprincipalpowerful types. Other essential properties, that Ehrenfeucht theo-ries possess, are the non-symmetry of the semi-isolation relation on
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nonempty sets of realizations of powerful types, and also the in-nite weight of nonprincipal powerful types in simple theories (seeA. Pillay [143]; B. Kim [105]). The principles of systematizationof structural properties of Ehrenfeucht theories and their powerfultypes have created in the candidate dissertation by the author [23].
A. H. Lachlan [110] has proved that structures of innite pseu-doplanes are contained in models of ω-categorical stable non-superstable theories. A. Pillay [145] has obtained a similar resultfor stable non-1-based theories. Thus, the positive solution of theLachlan problem is possible only in the class of theories interpretingpseudoplanes.
Interrelations of types in theories are dened in many aspectsby Rudin | Keisler preorders (see M. E. Rudin [157]). These pre-orders have nitely many equivalence classes for Ehrenfeucht theo-ries. D. Lascar [111][113] has investigated various Rudin | Keislerpreorders and has shown, that any powerful type corresponds to thegreatest equivalence class w.r.t. Rudin | Keisler preorder.
E. Hrushovski [93], using a modication of generic Jonsson |Frasse construction (see R. Frasse [68], [10]; B. Jonsson [100],[101]), has disproved the Zilber Conjecture constructing examplesof strongly minimal not locally modular theories in which in-nite groups are not interpreted. His original construction, whichserved as a basis for building of appropriate examples and solv-ing other known model-theoretic problems, has given an impetusto intensive studies of both the Hrushovski construction togetherwith its various (in a broad sense) modications, capable of creat-ing \pathological" theories with given properties (see J. T. Bald-win [2], [34], [35], [36], [46], [47]; A. Hasson [14], [75], [76], [79];A. S. Kolesnikov [16]; J. T. Baldwin, S. Shelah [39]; J. T. Bald-win, K. Holland [41], [42], [44]; A. Baudisch [48], [49]; A. Baudisch,A. Martin-Pizarro, M. Ziegler [53][52]; M. J. de Bonis, A. Nesin[57]; O. Chapuis, E. Hrushovski , P. Koiran, B. P. Poizat [62];D. M. Evans [64], [66], [67]; D. M. Evans, M. E. Pantano [65];A. Hasson, M. Hils [77]; A. Hasson, E. Hrushovski [78]; B. Her-wig [81], [82], [83]; B. Herwig, D. Lascar [84]; K. Holland [85],[86]; E. Hrushovski [89], [91], [92]; E. Hrushovski, B. I. Zilber [94];K. Ikeda [96], [97]; A. A. Ivanov [99]; A. S. Kechris, C. Rosendal[102]; B. Kim, A. S. Kolesnikov, A. Tsuboi [106]; N. Peateld,B. I. Zilber [136]; A. Pillay, A. Tsuboi [147]; B. P. Poizat [148], [149];S. Shelah [166]; S. Solecki [167]; V. V. Verbovskiy [199]; V. V. Ver-
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bovskiy, I. Yoneda [200]; A. M. Vershik [201]; A. Villaveces, P. Zam-brano [203]; I. Yoneda [207]; M. Ziegler [208]; B. I. Zilber [209],[210], [211]), and axiomatic bases, allowing to determine applicabil-ity bounds for that construction (see A. C. J. Bonato [5]; R. Aref'ev,J. T. Baldwin, M. Mazucco [28]; J. T. Baldwin [38][45]; J. T. Bald-win, N. Shi [37]; A. Baudisch [50]; Z. Chatzidakis, A. Pillay [63];D. M. Evans [64]; J. B. Goode [71]; K. Holland [87]; K. Ikeda,A. Pillay, H. Kikyo A. Tsuboi, [98]; D. W. Kueker, M. S. Laskowski[108]; M. S. Laskowski [114]; B. P. Poizat [150]; M. Pourmahdian[151]; R. Rajani [155]; F. O. Wagner [204]).
Relatively the Lachlan problem, B. Herwig [81] has shown a fruit-fulness of the Hrushovski construction, realizing, using it, a smallstable theory with a type having the innite weight.
Now we pass to the statement of results of ve basic Chaptersof the book.
The rst Chapter begins (Section 1.1) with a syntactic char-acterization of the class of complete theories with nitely manycountable models on the basis of Rudin | Keisler preorders anddistribution functions of number of limit models. The basic partof this characterization is distributed to the class of Ehrenfeuchttheories. Section 1.2 is devoted to the denitions of basic cases ofinessential combinations and colorings of models, used below for de-scriptions of intermediate constructions, and also for the solutionof the Lachlan problem. We dene, in Section 1.3, the concept oftype reducibility, according to which a structure of type of predi-cate theory is invariant w.r.t. restrictions of saturated structures tothe sets of realizations of the type. It is shown, that the type re-ducibility property doesn't hold in stable Ehrenfeucht theories. Anexample, realizing an absence of the type reducibility in the class ofstable theories, has constructed. The concept of powerful digraphis dened in Section 1.4. These digraphs, alongside with powerfultypes, are always locally presented in Ehrenfeucht structures. Con-nections of powerful digraphs and powerful types (always presentedin Ehrenfeucht structures) have shown, and properties of structureswith powerful digraphs are investigated. The results, represented inChapter 1, are published in the works [169][175], [177].
We describe (became already classical) semantic generic con-structions in the second Chapter (Section 2.1). Used for the solutionof the Lachlan problem, syntactic generic constructions, generaliz-ing semantic constructions, are dened in Section 2.2. Properties
14
of various classes of syntactic generic constructions are consideredin Sections 2.3 | 2.5. Various kinds of fusions of generic construc-tions are investigated, that also used for constructions of requiredEhrenfeucht theories (Section 2.6). Basic results, described in theChapter 2, are stated in the works [180], [185].
In the third Chapter (Section 3.1), we realize (on the basis ofsyntactic generic construction and of inessential ordered coloring ofacyclic digraph) an example of unstable generic powerful digraph,having unbounded lengths of shortest routes and admitting an ex-pansions till a structure of nonprincipal powerful type. Then, onthe basis of generic powerful digraph, theories with powerful types(Section 3.2) and generic Ehrenfeucht theories with three countablemodels (Section 3.3) are constructed. A modication of genericconstruction is represented, allowing to realize all possible charac-teristics of Ehrenfeucht theories w.r.t. Rudin | Keisler preordersand distribution functions of number of limit models (Section 3.4).These characteristics are generalized in Section 3.5 for the class ofall small theories with nite Rudin | Keisler preorders modulo theVaught conjecture. The description of Rudin | Keisler preordersin the class of small theories is shown in Section 3.6. Modicationsof generic construction of Ehrenfeucht theories, based on non-densestructures of powerful digraphs, and also on structures of powerfultypes without powerful digraphs, are described in Section 3.7. Thebasic results of Chapter 3 are represented in the works [176], [179],[188], [189]. The rst three Chapters, excepting Section 2.6, formthe rst Chapter of the author's thesis for the doctor degree [24].
In the fourth Chapter, we describe (in three steps) examples ofstable generic powerful digraphs on the basis of generic Hrushovski |Herwig construction with prerank functions. At rst, generic con-struction is transferred on bipartite digraphs with colored arcs (Sec-tion 4.1). Then it is transferred from bipartite digraphs to di-graphs without furcations (Section 4.2) and, at last step, from di-graphs without furcations to powerful digraphs (Section 4.3). In Sec-tion 4.4, a lack of the simplied construction of Ehrenfeucht theoriesfrom Chapter 3 is explained, that, by the specicity of construction,besides unstability of structure of powerful digraph generates theformula unstability from the type unstability. Features of genericconstruction, allowing to build stable Ehrenfeucht theories, havewritten in Section 4.5. In Sections 4.6 | 4.9, we describe requiredstable Ehrenfeucht theories with all possible Rudin | Keisler pre-
15
orders and distribution functions of number of limit models on thebasis of stable generic powerful digraphs, using Hrushovski fusionsof generic constructions of powerful digraphs with generic construc-tions of countable family of undirected graphs. Thus, in particular,an existence of stable Ehrenfeucht theories is shown, that solves theLachlan problem. The results of Chapter 4 are stated in the works[178], [181], [182][186].
In the fth, nal Chapter, the family of hypergraphs of primemodels of arbitrary small theory is considered. A mechanism ofthe structural description of models of a theory by these familiesis represented. Thus it is proved, in particular, the key role ofgraph-theoretical constructions for examples of Ehrenfeucht theoriesobtained in the book. Besides, the results of previous Chapters aregeneralized for the class of all small theories. Chapter 5 is formedby the works [187], [190].
Below, we use without specications:| the model-theoretical notations and terminology in Handbook
of Mathematical Logic [3] and in the book by S. V. Sudoplatov,E. V. Ovchinnikova [26] (see also Yu. L. Ershov [7]; Yu. L. Er-shov, S. S. Goncharov [8]; Yu. L. Ershov, E. A. Palyutin [9];C. C. Chang, H. J. Keisler [6]; W. Hodges [15]; A. Pillay [19];B. P. Poizat [20]; G. E. Sacks [21]; S. Shelah [22]; F. O. Wagner [27]);
| the Graph Theory terminology in the book by S. V. Su-doplatov, E. V. Ovchinnikova [25] (see also F. Harary [12]; [17];O. Ore [18]).
C h a p t e r 1CHARACTERIZATIONOF EHRENFEUCHTNESS. PROPERTIESOF EHRENFEUCHT THEORIES
§ 1.1. Syntactic characterization of the class of com-plete theories with nitely many countable mo-dels
In this Section, a syntactic characterization is furnished for theclass of elementary complete theories with nitely many countablemodels, which is the analog of a known theorem by C. Ryll-Nard-zewski [159], that the countable categoricity of a theory is equivalentto a niteness of number of n-types of the theory for every natu-ral n and xed set of free variables. Establishing characterization isbased on classifying the theories by Rudin | Keisler preorders anddistribution functions of a number of models limit over types.
Below, we denote innite models of elementary theories by M,N , . . . (possibly with indexes), and their universes, by M, N, . . ..The type of tuple a in the model M will be denoted by tpM(a) orby tp(a) if the model is given. The set of all types of theory T overthe empty set will be denoted by S(T ) or by S(∅). Considering theset of n-types of T , this set will be denoted by Sn(T ) or by Sn(∅)
A number of pairwise non-isomorphic models of theory T ina power λ is denoted by I(T, λ). A theory T is Ehrenfeucht if1 < I(T, ω) < ω.
Unless specied otherwise, we deal with just countable completetheories. Additionally in this Section, all considered theories arecountable.
17
DEFINITION [56]. A type p(x) ∈ S(T ) is said to be powerfulin a theory T if every model M of T realizing p also realizes everytype q ∈ S(T ), that is M |= S(T ).
The availability of a powerful type implies that T is small , thatis, the set S(T ) is countable, and hence for any type p ∈ S(T ) andits realization a, there exists a model Ma prime over a. Since allprime models over realizations of p are isomorphic, we often denotesuch by Mp.
The condition that p(x) is a powerful type is equivalent to everytype in S(T ) being realized in Mp, that is, Mp |= S(T ). Every typeof ω-categorical theory T is powerful.
Lemma 1.1.1. [56]. Every Ehrenfeucht theory T has a powerfultype.
P r o o f. Assuming on the contrary, by induction, there existsa sequence of types pn ∈ S(∅), such that pn ⊂ pn+1 and Mpn omitspn+1. As Mpm 6' Mpn for m 6= n, then I(T, ω) ≥ ω. ¤
As an illustration we consider the following Ehrenfeucht exam-ples [198] of theories Tn, n ∈ ω, with I(Tn, ω) = n ≥ 3.
E x a m p l e 1.1.1. Let Tn be the theory of a model Mn,formed from a model 〈Q; <〉 by adding of constants ck, k ∈ ω, suchthat lim
k→∞ck = ∞, and by unary predicates P0, . . . , Pn−3 which form
a partition of the set Q of rationals, with
|= ∀x, y ((x < y) → ∃z ((x < z) ∧ (z < y) ∧ Pi(z))),
i = 0, . . . , n−3. The theory Tn has exactly n pairwise non-isomorphicmodels:
a prime model Mn ( limk→∞
ck = ∞);prime models Mn
i over realizations of types pi(x) ∈ S1(∅), iso-lated by sets of formulas ck < x | k ∈ ω∪Pi(x), i = 0, . . . , n−3( limk→∞
ck ∈ Pi);a saturated model Mn (the limit lim
k→∞ck is irrational). ¤
DEFINITION [143]. A tuple a semi-isolates a tuple b (over ∅)if there exists a formula ϕ(x, a) ∈ tp(b/a) for which ϕ(x, a) ` tp(b).In this case we say that the formula ϕ(x, a) witnesses that b is semi-isolated over a.
18
If p ∈ S(T ) then SIp denotes the semi-isolation relation on real-izations of p:
SIp = (a, b) | |= p(a) ∧ p(b) and a semi-isolates b.Notice that the relation SIp forms a preorder for any type p ∈
S(T ). Indeed, if for realizations a, b and c of p, a formula ϕ(x, y) wit-nesses that b is semi-isolated over a and a formula ψ(y, z) witnessesthat c is semi-isolated over b, then the formula ∃y (ϕ(x, y)∧ψ(y, z))witnesses that c is semi-isolated a.
A preorder SIp is called a semi-solation preorder on set of real-ization of type p.
Lemma 1.1.2. [143]. If p ∈ S(T ) is a nonprincipal powerfultype, then nonempty relation SIp is non-symmetric.
P r o o f. Assuming on the contrary, SIp is an equivalence re-lation. Then all realizations of the type p in a model Ma, where|= p(a), are SIp-equivalent, because a semiisolates every tuple of el-ements from Ma. On the other hand, using Compactness Theoremand the nonprincipality of p, there exists a type q(x, y) ∈ S(T ) suchthat p(x)∪ p(y) ⊂ q(x, y) and (a′, b′) 6∈ SIp for any realizations a′ˆb
′
of q. So the type q is omitted in Mp, and we get a contradictionwith the powerfulness of p. ¤
Thus the availability of a non-principal powerful type p(x) pre-sumes the existence of a formula ϕ(x, y), l(x) = l(y), such that, forany (some) realization a of p, the following conditions hold:
(1) ϕ(a, y) ` p(y);(2) ϕ(x, a) 6` p(x), and moreover, there exists a tuple b which
realizes type p and is such that |= ϕ(b, a) and a does not semi-isolate b.
Every formula ϕ(x, y), satisfying conditions 1 and 2, is calleda formula, witnessing on non-symmetry of relation SIp.
The formula (x < y) witnesses on non-symmetry of relation SIpi
in the Ehrenfeucht examples.In what follows in this Section, unless otherwise stated, we deal
with a class of small theories only.Let p and q be types in S(T ). We say that the type p is domi-
nated by a type q, or p does not exceed q under the Rudin | Keisler
19
preorder (written p ≤RK q), if Mq |= p, that is, Mp is an elemen-tary submodel of Mq (written Mp ¹ Mq). Besides, we say thata model Mp is dominated by a model Mq, or Mp does not exceedMq under the Rudin | Keisler preorder , and write Mp ≤RK Mq.
Syntactically, the condition p ≤RK q (and hence also Mp ≤RK
Mq) is expressed thus: there exists a formula ϕ(x, y) such that theset q(y) ∪ ϕ(x, y) is consistent and q(y) ∪ ϕ(x, y) ` p(x). Sincewe deal with a small theory, ϕ(x, y) can be chosen so that, for anyformula ψ(x, y), the set q(y) ∪ ϕ(x, y), ψ(x, y) being consistentimplies that q(y) ∪ ϕ(x, y) ` ψ(x, y). In this event the formulaϕ(x, y) is said to be (q, p)-principal .
Types p and q are said to be domination-equivalent , realization-equivalent , or Rudin | Keisler equivalent (written p ∼RK q) ifp ≤RK q and q ≤RK p. Models Mp and Mq are said to bedomination-equivalent or Rudin | Keisler equivalent (writtenMp ∼RK Mq).
Clearly, domination relations form preorders, and domination-equivalence relations are equivalence relations.
If Mp and Mq are not domination-equivalent then they are non-isomorphic. Moreover, non-isomorphic models may be found amongdomination-equivalent ones.
In Ehrenfeucht examples, models Mnp0
, . . . ,Mnpn−3
are domina-tion-equivalent but pairwise non-isomorphic.
A syntactic characterization for the model isomorphism betweenMp and Mq is given by the following:
Proposition 1.1.3. For any types p(x) and q(y) of a small the-ory T , the following conditions are equivalent:
(1) models Mp and Mq are isomorphic;(2) there exist (p, q)- and (q, p)-principal formulas ϕp,q(y, x) and
ϕq,p(x, y) respectively, such that the set
p(x) ∪ q(y) ∪ ϕp,q(y, x), ϕq,p(x, y)is consistent;
(3) there exists a (p, q)- and (q, p)-principal formula ϕ(x, y), suchthat the set
p(x) ∪ q(y) ∪ ϕ(x, y)is consistent.
20
P r o o f. (1) ⇒ (2). Let Ma and Mb be prime models overrealizations a and b of types p(x) and q(y), respectively.
If there is an isomorphism between Ma and Mb, the existenceof (p, q)- and (q, p)-principal formulas ϕp,q(y, x) and ϕq,p(x, y), sat-isfying the condition that
p(x) ∪ q(y) ∪ ϕp,q(y, x), ϕq,p(x, y)is consistent, follows from the facts that Ma and Mb realize justprincipal types over a and b, respectively, and Ma = M
b′ for some
tuple b′ realizing type q(y).
(2) ⇒ (1). Now, assume that there exist (p, q)- and (q, p)-principal formulas ϕp,q(y, x) and ϕq,p(x, y) such that the set p(x) ∪q(y) ∪ ϕp,q(y, x), ϕq,p(x, y) is consistent. We argue to show thatMa and Mb are isomorphic. The existence of a (p, q)-principal for-mula ϕp,q(y, x) implies that Mb can be chosen to be an elementarysubmodel of Ma. On the other hand, the existence of a (q, p)-principal formula ϕq,p(x, y) and the consistency of p(x) ∪ q(y) ∪ϕp,q(y, x), ϕq,p(x, y) make it possible to choose Mb so that it iselementarily embedded in Ma in a way that a ˆ b is distinguishedconstantly. Since Ma is elementarily embedded in any model con-stantly containing a, Mb too is elementarily embedded in every suchmodel. The fact that every two prime models are isomorphic impliesthat Ma and Mb are isomorphic.
(2) ⇒ (3). Having (p, q)- and (q, p)-principal formulas ϕp,q(y, x)and ϕq,p(x, y), and consistent set
p(x) ∪ q(y) ∪ ϕp,q(y, x), ϕq,p(x, y),we get a required (p, q)- and (q, p)-principal formula ϕ(x, y) ϕp,q(y, x) ∧ ϕq,p(x, y).
The implication (3) ⇒ (2) is obvious. ¤Denote by RK(T ) the set PM of isomorphism types of models
Mp, p ∈ S(T ), on which the relation of domination is induced by≤RK , a relation deciding domination among Mp, that is, RK(T ) =〈PM;≤RK〉. We say that isomorphism types M1,M2 ∈ PM aredomination-equivalent (written M1 ∼RK M2) if so are their repre-sentatives.
Clearly, the preordered set RK(T ) has a least element, which isan isomorphism type of a prime model.
21
Proposition 1.1.4. If I(T, ω) < ω then RK(T ) is a nite pre-ordered set whose factor set RK(T )/∼RK , w.r.t. domination-equiva-lence ∼RK , forms a partially ordered set with greatest element.
P r o o f. That PM is a nite set is obvious, and the fact thatRK(T )/∼RK contains a greatest element follows from the existenceof a powerful type which dominates any type in S(T ). ¤
Below are two obvious remarks.
Remark 1.1.5. A theory T is ω-categorical i |RK(T )| = 1.
Remark 1.1.6. If |RK(T )| = 2, then any non-principal type ispowerful.
In the above-given Ehrenfeucht examples of theories Tn withI(Tn, ω) = n, each preordered set RK(Tn) consists of the least ele-ment and (n− 2) domination-equivalent elements corresponding tothe models Mn
p0, . . . ,Mn
pn−3. Thus all ordered sets RK(Tn)/∼RK
are two-element and linearly ordered.Recall that a model sequence (Mn)n∈ω is called an elementary
chain if Mn is an elementary submodel of Mn+1, n ∈ ω.An elementary chain (Mn)n∈ω is said to be elementary over
a type p ∈ S(T ) if Mn 'Mp for any n ∈ ω.
Proposition 1.1.7. If I(T, ω) < ω then for any countable modelM of a theory T there exists a type p ∈ S(T ) and an elementarychain (Mn)n∈ω over p such that M =
⋃n∈ω
Mn.
P r o o f. Let M be an arbitrary countable model of a smalltheory T . At rst we construct an elementary chain C of primemodels Mai over tuples ai, i ∈ ω, such that M =
⋃i∈ω
Mai . For
this purpose, we enumerate all elements of M: M = bk | k ∈ ω,and also all formulas of the form ϕ(x, c), c ∈ M : Φ = ϕm(x, cm) |m ∈ ω. We shall construct C inductively and, at any step k, somenite sequence of tuples a0, . . . , an will be dened, and also eachthat tuple will be connected with a nite set Xk
i , 0 ≤ i ≤ n, suchthat unions of these sets by all k w.r.t xed i will dene universesof models Mai . If the tuple ai is not dened before the step k thensets X l
i are supposed to be empty for any l < k.
22
At the initial step, we x the tuple a0 〈b0〉 and for the formulaϕm(x, b0) from Φ, having the minimal number and satisfying M |=∃x ϕm(x, b0), we nd a realization dm of a principal complete typep(x, b0) containing ϕm(x, b0). Now we set X0
0 b0, dm.Suppose that at the step k, we already have found tuples a0, . . . ,
an and have formed nite sets Xk0 , . . . , Xk
n satisfying the followingconditions:
1) all elements of ai are contained in the set of elements of ai+1,i < n and belong to Xk
i ;2) b0, . . . , bk ⊆ Xk
n;3) Xk
i ⊂ Xki+1, i < n− 1;
4) for chosen at the step k, a minimal w.r.t m and unconsideredbefore formula ϕm(x, cm), containing only elements of the maximalnonempty set Xk−1
j and satisfying M |= ∃xϕm(x, cm), a realizationdm ∈ M of a principal complete type p(x,Xk−1
j ∪ bk), containingϕm(x, cm), is found such that for any tuple ai with cm ∈ Xk−1
i
and for any tuple d ∈ Xk−1i ∪ dm the type tp(d/ai) is principal;
this realization is added in the minimal set Xki w.r.t i such that
cm ∈ Xk−1i .
At the step k + 1, we consider the element bk+1. If it belongsto Xk
n the sequence a0, . . . , an stays the same and we construct setsXk+1
i adding to Xki an element dm satisfying the conditions 3 and
4 for k + 1 instead of k.If bk+1 6∈ Xk
n and starting with some i0 ≤ n, all types tp(b/ai),b ∈ Xk
i ∪ bk+1, are principal, we again don't extend the sequencea0, . . . , an and add the element bk+1 to the set Xk
i0and to all con-
sequent sets Xki , i0 ≤ i ≤ n. Then we get sets Xk+1
i adding anelement dm satisfying the conditions 3 and 4 for k +1 instead of k.
If some type tp(b/an), b ∈ Xkn ∪ bk+1, is not principal, we add
to the sequence a0, . . . , an the tuple an+1 consisting of all elements ofthe set Xk
n∪bk+1. Then we add this set to the (initially empty) setXk
n+1 and form sets Xk+1i by adding a realization dm of a principal
complete type p(x,Xkn ∪ bk+1) containing the minimal (w.r.t m)
unconsidered before formula ϕm(x, cm) containing only elements ofXk
n and satisfying M |= ∃x ϕm(x, cm), such that for any tuple ai
with cm ∈ Xki and for any tuple d ∈ Xk
i ∪dm, the type tp(d/ai) isprincipal. We add the element dm to the minimal (w.r.t i) set Xk
i ,
23
and also to the consequent sets such that cm ∈ Xkj , i ≤ j ≤ n. Now
we set Xk+1n+1 Xk
n ∪ bk+1, dm.By construction, the sets Xi
⋃k∈ω
Xki are the universes of prime
models Mai over tuples ai. Moreover, we have Mai ≺ Mai+1 andM =
⋃iMai . If the number of indexes i is nite, the model M is
prime over the greatest tuple ai and we add the elementary chainof the models Mai to the countable chain taking the model Mcountably many times.
Since I(T, ω) < ω, we can choose, from the constructed sequence(Mi)i∈ω, an innite subsequence of models (Mij )j∈ω such that allits elements are isomorphic to a model Mp. This sequence is re-quired. ¤
A model M is said to be limit over a type p if M =⋃
n∈ωMn for
some elementary chain (Mn)n∈ω over p and M 6'Mp.
Proposition 1.1.8. A limit model over a type p exists i, forany (some) realization a of type p, there are a realization b of p inMa and a tuple c ∈ Ma such that tp(c/b) is a non-principal type.
P r o o f. Suppose that there exists a limit model M =⋃
n∈ωMn
over p, where Mn ' Mp, M0 = Ma, |= p(a), and there are nob ∈ p(M0) and c ∈ M0 such that tp(c/b) is a non-principal type.Then models Mn (and hence also M) realize just principal typesover any realizations of type p lying in Mn (in M). Hence the modelM is prime over a realization of p, which contradicts the assumptionthat M is limit.
Conversely, assume that for some a |= p1 there are tuples b ∈p(M0) and c ∈ M0 such that q(x, b) = tp(c/b) is a nonprinci-pal type. Our goal is to construct an elementary chain (Man)n∈ω
over p satisfying the following conditions: a0 = b, a1 = a, andtp(an+1/an) = tp(a/b). We argue to show that M =
⋃n∈ω
Man
and Mp are non-isomorphic. By way of contradiction, nd a tupled ∈ p(Man) such that M = Md. By the construction of M, how-ever, the type q(x, an) is omitted in the model Md but is realizedin the model M, a contradiction. ¤
1Here and below, we write a |= p as a re-notation of |= p(a).
24
Corollary 1.1.9. If the semi-isolation relation SIp on realiza-tions of type p in a model Mp is nonsymmetric then there existsa model limit over p.
P r o o f. By Proposition 1.1.8, it suces to notice that if a isa realization of p then there exists a tuple b ∈ p(Ma) such that bdoes not semi-isolate a, and hence tp(a/b) is a non-principal type. ¤
Corollary 1.1.10. If Mp and Mq are domination-equivalentnon-isomorphic models then there exist models that are limit overthe type p and over the type q.
P r o o f. Let a and b be realizations of p and c be a real-ization of q such that Mb ≺ Mc ≺ Ma (which exist in view ofMp ∼RK Mq). Since Mp 6' Mq, the type tp(a/c) is non-principalby Proposition 1.1.3. Hence a limit model over p exists by Propo-sition 1.1.8. The existence of a limit model over q is proved analo-gously. ¤
Proposition 1.1.11. If types p1 and p2 are domination-equiva-lent, and there exists a limit model over p1, then there exists a modelthat is limit over p1 and over p2.
P r o o f. We construct inductively an elementary chain(Man)n∈ω of models such that:
(a) models Man are prime over p1 for even n, and over p2 forodd n;
(b) the model⋃
n∈ωMan is prime neither over p1 nor over p2.
By Proposition 1.1.8, there exists a type q(x, a0), a0 |= p1, whichis not realizable in the model Ma0 but is realizable in some Mb ÂMa0 , b |= p1. Denote by Ma1 the prime model over a realizationa1 of type p2, which is an elementary extension of Mb (such existssince p1 and p2 are domination-equivalent). At even steps 2n + 2,we extend the model Ma2n+1 , a2n+1 |= p2, to a model Ma2n+2 ,a2n+2 |= p1, which realizes type q(x, a2n). At odd steps, we extendMa2n+2 to Ma2n+3 , a2n+3 |= p2. Clearly, the model
⋃n∈ω
Man is limitover p1 and over p2. ¤
Limit models M and N over a type p are said to be equivalent(written M ∼ N ) if there exist elementary chains (Mn)n∈ω and(Nn)n∈ω over p satisfying the following conditions:
25
1) M =⋃
n∈ωMn, N =
⋃n∈ω
Nn;
2) there exist constant expansions M′n+1 = 〈Mn+1, c〉c∈M ′
n
and N ′n+1 = 〈Nn+1, c〉c∈N ′
n, n ∈ ω, M′
0 = M0, N ′0 = N0, such that
M′n+1 ' N ′
n+1, n ∈ ω.The following proposition is obvious.Proposition 1.1.12. If M and N are limit models over a type p
then M' N i M∼ N .Let M ∈ RK(T )/∼RK be the class consisting of isomorphism
types of domination-equivalent models Mp1 , . . . ,Mpn . Denote byIL(M) the number of equivalence classes of models each of which islimit over some type pi.
Propositions 1.1.4, 1.1.7, 1.1.12 and Corollary 1.1.10 can be com-bined to yield
Theorem 1.1.13. For any countable complete theory T , the fol-lowing conditions are equivalent:
(1) I(T, ω) < ω;(2) T is small, |RK(T )| < ω and IL(M) < ω for any M ∈
RK(T )/∼RK .If (1) or (2) holds then T possesses the following properties:(a) RK(T ) has the least element M0 ((an isomorphism type of
a prime model) and IL(M0) = 0;(b) RK(T ) has the greatest ∼RK-class M1 (a class of isomor-
phism types of all prime models over realizations of powerful types)and |RK(T )| > 1 implies IL(M1) ≥ 1;
(c) if |M| > 1 then IL(M) ≥ 1.Moreover, the following decomposition formula holds: :
I(T, ω) = |RK(T )|+|RK(T )/∼RK |−1∑
i=0
IL(Mi),
where M0, . . . , ˜M|RK(T )/∼RK |−1 are all elements of the partially or-dered set RK(T )/∼RK .
Notice that, by Propositions 1.1.3 and 1.1.12, the conditionsspecied in item (2) of Theorem 1.1.13 admit of a syntactic rep-resentation; so, this theorem is an analog of the Ryll-Nardzewskitheorem providing a syntactic characterization of ω-categoricity.
26
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Let p1, . . . , pn ∈ S(T ) be types the prime models over which arerepresentatives of all isomorphism types in a nite preordered setRK(T ) of T . We say that the theory T possesses the consistentextension property of chains of models prime over tuples (CEP) if,for any type pi, every two limit models over pi are equivalent.
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©ª
0
0
1
TT
TT
····
····
TT
TT
•
• •
•
h
h
h h
0
1
0 0
Figure 1.2
27
Proposition 1.1.11 implies that if T insists on (CEP ) thenIL(M) ≤ 1 for any M ∈ RK(T )/∼RK . Since there exists no modelthat is limit over a principal type, (CEP ), for |RK(T )/∼RK | = 2,implies the existence of a unique (up to isomorphism) model M theisomorphism type of which does not lie in RK(T ) (and M, in thisevent, is saturated).
Thus the following theorem, based on Theorem 1.1.13, is valid.
Theorem 1.1.14. Let T satisfy (CEP ). Then the followingconditions are equivalent:
(1) I(T, ω) < ω;(2) T is small and |RK(T )| < ω.
In this event, we have the inequality
I(T, ω) ≤ |RK(T )|+ |RK(T )/∼RK | − 1,
which turns into equality for |RK(T )/∼RK | ≤ 2.
Theorem 1.1.14 implies
Corollary 1.1.15. For any complete theory T , the following areequivalent:
(1) I(T, ω) = 3;(2) T is small, possesses (CEP) and |RK(T )| = 2. ¤
§ 1.2. Inessential combinations and coloringsof models
In the rst paragraph of this Section, we dene operations ofinessential and almost inessential combinations of models, and alsoof theories. A bases of (almost) inessential combinations of theoriesare states, and also preservations of properties of smallness and λ-stability at transformation to (almost) inessential combinations oftheories are shown. A sucient condition is resulted for the inessen-tiality of combinations of theories in assumption of inessentiality ofcombinations of their models.
In the second paragraph, concepts of coloring of model, of col-ored model and of colored theory are dened, and the results of therst item are transferred for (almost) inessential colorings. An ex-ample is given, showing that inessential colorings of models don'timply that corresponding theories have an inessential colorings. Also
28
there is an example, showing a separability of the class of theo-ries with almost inessential colorings from the class of theories withinessential colorings.
In the third paragraph, a concept of ordered coloring is dened,the role of such colorings in constructions of Ehrenfeucht theoriesis investigated, and an example is given of ω-stable theory with anordered coloring, inducing a continuum of pairwise non-isomorphiclimit models over a type.1. Combinations of models and theories. Recall (seeU. Sae, E. A. Palyutin, S. S. Starchenko [162]) that a theory Tis said to be ∆-based , where ∆ is some set of formulas withoutparameters if any formula of T is equivalent in T to a Booleancombination of formulas of ∆.
A theory T is said to be almost ∆-based , where ∆ is a set offormulas without parameters if there exists a function f : ω →ω such that any formula ϕ(x1, . . . , xn) of T is equivalent in T toa formula of form
∃y1 . . .∃yf(n) ψ(x1, . . . , xn, y1, . . . , yf(n)),
where ψ(x1, . . . , xn, y1, . . . , yf(n)) is a Boolean combination of for-mulas of ∆.
Recall, that the set of all (complete and uncomplete) types overa set A is denoted by ⊆S(A).
A type q(x) ∈ ⊆S(A) is isolated or is dened by a set Φ(x, A)of formulas of q if Φ(x,A) ` q(x).
The following two assertions are clear.
Lemma 1.2.1. If a type q(x) ∈ ⊆S(A) is isolated by a setΦ(x,A) and the type Φ(x,A) is isolated by a set Ψ(x,A) then q(x)is isolated by Ψ(x,A).
Lemma 1.2.2. If |= Φ(a, b) then the type tp(a b) is isolated byΦ(x, y) i the type tp(b/a) is isolated by Φ(a, y) and the type tp(a)
is isolated by∃y
(∧i
ϕi(x, y))| ϕi(x, y) ∈ Φ(x, y)
.
Recall that a countable model M of T is weakly ω-universal ifany type over the empty set is realizable in M that is M |= S(T ).
Let ∆ be a set of formulas of a theory T , p(x) be a type of T ,lying in S(T ). The type p(x) is said to be ∆-based if p(x) is isolatedby a set of formulas ϕδ ∈ p, where ϕ ∈ ∆, δ ∈ 0, 1.
29
The following lemma, being a corollary of Compactness Theo-rem, noticed in U. Sae, E. A. Palyutin, S. S. Starchenko [162].
Lemma 1.2.3. A theory T is ∆-based i for any tuple a ofweakly ω-universal model of T , the tp(a) is ∆-based.
Lemma 1.2.4. A theory T is almost ∆-based i for any tuplea of weakly ω-universal model M of T there exists a tuple b ∈ M ,containing all coordinates of a and such that tp(b) is ∆-based.
P r o o f. Suppose a theory T is almost ∆-based and a is a tupleof weakly ω-universal model M of T . So tp(a) is isolated by someset
∃y(∧
i
ϕi(x, y)
)| ϕδi
i (x, y) ∈ ∆
.
Now using Compactness Theorem and weak ω-universality of M,there exists a tuple b ∈ M , extending a and satisfying all formu-las ϕδi
i (x, y). Since, by the condition, the set of formulas ϕδii (a, y)
isolates tp(b/a), using Lemma 1.2.2, we get the ∆-baseness of tp(b).Suppose now that for any tuple a of weakly ω-universal modelM
of T , there exists a tuple b ∈ M , extending a and such that tp(b) is∆-based. Then by Compactness Theorem, there exists a function f :ω → ω, bounding minimal lengthes of tuples b depending of lengthesof tuples a. On the other hand, we have that any consistent formulaϕ(x) is deducible from some consistent formula of form ∃y ψ(x, y),where ψ(x, y) is a conjunction of formulas and negations of formulasfrom ∆, l(y) ≤ f(l(x)). By Compactness Theorem we get thatthe formula ϕ(x) is equivalent to a disjunction of formulas of form∃y ψ(x, y), and so to a formula of form ∃y ψ(x, y), where ψ(x, y) is aBoolean combination of formulas of ∆. Thus the theory T is almost∆-based. ¤
Let M1 and M2 be models of disjoint languages Σ1 and Σ2
respectively such that M1 = M2. A model M of the languageΣ1 ∪ Σ2 is said to be a combination of models M1 and M2 if M =M1 and interpretations of language symbols of M coincide withcorrespondent interpretations in M1 and M2. We denote M byComb(M1,M2).
A theory T is said to be a combination of theories T1 and T2
over models Mi |= Ti, i = 1, 2, if T = Th (Comb(M1,M2)).Let a be a tuple in a model Comb(M1,M2). a type tpM(a) is
said to be an inessential combination of types tpM1(a) and tpM2
(a)
30
if tpM(a) is isolated by tpM1(a) ∪ tpM2
(a). The set of tuplesa ∈ M , such that tpM(a) is an inessential combination of tpM1
(a)and tpM2
(a), will be denoted by IECTM.A combination of models M = Comb(M1,M2) is said to be
inessential (written M = IEC(M1,M2)) if IECTM consists of alltuples in M. A combination of models M = Comb(M1,M2) issaid to be almost inessential (written M = AIEC(M1,M2)) if forany tuple a ∈ M , there exists a tuple b ∈ IECTM, extending a.
A combination T of theories T1 and T2 is said to be (almost)inessential if M = IEC(M1,M2) (accordingly M =AIEC(M1,M2)) for any model M |= T , where Mi is a restrictionof M to the language Σ(Ti), i = 1, 2.
Clearly, any inessential combination of theories is almost inessen-tial.
Lemma 1.2.5. Let T be a combination of theories T1 and T2,M be a weakly ω-universal model of T . Then the following condi-tions are equivalent:
(1) the combination T is (almost) inessential;(2) M = IEC(M1,M2) (M = AIEC(M1,M2)), where Mi is
a restriction of M to the language Σ(Ti), i = 1, 2.
P r o o f is obvious.
Theorem 1.2.6. Let T be a combination of ∆i-based theoriesTi, i = 1, 2. Then the following conditions are equivalent:
(1) the combination T is inessential;(2) T is (∆1 ∪∆2)-based.P r o o f. (1) ⇒ (2). Suppose that T is an inessential combina-
tion of theories T1 and T2, M be a weakly ω-universal model of T .By Lemma 1.2.3, it suces to show that for any tuple a ∈ M , itstype tpM(a) is (∆1 ∪∆2)-based. Denote by Mi a restriction of Mto the language Σ(Ti), i = 1, 2. Since by Lemma 1.2.5 we have M =IEC(M1,M2), the type tpM(a) is isolated by tpM1
(a) ∪ tpM2(a).
As Ti is ∆i-based then, by Lemma 1.2.3, the type tpMi(a) is iso-
lated by some set Φi(x) of formulas and negations of formulas of∆i, i = 1, 2. Then by Lemma 1.2.1, the type tpM(a) is isolated byΦ1(x) ∪ Φ2(x). Thus, the type tpM(a) is (∆1 ∪∆2)-based.
(2) ⇒ (1). Let T be a (∆1 ∪ ∆2)-based theory, i.e., any typetpM(a) of tuple a in a weakly ω-universal model M is isolated by
31
some set Φ(x) of formulas and negations of formulas of ∆1 ∪ ∆2.Since Σ(T1)∩Σ(T2) = ∅ then Φ(x) = Φ1(x)∪Φ2(x), where Φi(x) isthe set of formulas in Φ(x) of the language Σ(Ti). Here, by Lemma1.2.3, the set Φi(x) isolates a type tpMi
(a), whereMi is a restrictionof M to the language Σ(Ti). Since Φi(x) ⊂ tpMi
(a), then tpM1(a)∪
tpM2(a) isolates tpM(a). As M is a weakly ω-universal model, then
by Lemma 1.2.5, T is an inessential combination of T1 and T2. ¤
Theorem 1.2.7. Let T be a combination of ∆i-based theoriesTi, i = 1, 2. Then the following conditions are equivalent:
(1) the combination T is almost inessential;(2) T is almost (∆1 ∪∆2)-based.
P r o o f is similar to the proof of Theorem 1.2.6 using Lemma1.2.4 instead of Lemma 1.2.3. ¤
Recall, that a theory T is said to be λ-stable, where λ is aninnite cardinality if for any set A of power λ, the number of typesover A is not more than λ, that is |S(A)| ≤ λ.
Theorem 1.2.8. If T is an almost inessential combination oftheories T1 and T2 then T λ-stable (small) i T1 and T2 are λ-stable(small).
P r o o f. As T1 and T2 are restrictions of T , then the λ-stability(smallness) of T implies λ-stability (smallness) of T1 and T2.
Suppose that T1 and T2 are λ-stable. Consider a model M of T ,having the cardinality λ, and its elementary extension M′ of cardi-nality λ, including with any tuple a its extending tuple b ∈ IECTM′ .Denote by M′
i a restriction of M′ to the language Σ(Ti), i = 1, 2.As Ti are λ-stable, we have |S(Mi)| ≤ λ. On the other hand, analmost inessentiality of the combination of T1 and T2 implies
|S(M)| ≤ |S(M ′)| ≤ |S(M ′1)| · |S(M ′
2)| ≤ λ · λ = λ.
As considered model M is arbitrary, the theory T is λ-stable.Suppose now, that T1 and T2 are small, i.e., |S(T1)| = |S(T2)| =
ω. Then an almost inessentiality of the combination of T1 and T2
implies |S(T )| ≤ |S(T1)| · |S(T2)| = ω · ω = ω, that is the theory Tis small too. ¤
Let p(x) be a type in S(T ), Φ(x) be a set of formulas ϕn(x),n ∈ ω such that ` ϕn+1(x) → ϕn(x) and p(x) is isolated by Φ(x).
32
Consider a model M of T . A sequence (an)n∈ω of tuples in M issaid to be dening for p(x) (over Φ(x)) if |= ϕn(an) for any n ∈ ω.A dening sequence of type p(x) is said to be regular in M if thetype p(x) has a realization in M. Otherwise a dening sequence issaid to be irregular in M.
Clearly, any sequence of tuples may be dening for unique type.So it is possible not to indicate types correspondent to regular se-quences.
If (an)n∈ω is a regular in a model M, dening sequence of typep(x) and M |= p(a), then we say that a is a limit of sequence(an)n∈ω in M and it will be denoted by a ∈
(lim
n→∞ an
)M
. Here, the
set(
limn→∞ an
)M
coincides with the set p(M) of realizations of p(x)in M.
Proposition 1.2.9. Let M be an inessential combination ofweakly ω-universal models M1 and M2 such that any regular se-quence (an)n∈ω in Mi is regular in M2−i, i = 0, 1, and moreover
(lim
n→∞ an
)M1
∩(
limn→∞ an
)M2
6= ∅.
Then M is a weakly ω-universal model and Th(M) is an inessentialcombination of theories Th(M1) and Th(M2).
P r o o f. Consider an arbitrary type p(x) ∈ S(∅) of theoryTh(M) and show that p(x) is realizable in M. Indeed, let (an)n∈ω
be a dening sequence of type p(x) in M. Then (an)n∈ω is deningin M1 and in M2. A weak ω-universality of M1 and M2 impliesa regularity of this sequence in M1, and also in M2. Moreover,there exists a tuple a such that a ∈
(lim
n→∞ an
)M1
∩(
limn→∞ an
)M2
. Asthe combination of models is inessential, the set tpM1
(a)∪ tpM2(a)
isolates the type tpM(a). But tpM1(a)∪ tpM2
(a) ⊂ p(x), so p(x) =tpM(a) and a is a realization of type p(x) in M. Thus, M is aweakly ω-universal model.
An inessentiality of the combination of theories Th(M1) andTh(M2) follows from the weak ω-universality of M by Lemma1.2.5. ¤
33
2. Colored models. Let M be a model. Any function Col :M → λ ∪ ∞, where λ is a power and ∞ is a symbol of innity,is said to be a coloring of model M. Here, for any a ∈ M , a valueCol(a) is said to be a color of element a. A pair 〈M, Col〉 is said tobe a colored model .
Below, colored models 〈M, Col〉 will be identied with expan-sions of M by unary predicates Colµ = a ∈ M | Col(a) = µ,µ < λ. Clearly, a colored model 〈M, Col〉 is a combination ofM with a coloring of its universe, i.e., with a model 〈M, Col〉 =〈M ; Colµ〉µ<λ: 〈M, Col〉 = Comb(M, 〈M, Col〉).
A coloring Col of model M is said to be innerly inessential if,for any tuple a ∈ M , a type tp〈M,Col〉(a) is isolated by types of ain M, and also by colors of elements of a.
Clearly, an inner inessentiality of coloring Col of model M isequivalent to equality 〈M, Col〉 = IEC(M, 〈M, Col〉).
A coloring Col of model M is said to be innerly almost inessen-tial if, for any tuple a ∈ M , there exists a tuple b ∈ M extending aand such that tp〈M,Col〉(b) is isolated by type of b in M, and alsoby colors of elements of b.
An inner almost inessentiality of coloring Col of models M ischaracterized by an equality 〈M, Col〉 = AIEC(M, 〈M, Col〉).
For any model M′ |= Th(〈M, Col〉) a coloring Col′ : M ′ →λ ∪ ∞ is dened naturally by the following rules:
1) Col′(a) = µ if M′ |= Colµ(a);2) Col′(a) = ∞ if M′ 6|= Colµ(a) for any µ < λ.Below, models M′ will be denoted by 〈M′, Col′〉, and by M′ we
shall denote a restriction of model 〈M′, Col′〉 to language Σ(M).Any expansion T ′ of theory T by pairwise inconsistent unary
predicates Colµ, µ < λ, is said to be a colored theory . Clearly, anycolored theory is a theory of some colored model 〈M, Col〉, whereM |= T .
A coloring Col of model M is said to be (almost) inessential if,for any model 〈M′, Col′〉 of colored theory Th(〈M, Col〉), a corre-spondent coloring Col′ is innerly (almost) inessential.
The following example shows that an inner inessentiality of col-oring of model doesn't imply an inessentiality of the coloring.
E x a m p l e 1.2.1. Let M be a model consisting of constantsci
n | n ∈ ω, i ∈ 0, 1, 2, and expanded by substitution f actingby f(c0
n) = c22n, f(c2
2n) = c0n, f(c1
n) = c22n+1, f(c2
2n+1) = c1n, n ∈ ω.
Consider a coloring Col : M → 0, 1, 2, dened by Col(cin) = i,
34
n ∈ ω, i ∈ 0, 1, 2. Clearly, any coloring of M is innerly inessential,because a type p(x) of any tuple a ∈ M is isolated by some set offormulas (xj ≈ c
ijnj ) | 1 ≤ j ≤ l(x). But a coloring Col′ of weakly
ω-universal model 〈M′, Col′〉 |= Th(〈M, Col〉) is not inessential.Indeed, consider elements ak ∈ M ′ such that|= Col2(ak) ∧ ¬(ak ≈ c2
n) ∧ ∃xk (Colk(xk) ∧ (f(xk) ≈ ak)),
n ∈ ω, k = 0, 1. Clearly, tpM′(a0) = tpM′(a1) and tp〈M ′,Col′〉(a0) =tp〈M ′,Col′〉(a1), but tp〈M′,Col′〉(a0) 6= tp〈M′,Col′〉(a1), i.e.,
〈M′, Col′〉 6= IEC(M′, 〈M ′, Col′〉). ¤Recall, that for any set A of T the union of sets of solutions of
formulas ϕ(x, a), a ∈ A such that |= ∃=nx ϕ(x, a) for some n ∈ ω(accordingly |= ∃=1x ϕ(x, a)) is said to be an algebraic (denable)closure of A. An algebraic closure of A is denoted by acl(A) and itsdenable closure, by dcl(A).
Using the following assertion, it is easy to construct examples ofinnerly almost inessential colorings, being not innerly inessential.
Proposition 1.2.10. If a colored model 〈M, Col〉 contains tu-ples a, b and elements c, d such that tpM(a ˆ c) = tpM(b ˆ d),tp〈M,Col〉(a) = tp〈M,Col〉(b), c ∈ dcl(a), d ∈ dcl(b) and Col(c) 6=Col(d), then the coloring Col is not innerly inessential.
P r o o f. Notice, that tp〈M,Col〉(a) 6= tp〈M,Col〉(b), because anexistence of automorphism f of homogeneous extension of model〈M, Col〉, transforming a to b, implies f(c) = d, that is impossiblefor Col(c) 6= Col(d). Since tpM(a) = tpM(b) and tp〈M,Col〉(a) =tp〈M,Col〉(b), then a, b 6∈ IECTM. Thus, the coloring Col is notinnerly inessential. ¤
E x a m p l e 1.2.2. Consider a system Γ = 〈a, b, c, d; (a, b),(b, a), (c, d), (d, c)〉 and its innerly almost inessential coloring Coldened by equalities Col(a) = Col(b) = Col(c) = 0, Col(d) = 1.The coloring Col is not innerly inessential, as tpΓ(a) = tpΓ(c) andtp〈Γ,Col〉(a) 6= tp〈Γ,Col〉(c). ¤
Notice, that for any colored model 〈M′, Col′〉, the the-ory Th(〈M ′, Col′〉) is totally transcendental and ∆Col-based, where∆Col is a closure of (x ≈ y) ∪ Colµ(x) | µ < λ w.r.t. substitu-tions of variables.
The following Theorems hold in view of Theorems 1.2.61.2.8.
35
Theorem 1.2.11. Let Col be a coloring of model M of ∆-basedtheory T . The following conditions are equivalent:
(1) Col is an (almost) inessential coloring;(2) Th(〈M, Col〉) is an (almost) (∆ ∪∆Col)-based theory.Theorem 1.2.12. If Col is an almost inessential coloring of
model M of cardinality |Σ(M)|+ ω, then Th(〈M, Col〉) is λ-stable(small) i Th(M) is λ-stable (small).
3. Ordered colorings. Let M be a model of theory T , ϕ(x, y)be a formula of T . A coloring Col : M → λ ∪ ∞ (where λ is aninnite cardinality) is said to be ϕ-ordered if the following conditionshold:
(a) for any µ ≤ ν < λ there exist elements a, b ∈ M such that|= Colµ(a) ∧ Colν(b) ∧ ϕ(a, b);
(b) if µ < ν < λ then there are no elements c, d ∈ M such that|= Colµ(c) ∧ Colν(d) ∧ ϕ(d, c).
Recall, that a theory T is said to be transitive if T has unique1-type over the empty set.
A coloring Col of modelM is said to be n-inessential , n ∈ ω\0,if (M ′)n ⊆ IECT〈M′,Col′〉 for any model 〈M′, Col′〉 |= Th(〈M, Col〉).
Clearly, any inessential coloring is n-inessential for any n ≥ 1.Notice, that if Col : M → λ ∪ ∞ is a surjective 1-inessential
coloring of model M of transitive theory T , then the set of 1-typesof Th(〈M, Col〉) over ∅ consists of types pµ(x), µ ∈ λ∪∞, wherepµ(x) is a type isolated by formula Colµ(x), µ ∈ λ, p∞(x) is a non-principal type isolated by set of formulas ¬Colµ(x) | µ < λ.
In the theory T3 with three countable models, being the Ehren-feucht example, an expansion of a model of transitive the-ory Th(〈Q;<〉) by constants ck, k ∈ ω, can be interpretable asan inessential coloring Col given by the following conditions:
Col(a) =
0 if a < c0,2k + 1 if a = ck,2k + 2 if ck < a < ck+1.
It's easy to see that the coloring Col is ϕ-ordered, where ϕ(x, y) x < y. Moreover, a relation SIp∞ on the set of realizations of pow-erful type p∞ is non-symmetric, that is witnessed by the formula ϕ.In Ehrenfeucht examples, n ≥ 4, constant expansions of models〈Q; <, P0, . . . , Pn−3〉 can be also considered as colored models withinessential ordered colorings.
36
Consider a sucient conditions for a ϕ-ordered 1-inessential col-oring to imply a non-symmetry of relation SIp∞ witnessed by ϕ.
Proposition 1.2.13. Let ϕ(x, y) be a principal (i.e., isolating acomplete type) formula of transitive theory T , Col be an 1-inessentialϕ-ordered coloring of model M of theory T such that 〈M′, Col′〉 |=Th(〈M, Col〉) and 〈M′, Col′〉 |= ϕ(a, b) imply (a, b) ∈ IECT〈M′,Col′〉.Then for any (i.e., some) realization a of type p∞(x) the followingconditions hold:
1) if |= ϕ(a, b) then |= p∞(b) and a semi-isolates b;2) if |= ϕ(a, b) then b doesn't semi-isolate a.P r o o f. 1. Assume on the contrary that |= p∞(a), |= ϕ(a, b)
and 6|= p∞(b). Then for some µ the set ¬Colν(x) | ν < λ ∪ϕ(x, y), Colµ(y) is consistent and, in particular, the set ¬Colν(x) |ν ≤ µ ∪ ϕ(x, y), Colµ(y) is consistent too. So there exists α > µsuch that
|= ∃x, y (Colµ(y) ∧ Colα(x) ∧ ϕ(x, y)) .
It contradicts to point b) of the denition of ϕ-ordering of coloringCol. Thus, |= ϕ(a, b) implies |= p∞(b) and a semi-isolates b.
2. Assume on the contrary that |= p∞(a), |= ϕ(a, b) and b semi-isolates a. By the condition of Proposition, the formula ϕ(x, b) cannot witness to a semi-isolation of a over b. On the other hand, thereexists a formula ψ(x, y) such that |= ψ(a, b) and ψ(x, b) ` p∞(x).Moreover, the set p∞(x) ∪ p∞(y) ∪ ϕ(x, y) ∧ ψ(x, y) is consistent.By Compactness Theorem, a non-principality of p∞(x) implies aconsistency of p∞(x) ∪ p∞(y) ∪ ϕ(x, y) ∧ ¬ψ(x, y). It means thatthe set ¬Colµ(x) ∧ ¬Colµ(y) | µ < λ ∪ ϕ(x, y) doesn't semi-isolate a complete type. This one is a contradiction to the conditionsthat the formula ϕ(x, y) is principal in T and (a, b) ∈ IECT〈M′,Col′〉for any (a, b) with |= ϕ(a, b). Thus, |= ϕ(a, b) and |= p∞(a) implythat b doesn't semi-isolate a. ¤
Notice, that the conclusion of Proposition 1.2.13 is true if weassume that ϕ(x, y) is a disjunction of principal formulas.
Recall several notions of Graph Theory. An algebraic systemΓ = 〈X; Q〉 with a (nonsymmetric, symmetric) binary relation Q issaid to be a graph (accordingly a directed graph or, in abbreviatedform, a digraph, an undirected graph or, in abbreviated form, anundigraph). Furthermore, the set X is said to be a set of vertices andthe relation Q is a set of arcs od graph Γ. Any nonempty sequence
37
S = (a0, . . . , an) of vertices in a graph Γ, such that Γ |= Q(ai, ai+1),i = 0, . . . , n − 1, is said to be a route in Γ. Here, the route S isalso said to be a (a0, an)-route and the number n is the length ofS. A cycle in a digraph Γ is an arbitrary (a, a)-route of nonzerolength. If digraph doesn't have cycles it is said to be acyclic. Agraph 〈X;Q〉 is said to be connected if any two dierent verticesa, b ∈ X are connected by a (a, b)-route in graph 〈X; Q∪Q−1〉. Any(a, a)-route (a0, . . . , an) of nonzero length in an undirected graph Γ,such that there are no arcs (ai, ai+1) that are repeated or coincidewith (aj+1, aj) for ai 6= ai+1, is said to be a cycle in Γ. An undigraphΓ is acyclic Γ doesn't have cycles.
The following example, constructed on the base of free directedpseudoplane, found independently by A. Pillay [146] and by theauthor [169], shows, that dierent elementary chains over a sametype may generate non-isomorphic limit models forming continuumpairwise non-isomorphic limit models.
E x a m p l e 1.2.3. Consider a countable model M0 of a con-nected acyclic digraph 〈M0;Q〉 with acyclic undigraph 〈M0; Q∪Q−1〉such that every element has innitely many images and innitelymany preimages. The system M0 has constructed independentlyby A. Pillay [146] and by the author [169] for a realization of non-symmetric semiisolation relation in the class of stable theories. Thesystem M0 is said to be a free directed pseudoplane.
Expand the signature by new binary predicates Q0 and Q1, form-ing a partition of the predicate Q with the following condition: forany element a ∈ M0 there exist innitely many images and innitelymany preimages with respect to Q0 and to Q1. Dene then a 1-inessential Q-ordered coloring Col : M0 → ω ∪ ∞ of the resultingmodel so that every element of color n has the following:
(1) innitely many images of color µ relative to Q0 and to Q1
for any µ ≥ n (including ∞);(2) innitely many preimages of color m relative to Q0 and to
Q1 for any m ≤ n.The ω-stability of Th(〈〈M0; Q, Q0, Q1〉, Col〉) is followed from
its ∆-baseness (implied by an acyclicity of structure), where ∆ isthe least closed, relative substitutions of variables, set of formulaswithout more than two free variables such that this set contains aformula (x ≈ y) and satises the following condition: if ϕ(x, y) ∈ ∆then ∃z (ϕ(x, z) ∧ Qδ1
i (z, y) ∧ Colδ2n (z)) ∈ ∆, where δ1,∈ −1, 1,i, δ2 ∈ 0, 1, Q1
i (x, y) = Q(x, y), Q−1i (x, y) = Q(y, x), Col1n(z) =
38
Coln(z), Col0n(z) = ¬Coln(z). Here, the countable number of 1-types over every countable set A is guaranteed by the countablenumber of possibilities of distance distributions from elements of Ato realizations of types.
The ω-stability of Th(〈〈M0; Q, Q0, Q1〉, Col〉) implies an exis-tence of prime model Mp∞ over a realization of type p∞(x). ByProposition 1.2.13, the relation SIp∞ is nonsymmetric and it is wit-nessed by formulas Q0(x, y) and Q1(x, y).
We argue to show that there exist 2ω pairwise non-isomorphiclimit models over p∞. For this goal to be met, we construct in-ductively elementary chains (Mα¹n)n∈ω, α ∈ 2ω, over p∞. Wetake by Mα¹0 an arbitrary prime model over a realization aα¹0of p∞. If models Mα¹0, . . . ,Mα¹n are already constructed, andMα¹n is a prime model over a realization aα¹n of type p∞, thenas Mα¹n+1 we take a prime model over a realization aα¹n+1 of p∞,where Mα¹n ≺ Mα¹n+1 and |= Qα(n)(aα¹n+1, aα¹n). Denote themodel
⋃n∈ω
Mα¹n by Mα. Sequences α and β in 2ω are said to beequivalent if there exist k,m ∈ ω such that α(k +n) = β(m+n) forall n ∈ ω. Clearly, models Mα and Mβ i α and β are equivalent.Since every equivalence class is countable, there are 2ω equivalenceclasses. Choosing one model in each class yields 2ω pairwise non-isomorphic limit models over p∞. ¤
§ 1.3. Type reducibility and powerful typesIn this Section, we dene concepts of p-principal p-type and of
reducibility of theory over a type and prove, on the one hand, that,for a theory T without the strict order property and with a non-principal powerful type p, there is a non-p-principal p-type and Tis not reducible over p. On the other hand, we show an example ofω-stable theory, having a non-p-principal p-type such that this typeis realized in models Mp.
Let p(x) be a type in S(T ). A type q(x1, . . . , xn) ∈ S(T ) issaid to be a (n, p)-type if q(x1, . . . , xn) ⊇
n⋃i=1
p(xi). The set of all(n, p)-types of T is denoted by Sn,p(T ) and elements of Sp(T ) ⋃n∈ω\0
Sn,p(T ) are p-types.
Without loss of generality of the results, we will consider in thisSection for simplicity that p is a type in S1(∅).
39
A type q(y) in S(T ) is said to be p-principal if there is a formulaϕ(y) ∈ q(y) such that∪p(yi) | yi ∈ y ∪ ϕ(y) ` q(y).
The following lemma is obvious.Lemma 1.3.1. For any type p and natural n ≥ 1, the following
conditions are equivalent:(1) the set of (n, p)-types with free variables x1, . . . , xn is innite;(2) there is a non-p-principal (n, p)-type.
Let M be a countable saturated model of a theory T having apredicate language. Consider, induced by M, a subsystem p(M) =〈p(M); Σ(T )〉 of language Σ(T ) of T with the universe p(M) = a ∈M | |= p(a) and relations R(p(M)) = R(M) ∩ (p(M))µ(R), R ∈Σ(T ). Denote the theory Th(p(M)) by Tp.
A theory is said to be reduced over a type p if T and Tp admitsthe quantier elimination.
Proposition 1.3.2. If a small theory T is reduced over a typep, then, in a model Mp, any non-p-principal p-type is omitted.
P r o o f. Notice, that by the quantier elimination of T and Tp,there exists a bijection ·p : Sp(T ) → S(Tp) such that qp(p(M)) =q(M), q ∈ Sp(T ), and the restriction of this bijection on the set ofp-principal types implements a one-to-one correspondence with theset of principal types of Tp. Denote a prime model of Tp by M0, thatexists by the smallness of T . Assume that, in Mp, a non-p-principaltype q(y) in Sn
p (T ) is realized. Then there is a quantier-free formulaψ(x, y) of T such that
T ` ∃x (ϕ(x) ∧ ∃y ψ(x, y) ∧ ∀y (ψ(x, y) → χ(y)))
for any ϕ(x) ∈ p(x) and quantier-free formulas χ(y) ∈ q(y). Ithence follows that
Tp ` ∃x (∃y ψ(x, y) ∧ ∀y (ψ(x, y) → χ(y))),
where χ(y) is a quantier-free formula of type qp(y). Since Tp is atransitive theory admitting the quantier elimination, there is anelement a ∈ M0 such that
M0 |= ∃y ψ(a, y) ∧ ∀y (ψ(a, y) → qp(y)).
40
It means that the type qp(y) ∈ Sn(Tp) is realized in M0. Butqp is a nonprincipal type, since the correspondent p-type q is non-p-principal. Consequently, a nonprincipal type is realized in the primemodel M0, a contradiction. ¤
Fix a theory T and consider its Morleyzation, i.e., an expansionto a complete theory T ′ of language Σ(T )∪Rϕ | ϕ is a formula of Tsuch that Rϕ is a l(y)-ary predicate symbol with T ′ ` Rϕ(y) ↔ ϕ(y),where ϕ(y) is a formula of T . A restriction of T ′ to the completetheory of language Rϕ | ϕ is a formula of T is denoted by T ∗.Thus, we get an operation ·∗ : T → T ∗. It's easy to see an existenceof a one-to-one correspondence ·∗ : S(T ) → S(T ∗) for which a com-plete type q∗(y) ψ(y) | T ∗ ` Rϕ(y) → ψ(y) for some formulaϕ(y) ∈ q corresponds to the type q(y) ∈ S(T ). For this correspon-dence, the λ-stability, the simpleness (see [27]) and the smallness oftheories, and also the properties of isolation and of being powerfulfor types are preserved. So considering questions on existence ofnonprincipal powerful types in aforesaid classes of theories, it suf-ces to take theories of form T ∗.
Then Lemma 1.3.1 and Proposition 1.3.2 implyCorollary 1.3.3. If |Sn,p(T )| = ω and the small theory T ∗ is
reduced over the type p∗, then some (n, p)-type is omitted in themodel Mp.
Recall, that a theory T has the strict order property if thereexists a formula ϕ(x, y) of T and tuples ai, i ∈ ω such that thefollowing equivalence holds:
` ϕ(ai, y) → ϕ(aj , y) ⇔ i ≤ j.
Notice, that theories with formula-denable innite linear or-ders have the strict order property. In particular, the Ehrenfeuchtexamples have this property (see example 1.1.1). Here, for any pow-erful type p and for any natural n, the number of (n, p)-types withfree variables x1, . . . , xn is nite and, consequently, all p-types arep-principal.
The following proposition, contained implicitly in R. E. Woodrow[205], claries that the described situation is impossible for theorieswithout the strict order property.
Proposition 1.3.4. If p(x) is a nonprincipal powerful type oftheory T and T doesn't have the strict order property, then|S2,p(T )| = ω.
41
P r o o f. Consider a formula ϕ(x, y) of T witnessing the non-symmetry of relation SIp (such a formula exists by Lemma 1.1.2).We set ϕ0(x, y) (x ≈ y), ϕ1(x, y) ϕ(x, y), ϕn+1(x, y) ∃z(ϕn(x, z) ∧ ϕ(z, y)), n ∈ ω \ 0. It suces to show, that in asystem p(M∗) (where M∗ is a countable saturated model of T ∗)for any a ∈ p(M), inequalities Rϕn+1(a, p(M)) \Rϕn(a, p(M)) 6= ∅hold for every n ∈ ω. Assume on the contrary, that for some nan inclusion Rϕn+1(a, p(M)) ⊆ Rϕn(a, p(M)) is true. Considera formula ψ(x, y)
n∨i=0
ϕi(x, y). Then, by assumption, for any
a, b ∈ p(M), satisfying |= ϕ(a, b) and (b, a) 6∈ SIp, we get ` ψ(b, y) →ψ(a, y). Since the formula ψ(b, y) witnesses on the semi-isolationover b of every its realization, |= ψ(a, a) and (b, a) 6∈ SIp, then|= ∃y (ψ(a, y) ∧ ¬ψ(b, y)). Since tuples a and b realizes the sametype p, there exists a sequence (cn)n∈ω of realizations of p such that` ψ(ci, y) → ψ(cj , y) and |= ∃y(ψ(cj , y)∧¬ψ(ci, y)), i < j < ω. Thisrelations contradict an absence of the strict order property in T . ¤
In view of Corollary 1.3.3 and Proposition 1.3.4, we haveTheorem 1.3.5. If T a theory without the strict order property,
p(x) is a nonprincipal powerful type of T , then T ∗ is not reducedover p∗.
The following Proposition shows that a realizability on non-p-principal p-types in a model Mp implies the non-symmetry of rela-tion SIp.
Proposition 1.3.6. If a non-p-principal p-type q is realized ina model Ma, where a is a realization of p, then, for every element bi
of a realization b of q in Ma, the pair (a, bi) belongs to SIp and (bi, a)doesn't belong to SIp.
P r o o f. Let a be a realization of type p, ϕ(a, y) be a for-mula, isolating a non-p-principal p-type q(y). Assume, that someelement bi of a realization b of q(y) in Ma semi-isolates the ele-ment a. Consider a formula ψ(yi, x) witnessing on the semi-isolationof a over bi. Then the type q(y) is isolated by set ∪p(yi) | yi ∈y ∪ ∃x (ϕ(x, y) ∧ ψ(yi, x)). This is impossible, since the p-typeq(y) is not p-principal. ¤
Recall, that any maximal, w.r.t. inclusion, connected subgraphof graph Γ = 〈X;Q〉 is said to be a connected component of Γ. Eachconnected component C of Γ is uniquely dened by every its elementa ∈ C and is denoted by C(a,Γ), or by C(a,Q) if the universe X isgiven.
42
For a graph Γ = 〈X; Q〉, we dene inductively the followingrelations Qn, n ∈ Z: Q0 idX , Q1 Q, Qn+1 Qn Q, Q−n (Qn)−1, n ∈ ω.
E x a m p l e 1.3.1. We are going to construct an ω-stable the-ory, having a type p such that some non-p-principal p-type is real-izable in an elementary submodel Mp of model M.
The language Σ will consist of unary predicate symbols Coln,n ∈ ω, binary predicate symbols Q,R1, R2 and 3-ary predicatesymbol S.
The predicate Q denes on the universe M a free directed pseu-doplane, as in Example 1.2.3, with a transitive (that is connect-ing any two elements) automorphism group, with innitely manyconnected components C(a,Q) and with an 1-inessential Q-orderedcoloring Col, correspondent to symbols Coln, n ∈ ω, satisfying thefollowing conditions:
(1) innitely many images of color µ relative to Q for any µ ≥ n(including ∞);
(2) innitely many preimages of color m relative to Q for anym ≤ n.
The predicate R1 connects only elements a and b of same colors,for which |= ∃x (Q(x, a) ∧ Q(x, b)) holds, and denes, in a set ofsolutions of every formula Q(a, y), a sequence function with uniqueimage c1, unique preimage c2 6= c1 for each element b, such that|= Q(a, b), and without cycles.
The predicate R2, as well as Q, denes in M a free directedpseudoplane with a transitive automorphism group, with innitelymany connected components C(a,R2) and with an 1-inessential R2-ordered coloring Col, correspondent to symbols Coln, n ∈ ω, satis-fying the following conditions:
(1) innitely many images of color µ relative to R2 for any µ ≥ n(including ∞);
(2) innitely many preimages of color m relative to R2 for anym ≤ n.
Furthermore, every two dierent preimages of every element,w.r.t.
⋃n∈ω
(R2)n, lie in dierent connected components w.r.t. Q, andexactly two elements, b and c, lying in a same connected componentw.r.t. Q, are R2-connected with an image of each element a w.r.t.R2. These elements satises |= ∃x (Q(x, b)∧Q(x, c)), have the samecolor, and if Col(a) = n then, in the graph with relation R1∪(R1)−1,
43
the length of shortest (b, c)-route is not less than n. Moreover, weconsider, that for each pair of elements b, c of same color and with|= ∃x (Q(x, b) ∧ Q(x, c)), (b, c) ∈ Rn
1 ∪ (R1)−n and for any colorm ≤ n, there exists a common preimage of b and c w.r.t. R2, havingthe color m.
The predicate S connects all possible triples of elements a, b, csuch that |= R2(a, b) ∧R2(a, c).
As in Example 1.3.1 we state, that all requirements can be real-ized such that the theory T0 of described model is ∆-based, where∆ is the least closed (w.r.t. substitutions of variables) set of for-mulas without more then two free variables, containing the formula(x ≈ y) and satisfying the following condition: if ϕ(x, y) ∈ ∆ then∃z (ϕ(x, z) ∧ <δ1(z, y) ∧ Colδ2n (z)) ∈ ∆, where δ1 ∈ −1, 1, δ2 ∈0, 1, <1(x, y) = <(x, y), <−1(x, y) = <(y, x), < ∈ Q,R1, R2,Col1n(z) = Coln(z), Col0n(z) = ¬Coln(z). Using ∆-baseness, a rou-tine consideration of cases of connections of elements of tuples showsthe ω-stability of T0.
The set of formulas ¬Coln(x) | n ∈ ω isolates unique, beingin T0, nonprincipal 1-type. This type, denoted by p∞(x), is realizedby elements of innite color.
For any element a of innite color, the formula S(a, x, y) isolatesa non-p∞-principal (2, p∞)-type q(x, y) ∈ S(T0), that dened by setof formulas
∃z (Q(z, x) ∧Q(z, y) ∧ ¬Coln(z)) ∧ ¬Rn1 (x, y) | n ∈ ω.
Here for elements an of color n ∈ ω, formulas S(an, x, y) isolatetypes, approximating a description of type q(x, y). ¤
§ 1.4. Powerful digraphsIn this Section, we introduce the concept of a powerful digraph
and establish its \local" presence in the structure of any nonprinci-pal powerful type p. We also show, that, having the (2, p)-invarianceproperty of theory, a structure of powerful digraph is contained ina restriction of saturated structure to a structure of realizations ofnonprincipal powerful type, having the global pairwise intersectionproperty. We describe structures of transitive closures of saturatedpowerful digraphs, formed in models of theories with nonprincipal
44
powerful 1-types, when the number of nonprincipal 1-types is nite.Moreover, we prove, that the structure of powerful digraph, consid-ered in a model of simple theory [27], induces innite weight. Itmeans, that there are no powerful graphs in structures of knownclasses of simple theories (such as supersimple or nitely based the-ories), that don't contain Ehrenfeucht theories.
A countable acyclic digraph Γ = 〈X;Q〉 is said to be powerful ifthe following conditions hold:
(a) the automorphism group of Γ is transitive, that is any twovertices are connected by an automorphism;
(b) the formula Q(x, y) is equivalent in the theory Th(Γ) toa disjunction of principal formulas;
(c) acl(a) ∩ ⋃n∈ω
Qn(Γ, a) = a for each vertex a ∈ X;(d) Γ |= ∀x, y ∃z (Q(z, x) ∧ Q(z, y)) (the pairwise intersection
property).Clearly, in the classical examples of Ehrenfeucht theories, the
countable graph with the relation x < y of dense linear order ispowerful.
The following example, represented in the works by A. I. Mal'tsev[118] and E. E. Bouscaren, B. P. Poizat [58], denes a stable theoryof acyclic pairing function with a structure of powerful digraph.
E x a m p l e 1.4.1. Let M be a set with two functions f1, f2 :M → M such that (f1, f2) : M → M ×M is a bijection, for whichthere are no nonempty sequences i1, . . . , in and elements a ∈ Msuch that fin(. . . fi1(a) . . .) = a. The theory T Th(〈M ; f1, f2〉),being a theory of a locally free algebra2, is stable. Consider the free1-type p ∈ S1(T ), i.e., the type of elements a such that
fin(. . . fi1(a) . . .) = fjm(. . . fj1(a) . . .) ⇔ i1 . . . in = j1 . . . jm.
It's easy to see, that some countable set of realizations of p withthe relation, dened by formula (y ≈ f1(x)) ∨ (y ≈ f2(x)), formsa powerful digraph. ¤
In the following example of a theory with three countable models,which is similar to an example in the work by M. G. Peretyat'kin[137], the powerful digraph also includes the relation <.
2The stability of theories of locally free algebras is proved by O. V. Bele-gradek [55].
45
E x a m p l e 1.4.2. Let M = 〈M ;≤〉 be a lower semilatticewithout the least and greatest elements such that:
(a) for each pair of incomparable elements, their join does notexist;
(b) for each pair of distinct comparable elements, there is anelement between them;
(c) for each element a there exist innitely many pairwise in-comparable elements greater than a, whose inmum is equal to a.
Expand the system M by constants cn, n ∈ ω, such that cn <cn+1, n ∈ ω. The theory T of this expansion has exactly threecountable models: the prime model; the saturated model; the primemodel over the realization of the powerful type p∞(x), isolated bythe set cn < x | n ∈ ω of formulas. ¤
Apart from these examples, a rather rich class of pow-erful digraphs is formed by the acyclic digraphs 〈P ; Q〉 =〈P ; (p, p′) | p′ = pg0 on some line〉, corresponding to polygonome-tries pm(G, 〈P, L,∈〉, g0) on projective planes [172].
Let M be a model of a theory T , p(x) be a complete type of Tover the empty set, ψ(x, y) be a formula of T , where l(x) = l(y).Denote by p(M) the set of realizations of p(x) in M, and by Rp
ψ(M),the binary relation (a, b) ∈ (p(M))2 | M |= ψ(a, b).
The following statement shows that the powerful digraphs reside\locally" in the structure of each nonprincipal powerful type.
Proposition 1.4.1. If p(x) is a nonprincipal powerful type ofsome theory T and M is a countable saturated model of T then foreach formula ϕ(x) ∈ p(x), there exists a formula ψ(x, y) of T (wherel(x) = l(y)), satisfying the following conditions:
(1) for each a ∈ p(M) the formula ψ(a, x) is equivalent to adisjunction of principal formulas ψi(a, x), i ≤ n, such that ψi(a, x) `p(x), and |= ψi(a, b) implies, that b doesn't semi-isolate a;
(2) for every a, b ∈ p(M) there exists a tuple c such that |=ϕ(c) ∧ ψ(c, a) ∧ ψ(c, b).
P r o o f. By the condition and Lemma 1.1.2, there are real-izations a and b of p(x) in the model Ma such that b doesn'tsemi-isolate a. Since Mb ≺ Ma, for each realization c ∈ Mbthere is a principal formula χc(a, y) such that |= χc(a, c). Enu-merate all realizations of p(x) in Mb: p(Mb) = cn | n ∈ ω. Putχn(a, y) χcn(a, y) for n ∈ ω.
46
Fix some formula ϕ(x) ∈ p(x) and show that some formula∨mi=0 χi(x, y) can be taken as ψ(x, y). Clearly, each of those for-
mulas satises condition 1. Assuming that none of them satisescondition 2, by Compactness Theorem, we get the consistency ofthe set
r(x, y) p(x) ∪ p(x)∪¬∃z
((m∨
i=0
χi(z, x)
)∧
(m∨
i=0
χi(z, y)
)∧ ϕ(z)
)| m ∈ ω
.
Since p(x) is powerful, the type r(x, y) is realized in Mb by some tu-ples d1 and d2: Mb |= r(d1, d2). Thus, there exist formulas χd1
(x, y)and χd2
(x, y) such that |= χd1(a, d1) ∧ χd2
(a, d2), which contradictsthe consistency of r(x, y). Therefore, r(x, y) is inconsistent; hence,for some m0, we have the inconsistency of the set
p(x) ∪ p(y) ∪¬∃z
((m0∨
i=0
χi(z, x)
)∧
(m0∨
i=0
χi(z, y)
)∧ ψn(z)
)| n ∈ ω
.
Putting ψ(x, y) ∨m0
i=0 χi(x, y), we deduce the claim. ¤Call property 2 of Proposition 1.4.1, the local pairwise intersec-
tion property and denote it by (LPIP). If for a formula ψ(x, y) thestronger property is true:
2′) for every a, b ∈ p(M) there exists a tuplec ∈ p(M) such that|= ψ(c, a) ∧ ψ(c, b),we then call it the global pairwise intersection property for p(x) withrespect to ψ(x, y) and it will be denoted by (GPIP).
Whenever a formula ψ(x, y) with properties 1 and 2′exist, callthe digraph
⟨p(M);Rp
ψ(M)⟩
prepowerful .Recall, that theories T0 and T1 of languages Σ0 and Σ1 respec-
tively are said to be similar if for any models Mi |= Ti, i = 0, 1,there are formulas of Ti, dening in Mi predicates, functions andconstants of language Σ1−i such that the corresponding algebraicsystem of Σ1−i is a model of T1−i.
A theory T is said to be (n, p)-invariant if for each formula ψ(x)(where l(x) = n) of Tp the restriction of the Morleyzation of Tp to
47
the language Rψ is similar to the restriction of the Morleyzationof the theory of the structure of some formula-denable set ϕ(M)(where ϕ ∈ p) to the same language.
Show that the existence of a prepowerful structure on the set ofrealizations of a nonprincipal powerful type p of some (2, p)-invarianttheory implies the existence of a formula-dening a powerful digraphstructure on this set.
Proposition 1.4.2. If p(x) is a nonprincipal powerful type ofsome (2, p)-invariant theory T and
⟨p(M);Rp
ψ(M)⟩
is a prepowerfuldigraph then for some formula θ(x, y) with T ` θ(x, y) → ψ(x, y),the digraph
⟨p(M);Rp
θ(M)⟩
is powerful.P r o o f. The (2, p)-invariance of T implies that we can choose
to make principal the formulas Rpχ′i
(x, y) corresponding in T toχi(x, y) such that ψ(a, x) =
∨mi=0 χi(a, x), where χi(a, x) are prin-
cipal formulas for each a ∈ p(M) and all i = 1, . . . , m. Indeed, theelement a belongs to a prime modelM0 of the restriction of the Mor-leyzation of Tp to the language Rψ. Some realizations di of prin-cipal formulas, corresponding to the formulas χi(a, y) i = 0, . . . ,m,belong to the same model. Consider formulas χ′i(x, y) of languageRψ, corresponding to complete formulas of the types tp(a ˆ di),i = 0, . . . ,m0. Take as θ(x, y) some formula of T satisfying thefollowing conditions:
(1) the restriction of the Morleyzation of the theory of the struc-ture of some formula-denable set ϕ(M) (where ϕ ∈ p) to the lan-guage Rθ is similar to the restriction of the Morleyzation of Tp tothe same language;
(2) ` ((ϕ(x) ∧ ϕ(y)) → (θ(x, y) ↔ ∨mi=0 χ′i(x, y));
(3) ` θ(x, y) → ψ(x, y).Since the model M is saturated, the automorphism group of the
digraph Γ =⟨p(M);Rp
θ(M)⟩
is transitive.Notice, that for each a ∈ p(M) it follows from |= θ(a, b) that
b ∈ p(M) and b does not semi-isolate a. Thus, since the semi-isolation relation is transitive, the digraph Γ is acyclic.
The non-symmetry of the semi-isolation relation SIp, witnessedby the formula θ, implies the equality
acl(b) ∩⋃n∈ω
(Rpθ(M))n(Γ, b) = b
48
for each b ∈ p(M). Indeed, assuming that there exists
d ∈ acl(b) ∩⋃n∈ω
(Rpθ(M))n(Γ, b) \ b,
we deduce that b semi-isolates d in M. Thus, because the semi-isolation relation is transitive, the element b will semi-isolate a,where |= θ(a, b) and a ∈ p(M); this is a contradiction.
The (GPIP) for p(x) with respect to ψ(x, y) implies the sameproperty with respect to θ(x, y), and so, we have the pairwise inter-section property for Γ. Therefore, Γ is a powerful digraph. ¤
Notice, that under the assumptions of Proposition 1.4.2 theniteness of number of nonprincipal 1-types implies the relation
acl(a) ∩⋃n∈ω
(Rpθ(M))n(a,Γ) = a (1.1)
for each vertex a of the digraph Γ =⟨p(M);Rp
θ(M)⟩.
Indeed, assume that the type tp(b/a) is algebraic for some b ∈(Rp
θ(M))n(a,Γ), b 6= a. Then the original theory contains a semi-isolating formula θ(a, y) such that |= θ(a, b) ∧ ∃=ky θ(a, y) for somek ∈ ω. Since b does not semi-isolate a and the number of nonprin-cipal 1-types is nite, it follows that there exists a tuple c realizinga principal type such that |= θ(c, b) ∧ ∃=ky θ(c, y). This means thatthe nonprincipal type p(x) is realized in the prime model; this is acontradiction.
If the number of nonprincipal 1-x)-types is innite then (1.1)need not hold. To illustrate that, consider the following exampleof an ω-stable theory with a nonprincipal 1-type p0(x), having anonsymmetric semiisolation relation via a formula Q(x, y) such thatacl(a) =
⋃n∈ω
Qn(a,M) for each realization a of p0(x).
E x a m p l e 1.4.3. Denote by Ω the set of nonempty nitesequences α = 〈α0, α1, . . . , αn〉 with αi ∈ ω for i ≤ n and l(α) =α0 + 2.
Let T0 be a theory of language 〈P (1)α , Q(2)〉α∈Ω with the following
axioms:
49
1) if α = α′ ˆm ∈ Ω then
` (Pα′ ˆ (m+1)(x) → Pα′ ˆ m(x)
)∧∃≥ωx(Pα′ ˆ m(x) ∧ ¬Pα′ ˆ (m+1)(x)
);
2) if α1 = α′1 ˆ 0 and α2 = α′2 ˆ 0 are tuples in Ω and α′1 6= α′2then ` ¬∃x (Pα1(x) ∧ Pα2(x));
3) the relation Q forms the graph of a free (acyclic) unar withinnitely many preimages of each element;
4) ` ∀x, y ((P〈0,m〉(x) ∧ ¬P〈0,m+1〉(x) ∧ Q(x, y)) → (P〈0,m〉(y) ∧¬P〈0,m+1〉(y))) for m ∈ ω;
5) if |= P〈0,m〉(a) ∧ ¬P〈0,m+1〉(a) then the set of realizations ofthe formula Q(x, a) consists of innitely many realizations of theformula P〈0,m〉(x)∧¬P〈0,m+1〉(x), and innitely many realizations ofthe formulas P〈1,k,m〉(x) ∧ ¬P〈1,k,m+1〉(x) for each k ∈ ω;
6) if α = k ˆα′ ˆ l ˆm is a tuple in Ω, k ≥ 1, then
` ∀x, y
( (Pk ˆ α′ ˆ l ˆ m(x) ∧ ¬Pk ˆ α′ ˆl ˆ (m+1)(x) ∧Q(x, y)
) →
(P(k−1) ˆ α′ ˆ m(y) ∧ ¬P(k−1) ˆ α′ ˆ (m+1)(y)
) ),
m ∈ ω;7) if k 6= 0 and |= Pk ˆ α ˆ m(a)∧¬Pk ˆ α ˆ (m+1)(a), then the set of
realizations of the formula Q(x, a) consists of innitely many real-izations of the formulas P(k+1) ˆ α ˆ l ˆ m(x) ∧ ¬P(k+1) ˆ α ˆ l ˆ (m+1)(x)for each l ∈ ω.
The construction of a saturated model satisfying axioms 17,enables us to verify the completeness of T0. The ω-stability of T0
follows because each formula without parameters is equivalent to aBoolean combination of formulas of the form Pα(x), α ∈ Ω, and∃z (Qn1(x, z)∧ Qn2(y, z)), n1, n2 ∈ ω. Moreover, like Example 1.2.3,the countable number of 1-types over each countable set A is fol-lowed by the countable number of possibilities of distance distribu-tions from elements of A to realizations of types.
For the type p0(x) ∈ S1(∅), isolated by the setP〈0,m〉 | m ∈ ω
of formulas, the semi-isolation relation is not symmetric via theformula Q(x, y). For each a |= p0, the set of realizations of the
50
formula Q(x, a) is exhausted by the realizations of the type p0 andof the nonprincipal types p〈1,k〉(x) ∈ S1(∅), isolated by the setsP〈1,k,m〉 | m ∈ ω
, k ∈ ω. Since the relation Q forms the graph of a
free unar with innitely many preimages of each element, acl(a) =dcl(a) =
⋃n∈ω
Qn(a,M) for each element a of a model M |= T0. ¤In connection with the argument above, the problem seems in-
teresting of describing the powerful digraphs that can be expandedto the structures of powerful 1-types both in the case of nitelymany and innitely many nonprincipal 1-types.
Recall that a partially ordered set 〈X;≤〉 is said to be directeddownward (upward) if for each x, y ∈ X, there exists z ∈ X suchthat z ≤ x and z ≤ y (x ≤ z and y ≤ z).
Let us list the main possibilities that exhaust the structures ofthe transitive closures of powerful digraphs obtained from the struc-tures of nonprincipal powerful types p(x)for which the number ofnonprincipal l(x)-types is nite.
Theorem 1.4.3. Given a saturated powerful digraph Γ = 〈X; Q〉in which acl(a) ∩ ⋃
n∈ωQn(a,Γ) = a for each a ∈ X, the tran-
sitive closure TC(Γ) =⟨
X;⋃
n∈ωQn
⟩is isomorphic to a downward
directed set with a transitive automorphism group and one of thefollowing orders:
(1α) a dense partial order with maximal antichains containing αelements α ∈ (ω + 1) \ 0;
(2) a partial order with innitely many covering elements foreach element.
P r o o f. The re exivity and transitivity of the relation ≤ ofthe digraph TC(Γ) = 〈X;≤〉 are obvious. The antisymmetry of≤ follows from the acyclicity of Γ. The existence in the partiallyordered set TC(Γ) of the meet of two arbitrary elements follows fromthe pairwise intersection property. If the order ≤ is not dense thenthe existence of innitely many covering elements for each a ∈ Xfollows from the relation acl(a)∩ ⋃
n∈ωQn(a,Γ) = a, since in this
situation the consistent formula a < x ∧ ¬∃y (a < y ∧ y < x) doesnot belong to an algebraic type over a. ¤
Notice, that the dense partial orders with maximal antichainsof cardinality α, that we mentioned, are realized by replacing each
51
element in a dense linear order without endpoints by an equivalenceclass containing α pairwise incomparable elements.
Note that a partial order with innitely many covering elementsfor each element comes only from the powerful digraphs for whichthe formula Q(x, y) is not principal.
Indeed, if Q(x, y) is a principal formula then the truth of |=Q(a, b)∧Q(a, c)∧Q(c, b) for each a and some b, c and the existenceof automorphisms, xing a and connecting arbitrary elements inQ(a,Γ), imply that for each b in Q(a, Γ) there exists c in Q(a,Γ) ∩Q(Γ, b). Consequently, in the graph TC(Γ), between each pair ofdistinct elements there is another element. Therefore, we have
Corollary 1.4.4. If Γ = 〈X; Q〉 is a powerful digraph witha principal formula Q(x, y) then the relation
⋃n∈ω
Qn is a dense par-tial order.
Notice, that if the relation ≤ in the transitive closure of a sat-urated powerful digraph Γ = 〈X, Q〉 is not denable by formulasin the language of Γ (i.e., if the lengths of the shortest paths areunbounded) then by Compactness Theorem, in TC(Γ), over each a,there is an innite antichain belonging to Q(a,Γ). Therefore, thetheorem 1.4.3 implies
Corollary 1.4.5. Given a saturated powerful digraph Γ = 〈X;Q〉with unbounded lengths of shortest routes such that
acl(a) ∩⋃n∈ω
Qn(a, Γ) = a
for each a ∈ X, its transitive closure TC(Γ) =⟨
X;⋃
n∈ωQn
⟩is iso-
morphic to a downward directed set with a transitive automorphismgroup that has one of the following orders:
(1) a dense partial order with innite antichains;(2) a partial order with innitely many covering elements of each
element.Notice, that while the necessity of the local presence of pow-
erful digraphs in the structures of nonprincipal powerful types isproved, the question remains open of suciency, i.e., the possibilityof expansion of each powerful digraph to the structure of a powerfultype.
52
Recall several notions of Stability Theory related to the class ofsimple theories [3], [22], [27]. A formula ϕ(x, a) in a theory T is saidto be copied over a set A if there are a natural m and tuples an,n ∈ ω, such that the following conditions hold:
(1) tp(a/A) = tp(an/A), n ∈ ω;(2) the set ϕ(x, an) | n ∈ ω of formulas is m-inconsistent, i.e.,
for every w ⊂ ω of the cardinality m the formula∧
n∈wϕ(x, an) is
inconsistent in T .tuples a and b are said to be dependent over A if there exists a
formula ϕ(x, a) in T that is copied over A and satises |= ϕ(b, a).If tuples a and b are not dependent over A they are said to beindependent over A. Tuples, being dependent (independent) over ∅,are called simply dependent (independent). A sequence of tuples issaid to be independent if each tuple of this sequence is independentwith every tuple, formed by coordinates of other elements of thesequence.
Recall, that a type p(x) has innite own weight if there existsa realization a of p(x) and an innite independent sequence (an)n∈ω
of realizations of p(x) such that the tuples a and an are dependentfor each n ∈ ω.
The following Proposition shows that the powerful digraphs donot occur in the structures of known classes of simple theories thatdo not include Ehrenfeucht theories.
Proposition 1.4.6. If T = Th(Γ) is a simple theory of somepowerful digraph Γ = 〈X; Q〉 then the (unique) type p ∈ S1(∅) hasinnite own weight.
P r o o f. Show rst that if |= Qk(a, b) for some k > 0, then aand b are dependent. For that it suces to establish that the formulaQk(a, x) is copied over ∅. Indeed, there exists a number m ∈ ω suchthat for each element a0 there is at most m elements a1, . . . , am
satisfying the conditions Qk(ai, aj) for all 1 ≤ i < j ≤ m, becauseotherwise by the Compactness Theorem and the acyclicity of thedigraph Γ there is an innite sequence (an)n∈ω with the condition
|= Qk(ai, aj) ⇔ i < j,
which contradicts the simplicity of T .
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Dene a sequence (an)n∈ω inductively. Pick an arbitrary elementa0 in X. If a0, . . . , an−1 have been chosen then pick an satisfying thecondition Γ |= Qk(an−1, an), and belonging to the maximal numberof sets Qk(ai, Γ), i < n. The remarks above imply that the setQk(an, x) | n ∈ ω is m-inconsistent. Since every two elementsare connected by an automorphism, the formula Qk(a, x) is copiedover ∅.
Notice now that, by the pairwise intersection property, for allelements a1, . . . , an ∈ X there exists
a ∈ Q(Γ, a1) ∩Q2(Γ, a2) ∩ . . . ∩Qn(Γ, an),
and, in particular, every n elements comprising an independent se-quence depend on some a. Since every two elements are connectedby an automorphism and the integer n is unbounded, there existinnitely many elements that form an independent sequence anddepend on a. ¤
C h a p t e r 2GENERIC CONSTRUCTIONS
§ 2.1. Semantic generic constructionsThe construction of a generic structure with requisite properties
begins by dening a class K0 of nite structures of a countable predi-cate language. The class K0 is endowed with a partial order relation6 which is invariant under the transition to isomorphic structures,connoting the property of being a self-sucient structure, or strongsubstructure, and satisfying the following axioms:
(1) if A 6 B, then A ⊆ B;(2) if A 6 C, B ∈ K0, and A ⊆ B ⊆ C, then A 6 B;(3) ∅ is the least element of the system (K0;6);(4) (the amalgamation property) for any structuresA,B, C ∈ K0,
having embeddings f0 : A → B and g0 : A → C such that f0(A) 6 Band g0(A) 6 C, there are a structure D ∈ K0 and embeddingsf1 : B → D and g1 : C → D for which f1(B) 6 D, g1(C) 6 D andf0 f1 = g0 g1.
With the class K0 determined from nite structures of K0 usingamalgamation (i.e., embedding the structures B and C over A instructures D so as to comply with the amalgamation property), weconstruct a countable (K0;6)-generic model M step by step so asto satisfy the following:
(a) for any nite substructure A ⊆M, there is a structure B ∈K0, A ⊆ B ⊆ M, for which B 6 M, i.e., B 6 B′ for any structureB′ ∈ K0 with B ⊆ B′ ⊆M;
(b) for any nite substructure A ⊆M and any structure B ∈ K0
such that A 6 B, there is a structure B′ 6 M for which B 'A B′.
55
Thus, the following Theorem holds (Theorem 2.12 in work byJ. T. Baldwin and N. Shi [37]).
Theorem 2.1.1. For any partially ordered class (K0; 6), satis-fying conditions 14, there exists a (K0;6)-generic model.
The scheme above represents a semantic approach to construct-ing a generic modelM and the corresponding generic theory Th(M).
The utility of the semantic approach for realizations of desiredmodel-theoretic properties has conrmed by numerous examples(see the bibliography, re ected in the historical survey) for the caseswhen each predicate is not denable via other ones.
§ 2.2. Syntactic generic constructionsIn constructing generic models in which some predicates are
a priori denable via other ones, it is more preferable (and some-times inevitable) to use a syntactic approach,, within which com-plete or incomplete types over nite sets containing some externalinformation on elements are treated rather than nite structures.
The syntactic approach for creating generic theories, written inthis Section, generalizes the semantic approach, described above,and also leads to creating generic models. This approach will beused below, in third and forth Chapters for creating generic theo-ries, representing all possible stable, and also unstable Ehrenfeuchttheories w.r.t. Rudin | Keisler preorders and distribution functionsof number of limit models. Examples of this and following Sectionsshow, that the syntactic approach forms a proper generalization ofsemantic one for constructions of generic models.
Below in this Section, we write X, Y, Z, . . . for nite sets of vari-ables, and denote by A,B, C, . . . nite sets of elements, as well asnite sets in algebraic systems, or else the algebraic systems with -nite universes themselves; Φ(A),Ψ(B),X(C), . . . stand for diagrams,that is, complete or incomplete types over corresponding sets havingno free variables.
For a type Φ(A), we denote by [Φ(A)]AX the type Φ(X), obtainedvia some bijective substitution into Φ(A) of variables in X for con-stants in A, and denote by [Φ(A)]AB the type Φ(B) obtained viaa bijective substitution into Φ(A) of constants in B for constantsin A.
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We x an at most countable language Σ and consider a class T0
of (complete or incomplete) types Φ(A) over nite sets A such thatϕ(a) ∈ Φ(A) or ¬ϕ(a) ∈ Φ(A) for any quantier-free formula ϕ(x)and any tuple a ∈ A. Suppose that the class T0 is equipped witha partial order 6, closed under bijective substitutions [Φ(A)]AA′ ofpairwise distinct constants in A′ for constants in A into types Φ(A) ∈T0. Furthermore, we assume that results of bijective substitutions[Φ(A)]AX of sets X of variables for constants in A into types Φ(A) ∈T0 (over all sets A) form a countable set.
A partially ordered class (T0; 6) is said to be generic if T0 isclosed under intersections and satises the following:
(i) if Φ 6 Ψ, then Φ ⊆ Ψ;(ii) if Φ 6 X, Ψ ∈ T0, and Φ ⊆ Ψ ⊆ X, then Φ 6 Ψ;(iii) some type Φ0(∅) is the least element of the system (T0;6);(iv) (the t-amalgamation property) for any types Φ(A), Ψ(B),
X(C) ∈ T0, if there exist injections f0 : A → B and g0 : A → Cwith [Φ(A)]Af0(A) 6 Ψ(B) and [Φ(A)]Ag0(A) 6 X(C), then there are atype Θ(D) ∈ T0 and injections f1 : B → D and g1 : C → D forwhich [Ψ(B)]Bf1(B) 6 Θ(D), [X(C)]Cg1(C) 6 Θ(D) and f0f1 = g0g1;
(v) (the local realizability property) if Φ(A) ∈ T0 and Φ(A) `∃x ϕ(x) (respectively, t is a term of language Σ ∪ A containing nofree variables), then there are a type Ψ(B) ∈ T0, Φ(A) 6 Ψ(B),and an element b ∈ B for which Ψ(B) ` ϕ(b) ((t ≈ b) ∈ Ψ(B));
(vi) (the t-uniqueness property) for any types Φ(A), Ψ(A) ∈ T0
if the set Φ(A) ∪Ψ(A) is consistent then Φ(A) = Ψ(A).A type Φ is called a strong subtype of a type Ψ if Φ 6 Ψ.A type Φ(A) is said to be (strongly) embeddable in a type Ψ(B)
if there is an injection f : A → B such that [Φ(A)]Af(A) ⊆ Ψ(B)([Φ(A)]Af(A) 6 Ψ(B)). The injection f , in this instance, is called a(strong) embedding of type Φ(A) in type Ψ(B) and is denoted byf : Φ(A) → Ψ(B).
A type Φ(A) is said to be (strongly) embeddable in a model Mif Φ(A) is (strongly) embeddable in some type Ψ(B), where M |=Ψ(B). The corresponding embedding f : Φ(A) → Ψ(B), in thisevent, is called a (strong) embedding of type Φ(A) in model M andis denoted by f : Φ(A) →M.
57
Let T0 be a class of types, P be a class of models, and M bea model in P. The class T0 is conal in the model M if, for eachnite set A ⊆ M , there are a nite set B, A ⊆ B ⊆ M , and a typeΦ(B) ∈ T0 such that M |= Φ(B). The class T0 is conal in P ifT0 is conal in every model of P. We denote by T0 the class of allmodels M with the condition that T0 is conal in M, and by P asubclass of T0 such that each type Φ ∈ T0 is true for some modelin P.
Now we extend the relation 6 from the generic class (T0,6) toa class of subsets of models in the class T0.
Let M be a model in T0, A and B be nite sets in M withA ⊆ B. We call A a strong subset of the set B (in the model M),and write A 6 B, if there exist types Φ(A), Ψ(B) ∈ T0, for whichΦ(A) 6 Ψ(B) and M |= Ψ(B).
A nite set A is called a strong subset of a set M0 ⊆ M (in themodel M), where A ⊆ M0, if A 6 B for any nite set B such thatA ⊆ B ⊆ M0 and Φ(A) ⊆ Ψ(B) for some types Φ(A),Ψ(B) ∈ T0
with M |= Ψ(B). If A is a strong subset of M0 then, as above, wewrite A 6 M0. If A 6 M in M then we refer to A as a self-sucientset (in M).
Notice, that by the t-uniqueness property, the types Φ(A) andΨ(B) specied in the denition of strong subsets are dened uniquely.A type Φ(A) ∈ T0, corresponding to a self-sucient set A in M, issaid to be a self-sucient type (in M).
The following statement, generalizing Lemma 2.8 in the work byJ. T. Baldwin and N. Shi [37], shows that, for nite sets correspond-ing to generic classes whose types are generated by uniformly nitetypes, the condition of being self-sucient is type denable.
Proposition 2.2.1. Let T0 be a generic class consisting of typesΦ(A), which are deduced from nite subtypes whose cardinalitiesare uniformly bounded in terms of cardinalities |A|, and let M bea model in the class T0. Then for each nite set A 6 M , there ex-ists a type ΓA(X) such that M |= ΓA(A), and for each set A′ ⊆ M ,M |= ΓA(A′) implies A′ 6 M .
P r o o f. Let A be a self-sucient set in M, Φ(A) be a typein T0, M |= Φ(A). For any set B, A ⊆ B ⊆ M , and any typesΨ(B) ∈ T0 with Φ(A) 66 Ψ(B), we denote by Ψ0(X, Y ) the minimal(by inclusion) nite type from which a type [[Ψ(B)]AX ]B\AY , whereX ∩ Y = ∅, is deducible. The required type is
ΓA(X) Φ(X) ∪ ∀Y ¬ ∧Ψ(X, Y ) | A ⊆ B, Φ(A) 66 Ψ(B). ¤
58
A class (T0; 6) possesses the joint embedding property (JEP) iffor any types Φ(A), Ψ(B) ∈ T0, there is a type X(C) ∈ T0 such thatΦ(A) and Ψ(B) are strongly embeddable in X(C). Clearly, everygeneric class has JEP.
A model M ∈ P has nite closures with respect to the class(T0;6) if any nite set A ⊆ M is contained in some self-sucientset in M. A class P has nite closures with respect to the class(T0;6) if each model in P has nite closures.
Clearly, a countable model M has nite closures with respectto (T0; 6) i M =
⋃i∈ω
Ai for some self-sucient sets Ai with Ai 6Ai+1, i ∈ ω.
A countable model M ∈ T0 is (T0;6)-generic if it satises thefollowing conditions:
(a) M has nite closures;(b) if A 6 M , Φ(A), Ψ(B) ∈ T0, M |= Φ(A) and Φ(A) 6 Ψ(B),
then there exists a set B′ 6 M such that A ⊆ B′ and M |= Ψ(B′).Similarly to the construction of a (K0;6)-generic model, given
any generic class (T0; 6), we can embark on a step-by-step construc-tion of a (T0; 6)-generic model M using t-amalgamations and localrealizabilities.
Thus we have following:
Theorem 2.2.2. For any generic class (T0; 6), there exists a(T0;6)-generic model.
A theory Th(M) of a (T0;6)-generic model M is said to be(T0;6)-generic. A theory T is generic if T is (T0; 6)-generic, forsome generic class (T0; 6).
A model M is (T0; 6)-universal if each type in T0 is stronglyembeddable in M.
A model M is (T0;6)-homogeneous if for any self-sucient setsA and B in M such that, for corresponding types Φ(A) and Ψ(B)witnessing self-suciency, the equality [Φ(A)]AB = Ψ(B) and the ex-istence of a bijective mapping f realizing a substitution [Φ(A)]AB,equal to Ψ(B), insists on there being an automorphism of M, con-taining f .
Clearly, a (T0; 6)-generic model is (T0; 6)-universal and(T0;6)-homogeneous.
A generic class (T0; 6) consisting of quantier-free types is saidto be quantier free.
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The next theorem shows that constructing any (K0; 6)-genericmodel can be reduced to constructing some (T0; 6)-generic one.
Theorem 2.2.3. For any (K0; 6)-generic model M, there ex-ists a quantier-free class (T0; 6′) such that M is (T0; 6′)-generic.
P r o o f. The required class is the quantier-free generic class(T0;6′), where T0 Φ(A) | Φ(X) is the quantier-free typetpqf (A), A ∈ K0, Φ(A) 6′ Ψ(B) ⇔ A 6 B. ¤
Any generic class (T0; 6), which consists of types Φ(A), corre-sponding to nite structures A with universes A, allows us to con-struct a class K0, which consists of all nite structures isomorphicto structures A satisfying quantier-free subtypes Φ(A)qf of typesΦ(A) ∈ T0. Having dened the class K0 we specify a relation 6′with the following condition: A 6′ B ⇔ Φ(A) 6 Ψ(B) for someΦ(A),Ψ(B) ∈ T0 such that A |= Φ(A)qf and B |= Ψ(B)qf .
The class (K0;6′) obtained meets the above conditions (1)(4),which are imposed on a class generating a (K0; 6)-generic model.Moreover, the (K0; 6′)-generic model may fail to be isomorphic tothe (T0; 6)-generic model. The fact that the (T0; 6)-generic modelis uniquely reconstructed from the class (K0; 6′) implies that forevery structure A ∈ K0 and for corresponding types Φ(A) ∈ T0, thetypes will bear information on a number of all possible extensions Bof the structure A with A 6′ B, and on interrelations of elements ofthose extensions. If such information is available and is used, thenthe (T0; 6)-generic model is dened uniquely up to isomorphism.
§ 2.3. Self-sucient classesA generic class (T0; 6) is self-sucient if the following axiom
holds:(vii) if Φ,Ψ,X ∈ T0, Φ 6 Ψ, and X ⊆ Ψ, then Φ ∩X 6 X.Throughout this and two below Sections, we denote by (T0; 6)
a self-sucient generic class, by M a (T0; 6)-generic model, by T atheory Th(M), and by K a subclass of T0 consisting of all modelsof the theory T .
In order to illustrate the last condition (Mod(T ) ⊆ T0), wepresent two classes of examples.
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Clearly, any quantier-free generic class (T0; 6) of nite lan-guage, which is closed under restrictions of types Φ(A) ∈ T0 to anysubset of A, satises Mod(T ) ⊆ T0. The last-mentioned conditionis also met for any self-sucient class (T0; 6), having the followingt-covering property:
(viii) each type Φ(X) of theory T is deduced from some type[ΨΦ(B)]BX∪Y , where ΨΦ(B) ∈ T0.
Let Φ and Ψ be types in the self-sucient class (T0;6). A pair(Φ,Ψ) is minimal if Φ ⊆ Ψ, Φ 66 Ψ, and for any type Ψ′ ∈ T0, thecondition Φ ⊆ Ψ′ $ Ψ implies Φ 6 Ψ′.
Let M be a model in the class T0 and S be a set in the modelM. We say that the set S is closed in M and write S 6 M if, forany minimal pair (Φ(A), Ψ(B)) with M |= Ψ(B), A ⊆ S impliesB ⊆ S.
Now we show that the conditions S 6 M and A 6 M are com-patible for the case where S is a nite set and some type Φ(S)belongs to the class T0 with M |= Φ(S).
Proposition 2.3.1. Let M be a model in the class T0 and Φ(A)be a type in the class T0 with M |= Φ(A). The following conditionsare equivalent:
(1) A is a self-sucient set in M;(2) for any minimal pair (Φ′(A′),Ψ(B)) with M |= Ψ(B), A′ ⊆
A implies B ⊆ A.P r o o f. (1) ⇒ (2). Let A be a self-sucient set in M,
(Φ′(A′),Ψ(B)) be a minimal pair with M |= Ψ(B), and A′ ⊆ A.By axiom (vii), we have Φ(A) ∩ Ψ(B) 6 Ψ(B). By the minimalityof (Φ′(A′), Ψ(B)), we obtain A′ ⊆ A ∩ B and B = A ∩ B, that is,B ⊆ A.
(2) ⇒ (1). Assume that A 66 M , that is, there is a nite set B ⊆M such that A ⊂ B, and for corresponding types Φ(A), Ψ(B) ∈ T0,we have Φ(A) 66 Ψ(B). Now the niteness of B and axiom (vii)imply that there exists a minimal pair (Φ(A),Ψ0(B0)) satisfyingM |= Ψ0(B0), with B0 6⊆ A. The last fact clashes with condi-tion (2). ¤
Notice, that axiom (vii) gives rise to the following property ofmodels (similar to this axiom): if M,N ,N ′ ∈ T0, M 6 N , andN ′ ⊆ N , then M ∩N ′ 6 N ′.
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The next statement generalizes Lemma 2.18 in the workby J. T. Baldwin and N. Shi [37].
Proposition 2.3.2. The following conditions are equivalent:(1) class (T0; 6) does not contain an innite ascending chain
of minimal pairs;(2) class K has nite closures;(3) every ω-saturated model in K has nite closures;(4) some ω-saturated model in K has nite closures.P r o o f. (1) ⇒ (2). If the class K has no nite closures, that
is, if some nite set A of some model M ∈ K cannot be extendedto a self-sucient set, then (in view of K ⊆ T0) there exists a niteset B such that A ⊆ B ⊆ M , M |= Φ(B), Φ(B) ∈ T0, and norcan B be extended to a self-sucient set. Then, by induction, wemay construct an innite ascending chain of minimal pairs in T0
beginning with some pair (Φ(B), Ψ(C)).(2) ⇒ (3) are (3) ⇒ (4) obvious.(4) ⇒ (1). The fact that there exists an innite ascending chain
of minimal pairs in (T0; 6) implies that such is embeddable in anyω-implies that such is embeddable in any K, which is impossible formodels with nite closures. ¤
A consequence of Proposition 2.3.2 is
Corollary 2.3.3. If a generic model M is saturated, then theclass K has nite closures.
Let M and N be some models in the class T0, S be a closedset in the model M. An injection f : S → N is called a strongembedding of S in N if f(S) is a closed set in N , and for any typeΦ(A) ∈ T0 such thatM |= Φ(A) and A ⊆ S, we haveN |= Φ(f(A)).
We say that a generic class (T0;6) has amalgamation over closed(self-sucient) sets if, for any models M0,M1 ∈ K and any closed(self-sucient) set S in some model of the class K, the existence ofstrong embeddings f : S → M0 and g : S → M1 implies that thereexist a model N |= T and elementary embeddings f ′ : M0 → Nand g′ : M1 → N such that f f ′ = g g′. In this instance we alsosay that the (T0; 6)-generic theory T has amalgamation over closed(self-sucient) sets.
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The next theorem generalizes Lemma 2.21 in the workby J. T. Baldwin and N. Shi [37].
Theorem 2.3.4. Let (T0; 6) be a self-sucient class, M bea (T0;6)-generic model, and K be the class of all models of T =Th(M) which has nite closures. The following conditions are equiv-alent:
(1) theory T has amalgamation over closed sets;(2) theory T has amalgamation over self-sucient sets;(3) M is an ω1-universal model;(4) M is an ω-saturated model.P r o o f. (1) ⇒ (2) Is obvious. (2) ⇒ (1) Follows from the
compactness theorem and the condition that the class K has niteclosures.
(2) ⇒ (3). Let N be a countable model of T . We show that Nis elementary embeddable in M. We represent N as a union
⋃i∈ω
Ai
of an ascending 6-chain of self-sucient sets Ai, i ∈ ω, in N . SinceM is a generic model, the sets Ai are strongly embeddable in Mvia some strong embeddings fi : Ai → M , so that fi ⊆ fi+1, i ∈ ω.Let f be the embedding
⋃i∈ω
fi and N ′ be the image f(N). It is easy
to see that N ′ is a universe of a submodel N ′ of M. We claim thatN ′ 4 M. We claim that (N ′, a) ≡ (M, a) for any tuple a ∈ N ′.Since the class K has nite closures, the tuple a can be extended tosome tuple b enumerating the image f(Ai) of some set Ai. The self-suciency of Ai implies the self-suciency of f(Ai) in both of themodels N ′ and M. Since T enjoys amalgamation over self-sucientsets, we have (N ′, b) ≡ (M, b). Consequently (N ′, a) ≡ (M, a).
(3) ⇒ (4). Let M be an ω1-universal model. Since K has niteclosures, it is enough to show that all 1-types over self-sucient setsin M are realized in M. Let A be a self-sucient set in M, Φ(A)be a type in T0 for which M |= Φ(A), and p be a type in S1(A).Consider a countable elementary extension N of M in which thetype p is realized by some element a, letting f be an elementaryembedding of model N in model M. Since f(A) is self-sucient inM and M |= Φ(f(A)), there exists an automorphism g of M, map-ping f(A) to A. Consequently the element g(f(b)) is the requiredrealization of p in M.
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(4) ⇒ (2). We x strong embeddings f : A → M0 and g :A → M1, and also a type Φ(A) ∈ T0 for which M0 |= Φ(f(A)) andM1 |= Φ(g(A)). By Compactness Theorem, we may assume thatM0 and M1 are countable models. Since M is saturated, there areelementary embeddings f1 : M0 → M and g1 : M1 → M. Hencef1(f(A)) and g1(g(A)) are self-sucient sets in M, and we haveM |= Φ(f1(f(A))) and M |= Φ(g1(g(A))). Therefore there is anautomorphism h of M mapping f1(f(A)) to g1(g(A)). The maps g1
and f1 h witness that M is the required amalgam of the modelsM0 and M1. ¤
Let K be a class having nite closures, M be a model in K,and S be a set in M. The least (by inclusion) closed set in M,containing S, is called an intrinsic closure of S in M and is denotedby iclM(S), or by S, if it is clear from the context which of themodels M is in point. If the set S is nite then it is referred toas a self-sucient closure of the set S. A type in the class T0,corresponding to the self-sucient closure A of a set A, is denotedby Φ(A). If Φ(A) ∈ T0 and M |= Φ(A), then the type Φ(A) iscalled a self-sucient closure of the type Φ(A).
Theorem 2.3.5. If the class K has nite closures then for anymodel M∈ K and any nite set A ⊆ M there exists a self-sucientclosure A of A. Moreover, A ⊆ aclM(A).
P r o o f. Let A1 and A2 be self-sucient sets in M contain-ing A. By axiom (vii), their intersection A1 ∩ A2 is also a self-sucient set in M containing A. Cardinalities of self-sucient setsare nite; so there is a unique self-sucient set in M containing Aand having the least cardinality.
We show that A ⊆ aclM(A). Let N be an ω-saturated elemen-tary extension of M. Assume that p tp(A/A) is a non-algebraictype. Then there exists a realization A′ of p in N which is distinctfrom A. However iclM(A) = iclN (A) = A. Hence the existenceof A′ contradicts the uniqueness of iclN (A). ¤
Corollary 2.3.6. If the class K has nite closures then thegeneric model M is homogeneous.
P r o o f. Let a and b be two tuples of the same type in M, andlet A and B be sets consisting of elements of a and b respectively.Then the conditions A ⊆ aclM(A) and B ⊆ aclM(B) imply that
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Φ(B) = Ψ(B) for types Φ(A), Ψ(B) ∈ T0, where M |= Φ(A) andM |= Ψ(B)). Since M is a generic model, there exists an automor-phism f ∈ Aut(M) mapping A to B so that f(A) = B. ¤
§ 2.4. Genericity of countable homogeneous modelsA generic class (T0; 6) is (minimal) hereditary if T0 consists of
(minimal by inclusion) types Φ(A) containing all possible formulasdescribing a number of copies of a system of elements of a set Bover a system of elements of a set A, and interrelations of elementsof copies for each set B ⊇ A, where a respective type Ψ(B) belongsto T0 and satises Φ(A) 6 Ψ(B).
Theorem 2.4.1. Every at most countable homogeneous (sat-urated) algebraic system M is a (T0; 6)-generic model for somehereditary generic class (T0; 6) (with the t-covering property).
P r o o f. Let M be a countable homogeneous algebraic system.The required class is the hereditary class (T0;6), where T0 consistsof all copies of complete types Φ(A) for every nite set A ⊆ M , and6 is an inclusion relation. If M is a saturated model, then the class(T0;6) possesses the t-covering property, since the theory Th(M)is small. ¤
By virtue of the fact that each complete countable theory hasan at most countable homogeneous model, Theorem 2.4.1 yields thefollowing:
Corollary 2.4.2. Every complete countable theory is generic.We know that the property of T being small implies that there
exists a countable saturated model of T . By Theorem 2.4.1, there-fore, the theory T is (T0; 6)-generic for some generic class (T0; 6)with the t-covering property.
Conversely, if T is a (T0; 6)-generic theory for some genericclass (T0; 6) with the t-covering property, then the countability ofa number of types [Φ(A)]AX , where Φ(A) ∈ T0, and the t-coveringproperty imply that the set S(T ) of types for T , is countable, thatis, T is small.
We thus arrive atTheorem 2.4.3. For any complete countable theory T , the fol-
lowing conditions are equivalent:(1) T is small;(2) T is (T0; 6)-generic for some generic class (T0; 6) with the
t-covering property.
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A generic class (T0;6) is complete if there exists a type Φ(A) ∈T0 containing some complete theory of language Σ.
The property of (T0; 6) being complete implies that the set offormulas occurring in types of T0 generate a (T0; 6)-generic theory.At the same time, the condition of being hereditary for (T0; 6)cannot guarantee generation of a complete theory. For example,in the generic construction of an innite linearly ordered set via aminimal hereditary class (T0; 6), the formula ∀x, y ((x ≤ y) ∨ (y ≤x)) is not deduced from a set of formulas occurring in types Φ(X),where Φ(A) ∈ T0 for some A.
In the proof of Theorem 2.4.1 (on representability of any count-able homogeneous model as a generic one), use was made of com-plete generic classes. At the same time, generic classes, in solvingvarious problems, are dened for constructing an a priori unknowntheory. For a required theory to possess requisite properties, there-fore, it is preferable that our generic classes involve types containinga minimum of relevant information.
Let (T0; 6) and (T′0; 6′) | be generic classes of languages Σ
and Σ′, respectively, with Σ ⊆ Σ′. We say that the class (T′0; 6′)
dominates the class (T′0; 6′), and write T0 E T′
0, if for any typeΦ(A) ∈ T0 there is a type Φ′(A′) ∈ T′
0 such that Φ(A) ⊆ Φ′(A′), andthe condition of there being some nite systems, which are exten-sions over A, together with available information on interrelationsof elements in these extensions written in the type Φ(A), impliesthat the same extensions exist over A, and that similar informationis available on interrelations of elements in those extensions writtenin the type Φ′(A′).
Obviously, the relation E is a preorder on the class of genericclasses.
It is easy to see that if a model M is isomorphically embeddablein a modelM′ ¹ Σ, then the minimal hereditary class (T′
0; 6′), whichis equal to the closure of a set of types Φ′(B) for all possible nitesets B ∈ M ′ w.r.t. bijective substitutions of constants on which theinclusion relation 6′ is dened, dominates the minimal hereditaryclass (T0;6), which is equal to the closure of a set of types Φ(A)for all possible nite sets A ∈ M w.r.t. bijective substitutions ofconstants on which the inclusion relation 6 is dened.
At the same time, the condition T0ET′0 implies that the (T0; 6)-
generic model is isomorphically embeddable in the restriction of the(T′
0;6′)-generic model to the language Σ.
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Thus, we haveTheorem 2.4.4. Let M and M′ be countable homogeneous mod-
els of languages Σ and Σ′, respectively. The following conditions areequivalent:
(1) model M is isomorphically embeddable in model M′ ¹ Σ;(2) there are generic classes (T0; 6) and (T′
0; 6′) such that Mand M′ are, respectively, (T0; 6)- and (T′
0; 6′)-generic, andT0 E T′
0.
§ 2.5. The uniform t-amalgamation property and sat-urated generic models
Let (T0; 6) be a self-sucient class satisfying the following con-ditions:
a) for any type Φ(A) ∈ T0, the type Φ(A) yields a formulaχΦ(A) describing the self-sucient condition for the closure Φ(A);moreover, χΦ(A) also contains a formula which is deducible fromΦ(A) and describes an upper bound for the cardinality of the set A;
b) for any self-sucient types Φ(A) and Ψ(B), where Φ(A) 6Ψ(B), and for any formula ψ(X,Y ) in Ψ(X ∪ Y ) (here, X andY are disjoint sets of variables, bijective with sets A and B \ Arespectively), there exists a formula ϕ(X) which is deducible fromΦ(X) and is such that the following formula holds true in M:
∀X ((χΦ(X) ∧ ϕ(X)) → ∃Y (χΨ(X,Y ) ∧ ψ(X, Y ))).
If the above conditions are satised then we say that the class(T0;6) has the uniform t-amalgamation property .
Notice, that the concept of uniform t-amalgamation generalizesuniform amalgamation as dened in work by J. T. Baldwin andN. Shi [37].
As distinct from the uniform amalgamation property, the uni-form t-amalgamation property, if satised, is not supposed to in-volve a function f ∈ ωω, representing uniform upper bounds f(n)for the cardinalities of self-sucient closures A depending only oncardinalities n of given sets A.
As an example of a generic class that possesses the uniformt-amalgamation property but lacks in the above-mentioned upperbound, we may take a class of types corresponding to nite acyclic
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undigraphs with unbounded degrees of all elements. In fact, takingelements an and bn, that are connected by shortest routes of lengthn, we have unbounded cardinalities of self-sucient closures, thatformed by adding of elements of shortest routes: |an, bn| = n + 1.
The following theorem generalizes Theorem 2.28 in work byJ. T. Baldwin and N. Shi [37].
Theorem 2.5.1. If (T0; 6) is a self-sucient class having theuniform t-amalgamation property and the class K has nite closures,then the (T0; 6)-generic model M is ω-saturated. Moreover, anynite set A ⊆ M is extendable to its self-sucient closure A ⊆ M ,the type tp(A) contains the type Φ(Y ) for a self-sucient type Φ(A),and Φ(Y ) ` tp(A).
P r o o f. Let M be a (T0; 6)-generic model and N be an ω-saturated model of Th(M). We show that the models M and Nare L∞,ω-equivalent. To do this, it suces to establish that betweenM and N there are nite partial isomorphisms f : A∼→A′ whichare mutually extendable for any self-sucient sets A ⊆ M and A′ ⊆N realizing self-sucient types Φ(A) and Φ′(A′), where Φ′(A′) =Φ(A′).
Let f : A∼→A′ be a nite partial isomorphism satisfying theconditions above. Consider a self-sucient type Ψ′(B′) ∈ (T0; 6)with Φ′(A′) 6 Ψ′(B′) and N |= Ψ′(B′). Since M is a (T0; 6)-generic model, M has an isomorphic copy B of B′ over A′ which isrealized over A and is such that M |= Ψ(B), Ψ(B) = Ψ′(B), andΦ(A) 6 Ψ(B), for a self-sucient type Ψ(B). This means that therequired extension g : B∼→B′ of the partial isomorphism f exists.
Consider a self-sucient type Ψ(B) ∈ (T0; 6) with Φ(A) 6Ψ(B) and M |= Ψ(B). Since the formulas
∀x ((χΦ(X) ∧ ϕ(X)) → ∃Y (χΨ(X,Y ) ∧ ψ(X, Y ))).
are true in M, they are also true in N . The set (χΨ(A′, Y ) ∪ψ(A′, Y ) | ψ(A,B \A) ∈ Ψ(B) is locally satisable and the modelN is ω-saturated; so the set at hand is satisable in N , that is, thereexists a set B′ ⊆ N satisfying N |= Ψ′(B′), Ψ′(B′) = Ψ(B′), andΦ′(A′) 6 Ψ′(B′), for a self-sucient type Ψ′(B′). Hence again weare faced up to the required extension g : B∼→B′ of f .
In view of the possibility for extending isomorphisms f : A∼→A′,so as to preserve formulas of corresponding self-sucient types Φ(A)and Φ′(A′), on the basis of the back-and-forth argument, we con-
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clude that the model M with a constantly distinguished set A isisomorphic to a countable elementary submodel of the model Nwith a constantly distinguished set A′. Since the sets A and A′ arechosen arbitrarily and the model N is saturated, we see that anytype over a nite set in M is realized in, that is, M is a saturatedmodel.
A possibility for extending partial isomorphisms f : B∼→B′ soas to preserve formulas of corresponding self-sucient types Ψ(B)and Ψ′(B′), implies that if Ψ(B) = Ψ′(B) then there exists an au-tomorphism of M extending a given partial isomorphism betweenB and B′. Hence, tpM(B) = tpM(B′). Since any type Φ(A) canbe extended to its self-sucient closure Φ(A) (see Theorem 2.3.5above), any nite set A ⊆ M is extendable to its self-sucient clo-sure A ⊆ M so that the type tp(A) contains the type Φ(Y ) for aself-sucient type Φ(A), and Φ(Y ) ` tp(A). ¤
Theorem 2.5.1, combined with Compactness Theorem andLemma 1.2.3, implies that, for any selfsucient class (T0;6) withuniform t-t-amalgamation, if the class K has nite closures, then the(T0;6)-generic theory T = Th(M) is ∆(T0)-based, where ∆(T0)is a set consisting of all possible formulas obtained by existentiallyquantifying over conjunctions
n∧i=1
ϕi(X) of formulas ϕi(X) ∈ Φ(X),i = 1, . . . , n, where Φ(A) ∈ T0 for some set A. Thus Theorem 2.5.1gives rise to the following:
Corollary 2.5.2. If P is some property of formulas preservedunder Boolean combinations of the formulas, (T0; 6) is a self-sucient class possessing the uniform t-amalgamation property, andthe class K has nite closures, then any formula of (T0; 6)-generictheory possesses P i any formula in ∆(T0) P has the property P .
In work by V. Harnik and L. Harrington [72] was stated that anyBoolean combination of stable formulas is itself a stable formula.Based on Corollary 2.5.2 we derive
Corollary 2.5.3. If (T0; 6) is a self-sucient class possessingthe uniform t-amalgamation property, and the class K has niteclosures, then the (T0;6)generic theory is stable i any formula inthe set ∆(T0) is stable.
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§ 2.6. On the nite closure property in fusions ofgeneric classes
E. Hrushovski [91] dened a mechanism of fusion of two generictheories for obtaining a strongly minimal theory, having a struc-ture with elds of two dierent characteristics. His technique hasdeveloped essentially last time by questions of fusions of elds andof fusions of vector spaces having various required properties (seeA. Baudisch, A. Martin-Pizarro, M. Ziegler [53], [51], [54], [52];A. Hasson, M. Hils [77]; K. Holland [85], [86]; M. Ziegler [208]). Con-sidered in Section 1.2 combinations and colorings of models, whenthese models are countable and homogeneous, are interpretable aspartial cases of fusions for correspondent generic classes.
As shown in Section 2.3, any self-sucient generic class generatesoperations of self-sucient closures on its generic models. Afterfusion of generic closures, these operations, via a transitive closure,are extended to the operation of self-sucient closure on a genericmodel of this fusion. Thus, the system of nite closures arises,generating new and more general operation of nite closures.
Since a saturation of generic model is conditioned by formula-denability of operation of self-sucient closure A of each niteset A, a natural question arises on possibility of creating for fu-sion of generic models, having formula-denable operations of self-sucient closures, with formula-denability of resulted operation ofself-sucient closure.
In this Section, we shall formulate exactly this problem on fusionof generic classes and shall propose sucient conditions for existenceof such fusions with that models possess nite closures.
Let (T; 6) be a generic class. We say that (T;6) has the -nite closure property if any model of (T;6)-generic theory has niteclosures.
The following theorem proposes a characterization of the niteclosure property for generic classes in terms of domination relation.
Theorem 2.6.1. A generic class (T; 6) of language Σ has thenite closure property i (T; 6) is dominated by a generic class(T′; 6′) of language Σ, satisfying the following conditions:
(1) each type Φ(A) of T′ contains a description of some its min-imal self-sucient extension and has a restriction to a type, in T,over A;
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(2) each type p(x) ∈ S(∅) of (T′; 6′)-generic theory has an ex-tension q(y) ∈ S(∅), containing a type [Φ(A)]AY , where Y is the setof coordinates of tuple y and Φ(A) ∈ T′.
P r o o f. Suppose, that the class (T;6) has the nite closureproperty. Then each nite set in a model of (T; 6)-generic the-ory T is extensible to a self-sucient set. Since the set of all types[Φ(A)]AX , correspondent to types Φ(A) ∈ T, is countable, all possiblepairwise inconsistent extensions of types Φ(A) to types Ψ(A), con-taining descriptions of their self-sucient extensions, form requiredgeneric class (T′; 6′), where 6′ inherits the relation 6.
Conversely, suppose that the generic class (T;6) is dominatedby a generic class (T′; 6′) of the same language and such that anytype Φ(A) of T′ contains a description of some its minimal self-sucient extension and has a restriction to a type, in T, over A,and also each type p(x) ∈ S(∅) of (T′;6′)-generic theory T has anextension q(y) ∈ S(∅), containing a type [Φ(A)]AY , where Y is the setof coordinates of tuple y and Φ(A) ∈ T′. Then any (T′; 6′)-genericmodel is (T; 6)-generic. Since any type p ∈ S(T ) contains an infor-mation on existence of self-sucient extensions of its realizations,we have the nite closure property for the generic class (T; 6). ¤
In view of the condition (2) in Theorem 2.6.1, if the class(T;6) has the nite closure property, then there are countably manyrestrictions of types q(y) ∈ S(∅) to types [Φ(A)]AY .
The generic class (T′; 6′), described in Theorem 2.6.1, is said tobe a generic class, witnessing on the nite closure property for theclass (T;6). Similar addition of external information on propertiesof types of given generic class (T;6), forming an expansion of genericclass, is said to be a witness on correspondent property.
Let (T0;60), (T1; 61), and (T2; 62) be generic classes of lan-guages Σ0, Σ1, and Σ2 respectively, Σ0 = Σ1 ∩ Σ2, 60 = 61 ∩ 62.A generic class (T3;63) of language Σ1 ∪ Σ2, such that (T3; 63) ¹Σi = (Ti; 6i), i = 1, 2, is said to be a fusion of classes (T1; 61) and(T2;62) over (T0;60). Here, a (T3; 63)-generic model (theory) isa fusion of (T1;61)- and (T2; 62)-generic models (theories).
Fusions of (T1;61) and (T2; 62) over (T0; 60) will bedenoted by
(T1; 61) F(T0;60) (T2; 62).
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A fusion of (T1; 61)-generic model M1 (theory T1) and(T2;62)-generic model M2 (theory T2) over (T0; 60)-generic modelM0 (theory T0) is denoted by M1FM0M2 (respectively T1FT0 T2).
Clearly, fusions of classes (T1;61) and (T2; 62) is allowed tonot exist (if, for instance, a chain of self-sucient closures for a setrelative to 61 and to 62 is not stabilized), and if it exists, then it canbe not unique. Moreover, a presence of the nite closure propertyor the uniform t-amalgamation for the classes (T1; 61) and (T2; 62)doesn't imply correspondent properties hold for fusions.
Besides notice, that the nite closure property may be satisedboth with uniform estimates for powers of closures, depending oncardinalities of initial nite sets, and in a case of absence of such esti-mates, provided that powers and structures of closures are describedin types of given nite sets. Generic classes with power estimatesare called PE-classes and generic classes without such estimates areNPE-classes.
All examples of generic classes are PE-classes, that similarto Hrushovski example generated non-negative predimension func-tions δ and have saturated generic models (see surveys by J. T. Bald-win [38], [40] and B. P. Poizat [150]). Generic classes of free acyclic(it will be considered in Chapter 3) and of cubic theories, being NPE-classes, are described in the works by the author [176] and [184].
In view of Theorem 2.6.1, the nite closure property for fusionsof generic classes is obviously characterized in terms of expansionsof generic classes.
Let Mi be a (Ti;6i)-generic model, i = 0, 1, 2, M3 bea (T1; 61) F(T0;60) (T2; 62)-generic model, where (Ti; 6i) and(T1;61) F(T0;60) (T2; 62) are self-sucient generic classes.
Clearly, M0 is elementary embeddable in M1 ¹ Σ0 and in M2 ¹Σ0; the models M1 and M2 are elementary embeddable in M3 ¹ Σ1
and in M3 ¹ Σ2 respectively. So we shall assume, that M0 is anelementary submodel of M1 ¹ Σ0 and of M2 ¹ Σ0; M1 and M2
are elementary submodels of M3 ¹ Σ1 and of M3 ¹ Σ2 respectively.Moreover, M3 = M1 FM0 M2.
Denote the operations of self-sucient closures in Mi by Cli,i = 1, 2, 3.
Clearly, for any nite set A ⊆ M3, the inclusion Cl3(A) ⊇ ⋃n∈ω
An
holds, where A0 = A, An+1 = Cl1(Cl2(An)). Moreover, since the set
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Cl3(A) is nite, the chain of sets An, n ∈ ω, is stabilized, startingwith some n. That number is said to be an iterative number and itis denoted by nA(M3) or by nA.
If Cl3(A) =⋃
n∈ωAn for any nite set A ⊆ M3, we say that the
operation Cl3 is generated by Cl1 and Cl2 and write Cl3 = 〈Cl1, Cl2〉.Notice, that conditions of coincidence or non-coincidence of op-
erators Cl3 and 〈Cl1,Cl2〉 are witnessed by expansions of fusions ofgeneric classes.
Hrushovski style fusions of generic classes (Hrushovski fusions)(see E. Hrushovski [91]; A. Baudisch, A. Martin-Pizarro, M. Ziegler[53][52]; A. Hasson, M. Hils [77]; K. Holland [85], [86]; M. Ziegler[208]) are dened by non-negative linear predimension functions δi
for classes (Ti; 6i), i = 0, 1, 2, with non-negative linear predimen-sion functions
δ(A) = δ1(A) + δ2(A)− δ0(A),
of fusions. In general, these fusions don't have closures ofform 〈Cl1, Cl2〉, since closed, relative to Cl1 and Cl2, sets (withnondecreasing values δ1(A) and δ2(A)) can be not closed relativeCl3 (summarized number of weights of connections w.r.t. δ1(A)and δ2(A) can exceed the number of elements that used in calcula-tion of δ(A)). Moreover, iterative numbers nA can be unbounded:supnA = ∞.
Theories of Herwig graphs [81] and theories of digraphs, that willbe described in Chapter 4, can be also considered as Hrushovskifusions. Moreover, a countable graph structure, supplied by weightsof edges or arcs, allows to interpret resulted theories T as fusionsof countable set of theories Tk of languages I(2)
k , k ∈ ω, withClT 6= 〈Clk〉k∈ω, where ClT is a self-sucient closure in a genericmodel of T , and Clk are self-sucient closures in generic models ofTk, k ∈ ω.
Inessential combinations M3 of models M1 and M2 with iden-tical closures Cl1 and Cl2 generate identical closures Cl3. Anotherexamples of fusions for generic classes with Cl3 = 〈Cl1, Cl2〉 arerepresented in third Chapter.
Below, in this Section, we shall consider closure operations Cl3,generated by Cl1 and Cl2. Fix a fusion of generic classes
(T3;63) (T1; 61) F(T0;60) (T2; 62).
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The following statement represents an obvious (by CompactnessTheorem) characterization for the preservation of the nite closureproperty under transition from the classes (T1; 61) and (T2; 62) tothe class (T3;63).
Proposition 2.6.2. A generic class (T3;63) doesn't have thenite closure property i, in (T3;63)-generic model, there existsa sequence An, n ∈ ω, of nite sets having the same cardinality, suchthat closures Cl3(An) can be obtained by at least n iterations w.r.t.Cl1 and Cl2, and a description of unbounded number of iterationsfor these sets is consistent with (T3; 63)-generic theory.
For the illustration, we represent an example of fusion (T3; 63)of generic classes, for which Cl3 = 〈Cl1, Cl2〉 and the nite closureproperty is not satised.
E x a m p l e 2.6.1. Let (Ti;6i) be generic classes of graph lan-guages Q(2)
i , i = 1, 2, with types describing pairwise disjoint edgessuch that any vertex is either isolated or belongs to unique edge, be-ing not a loop. Moreover, we add the following requirements:
1) the number of edges and the number of isolated vertices areunbounded;
2) each nite graph with given numbers of edges and of isolatedvertices is represented by a type in Ti;
3) if a vertex a belongs to a set A, where Φ(A) ∈ Ti and thedescription Φ(A) contains an information that a belongs to an edge[a, b], then b ∈ A;
4) relations 61 and 62 coincides with the inclusion relations.Notice, that each self-sucient closure of nite set A in
a (Ti; 6i)-generic model is obtained by adding to each endpoint ofedge, belonging to A, an opposite endpoint of the edge.
Now we dene a fusion of the generic classes (T1;61) and(T2;62), allowing for each vertex to be either isolated, or belongto unique edge, or belong to two edges of dierent colors (Q1 andQ2), such that descriptions of types contain only information aboutnite chains, such that every nite length is represented.
Self-sucient sets of (T3;63)-generic model are nite sets, closedunder adding of opposite endpoints of edges. At the same time,a presence of unbounded chains means that there exists a countablemodel of (T3; 63)-generic theory, having an innite chain. Thereare no elements in such chain belonging to self-sucient sets, sincesuch sets should be nite. ¤
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For practical creation of operation Cl3 with non-identical clo-sures Cl1 and Cl2 and with preservation of the nite closure prop-erty, it is appropriate to use the minimization iterative principal,or MI-principal, by which iterative numbers nA are minimal. Thisminimization can be majorized by estimates f of numbers nA de-pending on cardinalities of |A|: nA ≤ f(|A|). if there exists a ma-jorizing estimate for number of iterations f for all sets A, includedin self-sucient types Φ(A) ∈ T3, and this estimate is preserved forself-sucient amalgams in the class T3, then this estimate will bevalid for all models of (T3; 63)-generic theory. Having a majorizingestimate for generic class (T3; 63), the nite closure property forthis class will be satised. Thus we have the following statement:
Proposition 2.6.3. Let a class (T3; 63) coincide with a genericclass (T1; 61)F(T0;60) (T2; 62), Cl3 = 〈Cl1, Cl2〉. If classes (Ti; 6i)have the nite closure property, i = 1, 2, and there exists a majoriz-ing estimate for iterative numbers for the class (T3;63), then theclass (T3; 63) has the nite closure property too.
Now we consider a sucient condition for existence of minimalmajorizing estimate (nA ≡ 1) of fusion
(T3;63) (T1; 61) F(T0;60) (T2; 62),
where closures Cl1 and Cl2 are, possibly both, non-identical.Suppose, that on the universe of (T3; 63)-generic model M3,
there is (not necessary formula-denable) an equivalent relation E,satisfying the following conditions for any nite set A ⊆ M3:
(1) Cl1(A) =⋃
a∈A
Cl1(A ∩ E(a));
(2) Cl2(C) = C for any set C with Cl2(A) ⊆ C ⊆ ⋃a∈Cl2(A)
E(a).
Having these conditions, we say that (Cl1,Cl2) is an E-stepped special closure system with minimality condition, or anESSM-system.
If (Cl1,Cl2) is an ESSM-system, then there exists a mini-mal majorizing estimate for iterative numbers of self-sucient class(T3;63). Indeed, let A be a nite set in a model of (T3; 63)-generictheory. Then the set B Cl1(Cl2(A)) is Cl1-closed, since the oper-ation Cl1 is transitive, and the inclusion B ⊆ ⋃
a∈Cl2(A)
E(a) implies
that B is Cl2-closed.
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The following generalization of concept of ESSM-system guar-antees an existence of majorizing estimate for iterative numbers inthe fusion (T3; 63).
Suppose, that on the universe of (T3;63)-generic model M3
there is (not necessary formula-denable) an equivalent relation E,satisfying the following conditions for any nite set A ⊆ M3, whereM3 |= Φ(A) for some type Φ(A) ∈ T3:
(1) Cl1(A) =⋃
a∈A
Cl1(A ∩ E(a));
(2) if C ⊆ ⋃a∈Cl2(A)
E(a) and Cl2(C) ⊆ ⋃a∈Cl2(A)
E(a), then
Cl2(C) = C;(3) there exists a nite number mA of E-classes E1, . . . , EmA ,
dened by some formula in Φ(A) such that Cl3(A) ⊆mA⋃i=1
Ei.
Having these conditions, we say that (Cl1, Cl2) is an E-steppedspecial closure system, or ESS-system.
If (Cl1,Cl2) is an ESS-system, then there exists a mini-mal majorizing estimate for iterative numbers of self-sucient class(T3;63). Indeed, let A be a nite set in a model of (T3; 63)-generictheory. Then the iterative number is bounded by mA +1, since eachiteration denes a subset of
mA⋃i=1
Ei, and since stabilization of numberof E-classes, containing the result of two sequential iterations, theconditions (1) and (2) imply that the resulted set is simultaneouslyCl1- and Cl2-closed.
Thus we have the following
Theorem 2.6.4. Let (T3; 63) be a class
(T1; 61) F(T0;60) (T2; 62),
(Cl1,Cl2) be an ESS-system, and the classes (Ti;6i), i = 1, 2, havethe nite closure property. Then the class (T3;63) has the niteclosure property too.
We denote the class (T3;63), being in Theorem 2.6.4,by (T1; 61) FESS
(T0;60) (T2;62).
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Let (Ti; 6i), (T′i; 6′
i), i = 1, . . . , n, be generic classes satisfyingthe following conditions:
1) (T′1;6′
1) = (T1;61);2) (T′
i+1;6′i+1) = (T′
i; 6′i)FESS
(T′i;6′i)∩(Ti;6i)(Ti;6i), i = 1, . . . ,
n− 1.The generic class (T′
n; 6′n) is denoted by (FESS)n
i=1(Ti; 6i).Theorem 2.6.4 implies, that the nite closure property is pre-
served under nite iterations of creation of generic classes on a baseof ESS-systems, i.e., by transition from classes (Ti; 6i), i = 1, . . . , n,to the class (FESS)n
i=1(Ti; 6i).
Corollary 2.6.5. Any class of form (FESS)ni=1(Ti; 6i) has the
nite closure property.
C h a p t e r 3GENERIC EHRENFEUCHT THEORIES
§ 3.1. Generic theory with a non-symmetricsemi-isolation relation
The construction, outlined in this Section (and in the next), es-tablishes the existence of a powerful digraph Γgen = 〈X,Q〉 withshortest routes of unbounded lengths, which | via some inessen-tial Q-ordered coloring | is expanded to a countable saturatedmodel M with a nonprincipal powerful type p∞(x) ∈ S1(∅) suchthat the digraph ⟨
p∞(M);Rp∞Q (M)
⟩
is isomorphic to Γgen, where
Rp∞Q (M) = (a, b) ∈ (p∞(M))2 | M |= Q(a, b).
We construct Γgen while simultaneously coloring it, using thesyntactic method, described in previous Chapter for constructinggeneric models.
Let Γ1 = 〈X1; Q1〉 be a colored subgraph of an acyclic coloreddigraph Γ2 = 〈X2;Q2〉 with coloring Col : X2 → ω∪∞, a and b bevertices in X1, S be an (a, b)-route, not entirely in Γ1. The route Sis external (over Γ1) if only endpoints in S belong to X1. Denote byW (Γ1, Γ2) the set of triples (a, b, n), a, b ∈ X1, n ∈ ω \ 0, 1, suchthat a and b are connected in Γ2 by a shortest (a, b)-route of lengthn, and moreover, every shortest (a, b)-route is external over Γ1. Atriple 〈X1, Q1,W1〉, where W1 = W (Γ1, Γ2), is called a c0-subgraphof the digraph Γ2 if the vertex set X1 is nite.
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The relation \is a c0-subgraph" is denoted by ⊆c0 , that is, ifthe set W1 exists, we write 〈Γ1,W1〉 ⊆c0 Γ2. A structure 〈Γ1,W1〉will often be treated independently; we call 〈Γ1,W1〉 a c-graph anddenote it also by 〈X1, Q1,W1〉, where Γ1 = 〈X1;Q1〉. Here, X1 iscalled the universe of the c-graph 〈Γ1,W1〉.
For a c-graph Γ1 = 〈X1, Q1,W1〉, cc(Γ1) denotes the minimaldigraph Γ ⊇ Γ1, which, for any triple (a, b, n) ∈ W1 contains ashortest (a, b)-route of length n, with every vertex in Γ\Γ1 being ofdegree 2.
We dene the relation ⊆c on the class of c-graphs. Thus, thec-graph Γ1 = 〈X1, Q1, W1〉 is called a c-subgraph of the c-graphΓ2 = 〈X2, Q2,W2〉, written Γ1 ⊆c Γ2, if X1 ⊆ X2, Q1 = Q2 ∩ (X1)2,and W1 = W (Γ1, cc(Γ2)).
Clearly, the relation ⊆c induces partial ordering on any set ofc-graphs.
Below in this Section, the c-graphs are denoted by A,B, . . . (pos-sibly with indices), and their corresponding universes | A,B, . . ..The empty set ∅ in this instance, is assumed to be the universe ofa c-graph having the form 〈∅,∅,∅〉. Also, we often write A ⊆c Nin place of A ⊆c0 N .
Denote by K∗ the class of all c-graphs A = 〈A,QA,WA〉 suchthat, for any vertices a, b ∈ A, the existence of an (a, b)-route (in thegraph 〈A; QA〉, or else as the condition that (a, b, n) ∈ WA) impliesCol(a) ≤ Col(b).
Clearly, any c-subgraph of a c-graph in K∗ is also a c-graphin K∗.
Denote by K the class of all colored acyclic digraphs whose everyc-subgraph belongs to K∗.
If A and B are c0-subgraphs of a digraph N in K, then sets A∩Band A∪B are universes of the c0-subgraphs of N , which we denoteby A ∩N B and A ∪N B, respectively.
Obviously, the value of A ∩N B does not depend on the choiceofN , whereas the value ofA∪NB may vary withN . In what follows,we drop the index N from the above-mentioned denotations if thereis clarity as to which digraph is being spoken of.
If A, B = 〈B, QB, WB〉, and C = 〈C,QC ,WC〉 are c-graphs, andA = B ∩ C, then we call the c-graph 〈B ∪ C, QB ∪ QC ,WB ∪ WC〉a free c-amalgam of the c-graphs B and C over A and denote it byB ∗A C.
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Clearly, the free c-amalgam B ∗A C exists for any c-graphs A,B,and C with A = B ∩ C. In this case c-graphs A, B, and C arec-subgraphs of the c-graph B ∗A C.
A one-to-one map f : A → B is called a c-embedding of the c-graph A = 〈A,QA,WA〉 into the c-graph B = 〈B,QB,WB〉 (writtenf : A →c B) if f is an embedding of the colored graph A = 〈A,QA〉into the colored graph B = 〈B,QB〉 such thatWB ∩ (f(A)× f(A)× ω) = (f(a1), f(a2), n) | (a1, a2, n) ∈ WA.
We say that c-graphs A and B are c-isomorphic if there is a c-embedding f : A →c B with f(A) = B. In this event f is called ac-isomorphism between A and B, and c-graphs A and B are calledc-isomorphic copies.
A one-to-one map f : A → N is a c-embedding of the c-graph Ainto the digraph N (written f : A →c N ) if f is a c-embedding ofthe c-graph A into a c0-subgraph f(A) of N with universe f(A).
Lemma 3.1.1. (amalgamation lemma). The class K∗ has thec-amalgamation property (c-AP), that is, for any c-c-embeddings f0 :A →c B and g0 : A →c C, where A,B, C ∈ K∗, there exist a c-graphD ∈ K∗ and cembeddings f1 : B →c D and g1 : C →c D such thatf0 f1 = g0 g1.
P r o o f. There is no loss of generality in assuming that A ⊆c Band A ⊆c C. Obviously, as D we can take the c-graph B ∗A C. ¤
Denote by K∗0 the subclass of K∗, generated from the set of
colored digraphs Γα,β,γ = 〈0, 1, 2, (0, 1), (0, 2), (1, 2)〉, whereCol(0) = α, Col(1) = β, Col(2) = γ, α ≤ β ≤ γ, γ ∈ ω ∪ ∞,by operations of taking c-subgraphs, c-c-isomorphic copies, free c-amalgams, by an operation which, for any c-graph A and any pair(a, b) of its vertices, with Col(a) ≤ Col(b), not connected by routesin the graph cc(A), allows the set WA to be added one arbitrarilychosen triple (a, b, m), where m is a natural number greater than themaximal of lengths of shortest routes in cc(A), and by the inverseoperation allowing an arbitrary triple (a, b, m) to be removed fromthe set WA of that c-graph A.
The operation allowing the set W to be added an informationon aforesaid routes is called a tracing, and the inverse operation ofremoving of information about that routes is a Detracing.
By denition, every c-graph A is endowed with some coloringCol : A → ω ∪ ∞. The function Col′ : A → ω ∪ ∞ is called
80
an admissible recoloring of A if, after replacing the function Colby Col′, we obtain a c-graph in the class K0. Denote the c-graphproduced by the recoloring by A(Col′).
Lemma 3.1.2. If A is a c-graph in the class K∗0 and Col′ is its
admissible recoloring then the c-graph A(Col′) belongs to K∗0.
P r o o f is by induction on the number of steps to be taken inconstructing a c-graph A from the graphs Γα,β,γ . ¤
Denote by K0 the class of all colored acyclic digraphs whoseevery nite subgraph forms a c-graph in K∗
0.
Theorem 3.1.3. There exists a countable, colored, saturated di-graph M∈ K0 satisfying the following:
(1) if f : A →c M and g : A →c B are c-embeddings, andB ∈ K∗
0, then there exists a c-embedding h : B →c M such thatf = g h;
(2) if A and B are c-isomorphic c-subgraphs of the digraph M,then tpM(A) = tpM(B);
(3) the coloring of the restriction M ¹ Q of the model M to thegraph language Σ = Q is inessential and Q-ordered;
(4) Q(x, y) is a principal formula in Th(M ¹ Q).P r o o f. Using the amalgamation lemma, we construct M as
the union of c-graphs (An)n∈ω, An ⊆c An+1, in K∗0. In so doing,
we require that the following condition is met: for any c-graphsA ⊆c An and B ∈ K∗
0 with A ⊆c B, there exists a copy C ⊆c Mof B over A such that C is a c-c-subgraph of the digraph Am, forsome m > n. Moreover, the colors of the elements of C \ A, undertaking graph copies over A, are distributed over A in any admissibleway, that is, so that for any vertices a, b ∈ C, the fact that an (a, b)-route exists would imply Col(a) ≤ Col(b). The possibility for suchdistributions of colors to be realized follows from Lemma 3.1.2.
The number of the requirements being countable entails the exis-tence of a countable model M satisfying all the conditions specied.
We claim that M is saturated. Let M′ be an ω-saturated modelof Th(M), A ⊆c M, A′ ⊆c M′, and f : A →c A′ be a c-isomorphism. If A′ ⊆c B′ ⊆c M′ then the construction of M impliesthat there exists a c-isomorphic copy B of the c-graph B′ over A′,which is realized over A in M. Hence, there is a c-isomorphismg : B →c B′, extending the c-isomorphism f .
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Now, letA ⊆c B ⊆c M and X and Y be disjoint sets of variables,which are in 1-1 correspondence with sets A and B\A. Assume thatthe formula ϕn(X) (ψn(X, Y ), respectively), n ∈ ω, describes thefollowing:
(a) nite colors of elements of A (of B);(b) negations of colors not exceeding n for elements of A (of B)
that are innite in color;(c) the existence and lengths of shortest routes connecting ele-
ments of A (of B);(d) the non-existence of routes of length at most n connecting
elements of A (of B), if the elements are not linked via routes.By the construction of M,
M |= ∀X (ϕn(X) → ∃Y ψn(X, Y )),
and henceM′ |= ∀X (ϕn(X) → ∃Y ψn(X, Y )).
This implies that the set ψn(A′, Y ) | n ∈ ω of formulas is lo-cally realizable in M′; hence, it is realizable in M′ since M′ is ω-saturated. Therefore there exist a c-graph B′ ⊆c M′, where A′ ⊆c
B′, and c-isomorphism g : B → B′ extending the c-isomorphism f .The possibility for extending any c-isomorphisms f : A →c A′
and the known back-and-forth method show that a model M withdistinguished constants forming universe of a c-graph A is isomor-phic to a countable elementary submodel of the model M′ withdistinguished constants forming universe of the c-graph A′. Sincethe cisomorphic c-graphs A and A′ are chosen arbitrarily, and M′is saturated, we conclude that M realizes any type over a nite set,M is saturated, and Th(M) is small.
The possibility for extending c-isomorphisms of c-graphs lyingin saturated models also implies that for c-isomorphic c-subgraphsA and B of the colored digraph M, there exists an automorphismof M, extending the initial c-isomorphism between A and B. Con-sequently, tpM(A) = tpM(B).
Since the type of any c-graph in M is dened by formulas con-taining at most two free variables and describing colors of elements,and since there exist sequences linking the elements, we concludethat the coloring of a digraph M ¹ Q is inessential. Since the num-bers of the colors, in moving over the route, remain non-decreasing,the coloring is Q-ordered.
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If elements a and b in M ¹ Q are connected by an arc, then thetype tp(a ˆ b) is dened by a formula Q(x, y), and hence Q(x, y) isprincipal in Th(M ¹ Q). ¤
Notice, that the proof of Theorem 3.1.3 in fact repeats the proofof Theorem 2.5.1 for the generic class T0 of types, correspondentto c-graphs. Moreover, we have shown, that the class T0 has theuniform t-amalgamation property.
The theory T0 Th(M) for the color digraph constructed in theproof of Theorem 3.1.3, is said to be K∗
0-generic, and its countablesaturated model M is K∗
0-generic.By the construction of T0, for any model M′ of T0, M′ |= Q(a, b)
implies (a, b) ∈ IECTM′ , that is, tpx ˆ y(aˆ b) is dened by Q(x, y),and also by the colors of the elements a and b. Thus, from Propo-sition 1.2.13 and Theorem 3.1.3, we infer
Corollary 3.1.4. The semi-isolation relation SIp∞(x) is non-symmetric.
Below, we state that the digraph Γgen M ¹ Q possesses theproperties specied at the beginning of the present Section.
Let A be a c-subgraph of M, and a, b be elements in A. Considera c-graph B obtained by adding toA an element c such that Col(c) ≤minCol(a), Col(b), and also arcs (c, a) and (c, b). Obviously, Bbelongs to the class K∗
0 and its copy extends A in M. Consequently
T0 ` ∀x, y (Colk(x)∧Colm(y) → ∃≥ωz (Coln(z)∧Q(z, x)∧Q(z, y)))
for any k,m, n with n ≤ mink, m. In particular, if a and b arerealizations of type p∞(x), then there exists a realization c |= p∞such that |= Q(c, a) ∧Q(c, b). Hence the digraph
Γ∞ =⟨p∞(M);Rp∞
Q (M)⟩
enjoys the pairwise intersection property. That the group Aut(Γ∞)is transitive is obvious. The formula Rp∞
Q (x, y) is principal in Th(Γ∞)by Theorem 3.1.3. By the construction of T0, aclΓ∞(a) = a forany a ∈ p∞(M). Consequently Γ∞ is a powerful digraph.
By construction, the digraph Γ∞ is isomorphic to the digraphΓgen, and the latter likewise is a powerful digraph. In view of beingconstructed, note, the digraph Γ−1
gen = 〈M ; Q−1〉 is isomorphic toΓgen. Thus, we have
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Corollary 3.1.5. The digraphs Γgen and Γ−1gen are powerful.
By the construction of M, for any elements a1, . . . , an, thereare elements bi ∈ Q(ai,M), i = 1, . . . , n, which are pairwise in-comparable under
⋃n∈ω
Qn. By the denition of K∗0, there exists an
element c ∈n⋂
i=1Q(bi,M), and hence
n⋂i=1
Q2(ai,M) 6= ∅. In view of
Compactness Theorem and the property of a digraph with Q2 beingacyclic, there is a sequence (an)n∈ω satisfying |= Q2(ai, aj) ⇔ i < j.This, combined with the fact that Γgen is acyclic, implies that theformula ϕ(x, y1 y2) Q2(y1, x) ∧ Q2(x, y2) has the tree property(see [3]). Now, appealing to the simplicity criterion in the book byF. O. Wagner [27] (Corollary 2.8.9), we arrive at
Corollary 3.1.6. Theory T0 is not simple.A question whether T0 lack in the strict-order property remains
open. In favor of the positive answer is the fact that there areno innite partial orders over relations Qn on the universe, and inconstructing the generic model, use is made of free amalgams only.
Theorem 3.1.7. (1) A type q of T0 is principal i every twodistinct elements, ai and aj, in any realization a of q are connectedby some (ai, aj)- or (aj , ai)-route, and all elements of the realizationsof q are nite in color.
(2) A type q of T0 is realized in the model Mp∞ i for any re-alization a of q, every two distinct elements, ai and aj, of a areconnected by some (ai, aj)- or (aj , ai)-route, and the following con-dition holds:
Let elements of a tuple a contain members in nite colors, as-sume that af is an element of nite color, which is the common pointat which terminate the routes, connecting all elements of nite col-ors with af , and suppose that among the elements of a are membersof innite colors and that a∞ is an element of innite color, whichis the common point at which commence the routes, connecting allelements of innite colors with af . Then there exists an (af , a∞)-route.
P r o o f. Let q be any type of the theory T0 and a be a realiza-tion of q.
(1) Suppose q is a principal type. Since the type p∞ is non-isolated, all elements ai ∈ a are nite in color. Assume that thereexist distinct elements ai, aj ∈ a which are not connected both by
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(ai, aj)- and (aj , ai)-routes. By the denition of K∗0, starting with
some n, there exist tuples an whose corresponding elements ani and
anj are connected by an (an
i , anj )-route of length n but not by shorter
routes, and all other lengths of the shortest routes between them arethe same as for elements in a; moreover, the elements of an coincidein color with the respective elements of a. Hence q is not isolatedby any formula, that is, it cannot be a principal type.
Assume, now, that every two distinct elements ai, aj ∈ a areconnected by some (ai, aj)- or (aj , ai)-route, and that all elementsof a are nite in color. By Theorem 3.1.3(2), then, a formula, de-scribing colors of elements as well as lengths of shortest routes, willisolate q, that is, q is a principal type.
(2) Let q be a type realized inMp∞ , a be a realization of type p∞,and ϕ(a, y) | be a consistent formula for which ϕ(a, y) ` q(y) and|= ϕ(a, a). We claim that all distinct elements of a tuple aˆa arecoupled by routes. Indeed, if not, that is, some distinct elementsai, aj ∈ a ˆ a are not linked by (ai, aj)- or (aj , ai)-routes, by thedenition of K∗
0 , then, starting with some n, there exist tuplesan an, whose respective elements an
i , anj are connected by an (an
i , anj )-
route of length n but not by shorter routes, and all other lengthsof the shortest sequences couplings are the same as for elements inaˆa; moreover, the elements of an ˆan coincide in color with therespective elements of aˆa. By the quantier elimination in T0, inview of Theorem 3.1.3(2), the formula ϕ(x, y) satises |= ϕ(an, an),starting with some n. Since |= p∞(an), the condition |= ϕ(an, an)con icts with ϕ(a, y) ` q(y). Thus all distinct elements in aˆa arecoupled by some routes.
Now, the property of Γgen being acyclic and the fact that thecoloring is ordered imply that there exists an element af ∈ a ofnite color which is maximal among all the elements of nite color(if any) and is such that all the routes, connecting af with theelements of nite colors in a, terminate at af .
At the same time, among all members of innite colors in a (ifany) is an element a∞ in innite color such that all routes connect-ing a∞ with the other elements of innite colors commence at a∞.
It remains to observe that the coloring being ordered implies theexistence of an (af , a)-route, and that the condition of a∞ beingsemi-isolated over a (since Mp∞ = Ma) entails the existence of an(a, a∞)-route. We have thus shown that the (af , a∞)-route exists,and that the condition specied in the statement that we are provingis necessary for q to be realized in Mp∞ .
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Suppose, now, that all distinct elements in a are coupled viaroutes. If a has no elements of innite colors, then q is principalby item (1); hence, it is realized in Mp∞ . If a contains elementsof innite colors then the property of being ordered for coloringsimplies that q is isolated by the set p∞(y∞) of formulas (where a∞is an element of innite color in a at which originate all routes,connecting a∞ with all elements a, of innite colors), and also bya formula describing nite colors of elements in a and lengths ofshortest routes, connecting elements in a. Hence q is realized inMa∞ . ¤
§ 3.2. Generic theories with nonprincipal powerfultypes
In this section we come up with a description of the constructionwhich allows us to build theories with non-principal powerful typesshaped as an expansion of the theory T0 for the color digraph Mfrom previous Section.
Obviously, since the lengths of the shortest sequences are not 185bounded, there exists a non-p∞-principal p∞-type in T0. In Example1.3.1, we described a mechanism for a non-p-principal p-type to berealized in model Mp. We use the trick from that Example fornding an expansion of T0, having a powerful 1-type dened by theset p∞(x).
A type r(y1, . . . , yk) in S(T0) is said to be (p∞, y1)-principal ifp∞(y1) ⊆ r(y1, . . . , yk) and ϕ(y1, . . . , yk) ∪ p∞(y1) ` r(y1, . . . , yk)for some formula ϕ(y1, . . . , yk) ∈ r.
Clearly, Mp∞ realizes exactly those types q(y2, . . . , yk) in S(T0)that are contained in (p∞, y1)-principal types r(y1, y2, . . . , yk) ∈S(T0).
Below, in order to turn p∞ into a powerful type, for every typeq(y2, . . . , yk) not contained in any (p∞, y1)-principal typer(y1, y2, . . . , yk), we introduce a new k-ary predicate Rq so that theset Rq(y1, . . . , yk) ∪ p∞(y1) is consistent and Rq(y1, . . . , yk) ∪p∞(y1) ` q(y2, . . . , yk).
With this goal in mind, we renumber the set q of all typesq(y1, . . . , yk) for tuples aq with mutually distinct coordinates forwhich the sets of elements of aq contain elements of nite col-ors whose number is not equal to 1, and Maq is not isomorphicto a prime model or to Mp∞ : q = qm(y1, . . . , ykm) | m ∈ ω.
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In this event, by Theorem 3.1.3, qm(y1, . . . , ykm) are dened by c-isomorphism types Am of their realizations am. In what follows,therefore, the types qm(y1, . . . , ykm) will be identied with the c-isomorphism types Am.
For every type qm(y) ∈ q and for a corresponding c-isomorphismtype Am, we x the set ΦAm
(y) of formulas ϕn(y), n ∈ ω, whichisolates qm(y). The formulas of this set describe the following:
(a) nite colors of elements in am;(b) negations of colors lesser than n for elements of innite colors
in am;(c) the existence and lengths of shortest routes connecting ele-
ments of am;(d) the non-existence of routes of length lesser than n connecting
elements of am, if the elements are not linked by routes.Now, consider a c-isomorphism type A of an arbitrary tuple
a = (a1, . . . , ak) on a set A = a1, . . . , ak of cardinality k which isnot in Mp∞ and contains s 6= 1 elements of nite color.1 Denoteby maxA the value maxCol(ai) | Col(ai) ∈ ω, ai ∈ A if the set Acontains elements of nite colors. Otherwise, that is, if all elementsof A are innite in color, we put maxA 0.
We dene (k + 1)-ary relations RA, as follows:(1) ` ∃y RA(x, y) ↔ ∧
n<maxA¬Coln(x);
(2) for any n ≥ maxA, the formula RA(x, y1, . . . , yk) ∧ Coln(x)is equivalent to a conjunction of ϕn(y) ∈ ΦA(y),2 and the formuladescribing the following properties:
(a) if 〈ai1 , . . . , air〉 (where i1 < . . . < ir) is a tuple of all ele-ments ai of innite color in a, and 〈aj1 , . . . , ajs〉 (where j1 < . . . < js)is a tuple of all elements aj of nite colors in a, then there ex-ist elements z0, . . . , zr−1 and u0, . . . , us−1 such that zr−1 = yir ,Q(zm−1, zm) ∧ Q(zm−1, yim), Col(zm−1) = n for m = 1, . . . , r− 1,
1The restriction s 6= 1 is introduced only for the sake of convenience of ourfurther reasoning. It does not diminish generality in treating c-isomorphismtypes for subsequent realizations of appropriate types in Mp∞ , since any sethaving one element of nite color can be added yet another element of nitecolor, and for any types q1 and q2, the conditions Mp∞ |= q2 and q1 ⊆ q2 implyMp∞ |= q1.
2Isomorphism types of tuples realizing the formula RA(a, y), Col(a) = n ≥maxA, approximate the description of a c-isomorphism type of a tuple a, andthe limit (as n → ∞) of these approximations corresponds to a description ofthe c-isomorphism type of a.
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us−1 = yjs , Q(um, um−1) ∧ Q(yjm , um−1), m = 1, . . . , s − 1,Col(um−1) = maxCol(um), Col(yjm) for 1 < m < s, and Col(u0) =n for s > 1; if s = 0 then x = z0; if r = 0 then x = u0; if r ≥ 1 ands ≥ 2 then ` Q(x, z0) ∧Q(x, u0);
(b) a c-graph consisting of elements x, y1, . . . , yk, z0, . . . , zr−1,u0, . . . , us−1 contains no arcs other than the arcs specied in (a)and in the description of A for the elements y; nor does it containexternal shortest routes of length at most n but for the externalshortest routes connecting elements of y and the elements describedin A.3
Item (2) implies that if A and A′ are sets that are non-coincidentor are coincident but are not c-isomorphic while the numberings arekept xed, then the corresponding relations RA and RA′ will bedisjoint, starting with some color w.r.t. the rst coordinate.
Notice, that predicates RA, where A are isomorphism typesof elements of innite colors, rene the graph structure while notincreasing sets of binary relations dened by projections such as
∃y1, . . . , yi−1, yi+1, . . . , yk (RA(x, y) ∧ ϕ(x)) (3.1)
and
∃x, y1, . . . , yi−1, yi+1, . . . , yj−1, yj+1, . . . , yk (RA(x, y)∧ϕ(x)), (3.2)
where ϕ(x) is a formula distinguishing some set of elements havinga nite or conite set of colors. In fact, by the denition of RA, anyrelation corresponding to a projection of form (3.1), is dened by aformula describing transitions from x to yi via some Q-routes. Therelations corresponding to projections of form (3.2) are dened byformulas describing the presence or absence of links between yi andyj via Q-routes of some bounded length.
The same eect is observed in taking projections of RA, whereA are isomorphism types of elements of nite colors.
If a tuple a contains elements of both innite and nite colors,then the relation RA, for a corresponding isomorphism type A, leadsto new binary relations arising via formulas ψ(x, yi) in form (3.1),where yi corresponds to an element of some nite color l. Moreover,
3This means that c-isomorphism types of sets A have a unique extension toc-isomorphism types including elements x, z0, . . . , zr−1, u0, . . . , us−1, where xsatises type p∞.
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satisability or non-satisability of the formula ψ(an, ali) at realiza-
tions an and ali of natural colors n and l, respectively, is character-
ized by a relation between the color n and the length of a shortest(al
i, an)-Q-route. Formulas of form (3.2) preserve, as noted, the for-mer graph structure.
We claim that with the above-mentioned renement of the graphstructure (via relations RA) in hand, the required expansion maybe realized by newly constructing the generic model M from thec-graphs expanded by nite records saying of positive links betweenelements via intermediate elements through projections of the rela-tions RA.
We start our construction by describing the class K∗1 of nite
structures endowed with nite records holding of interrelations ofthe elements satisfying conditions (1) and (2). Since the desiredgeneric model expands a K∗
0-generic model, we assume that everynite set A, which enters K∗
1 and is restricted to the graph signatureQ with coloring Col, forms a c-graph 〈A,Q,W 〉 in K∗
0. Moreover,introduction of the relations RAm
requires that the record W isadded positive information on interrelations of the elements w.r.t.projections ∃yl1 , . . . , ylt RAm
(x, y) in accordance with item (2).Before we end to dene structures in the class K∗
1, we observethe following. As shown in Theorem 3.1.3, a c-isomorphism typeof every c-graph A (i.e., information on colors of elements and oncouplings of the elements via shortest routes) determines type of theset A in generic model. In dening every relation RA the belongingof every tuple aˆa to that relation is specied either by a principalformula describing relations between the elements in prime modelor by a sequence of formulas (see item (2)) which locally describethe absence of links between some elements of a via routes, whilekeeping the links between the element a and the elements of a xedin length.
The last description | as noted above for binary relations |depends directly on the interrelations between colors of approxima-tions an (in prime model) of an element a (these approximations willbe called sources) and lengths of shortest routes between appropriateelements of approximations an (in prime model) of a tuple a (theseapproximations will be called successors). Namely, if the color num-ber of a source an does not exceed (unbounded as n →∞) lengthsof shortest routes between elements of successors an, then the rela-tion RAholds, provided that the elements z0, . . . , zr−1, u0, . . . , us−1,
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described in item (2), are in hand. But if the color number of an
is greater than is some (unbounded as n → ∞) length of shortestroutes between elements of an, then RA | under the same con-ditions | will fail. Below, the relation which the color number ofa source an associates with a collection of pairwise non-bounded (asn → ∞) lengths of shortest routes between elements of a succes-sor an", as well as relations between the sources and the elementsof nite colors in the successors, are for brevity referred to as CL-correlation.
In generic theory, all n-types are dened by 2-types describ-ing colors of elements and lengths of shortest routes; therefore, theCL-correlation may be characterized by formulas ρ(x, yi, yj), whichexpress the following:
(i) the possibility for transiting via a Q-route from an element x,corresponding to an, to an intermediate element z, which has thesame color as x and is separated from x, by a distance equal to themaximal of lengths of shortest (an, an
i )- and (an, anj )-routes, if ai
and aj are innite in color;(ii) the possibility for transiting via a Q−1-route from an ele-
ment x, corresponding to an, to an intermediate element z, whichhas the same color as x and is separated from x, by a distance equalto the maximal of lengths of shortest (an, an
i )- and (an, anj )-routes,
if ai and aj are nite in color;(iii) the possibility for transiting from an intermediate element z
to elements yi and yj by some ternary relation RA∗(z, yi, yj), if ai
and aj are both innite or nite in color together;(iv) the possibility for transiting via a Q-route of xed length
from an element x, corresponding to an, to an element yi, corre-sponding to an
j , and the impossibility of transiting via a Q−1-routeof length at most n from an element x to an element yj , where thecolor of aj is less than n.4
In fact, if the tuple an ˆ an belongs to the relation RA thenthe condition for the CL-correlation holds. By the construction ofa generic model, therefore, there are intermediate elements z whichare involved in the description of formulas ρ(x, yi, yj); these z arecolored as is an and are connected with yi and yj by the above-specied relations.
4The relation for (x, yi, yj) described in this item corresponds to a projectionof some relation RA leaving the coordinates x, yi and yj free.
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Conversely, if, for the tuple anˆan, we have descriptions of xedcolors of elements and lengths of shortest routes corresponding toRA, and also of the formulas ρ(x, yi, yj), then the coincidence ofthe color of intermediate elements z with that of x gives rise to theCL-correlation. Consequently, an ˆan belongs to RA by denition.
In what follows, we assume that projections of form (3.1) with el-ements yi, of nite colors likewise are dened by formulas ρ(x, yi, yj),with yi = yj .
Notice also, that using distinct relations RA in the formulas ρis equivalent to employing xed relations RA∗ , whose taking de-pends only on the conguration of colors and lengths of shortestsequences for pairs of elements. Thus the formulas ρ(x, yi, yj) withRA∗ are ternary indicators for positive or negative entries of the for-mulas ∃yl1 ...ylt RAm
(x, y) in the types of the theory for the genericmodel in question. We add these indicators to the descriptions ofc-isomorphism types Am in dening the relations RAm
themselves(the formulas ρ(yi, yj , yk), or add their negations conjunctively toϕn(y)) and to the general descriptions of the types of tuples. Theadding of ρ, in this instance, extends the very language determinedby the c-isomorphism types Am with the formulas ρδ(yi, yj , yk),δ ∈ 0, 1, added. This extended language remains countable dueto there being nitely many versions of adding positive formulas ρto every c-isomorphism type.
Ultimately we specify that the class K∗1 consists of all nite struc-
tures of the language Col∪Q∪RAm| m ∈ ω, which are obtained
from c-graphs belonging to K∗0 by adding the relations RAm
con-sistent with item (2) (with the c-isomorphism types Am extendedby ρ), and all possible admissible formulas ρ(ai, aj , ak).
Finite structures A with nite records WA, forming the class K∗1
are called c1-structures. Denote by K1 the class of all models of thelanguage Col ∪ Q ∪ RAm
| m ∈ ω, whose every nite subsetforms a c1-structure in K∗
1.The concept of a c1-embedding f : A →c1 B for c1-structures
A and B, under which an appropriate record WA (Wf(A) = WB ¹f(A)) is preserved, is a natural generalization of the concept of a c-embedding. Thereby we also dene the concept of a c1-embeddingf : A →c1 N of a c1-structure A into a model N in K1.
Two c1-structures, A and B, are said to be c1-isomorphic if thereexists a c1-embedding f : A →c1 B with f(A) = B.
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Theorem 3.2.1. There exists a countable saturated modelM∈ K1 satisfying the following conditions:
(a) if f : A →c1 M and g : A →c1 B are c1-embeddings, andB ∈ K∗
1, then there exists a c1-embedding h : B →c1 M such thatf = g h;
(b) if A and B are c1-isomorphic c1-structures in M, thentpM(A) = tpM(B);
(c) the restriction of M to the language Col∪Q is a K∗0-generic
model;(d) every formula RA(a, y), where |= p∞(a), is principal, and
a c1-isomorphism type of every realization of RA(a, y) coincides witha c1-isomorphism type A.
P r o o f. The existence of a countable saturated modelM in K1
satisfying (a)-(c) is proved by following essentially the same line ofargument as we used in proving Theorem 3.1.3. In so doing, to thedescriptions of the formulas ϕn(X) and ψn(X, Y ), for every pair(a, b) of vertices, we add the following:
(a) positive information on links of triples of elements via for-mulas ρ, if the links in question in c1-structures exist;
(b) negations of the links of triples of elements via formulas ρ inwhich lengths of shortest routes to intermediate elements z do notexceed n, if the links in question in c1-structures do not exist.
We claim that for any realization a of type p∞ and for any pred-icate symbol RA, the formula RA(a, y) is principal. Let b and c bearbitrary tuples for which |= RA(a, b)∧RA(a, c). By the denitionof a relation RA, we then see that the c1-structures B and C, consist-ing of the elements aˆb and aˆc, are c1-isomorphic. Since the typestpM(B) and tpM(C) coincide, there exits an automorphism xingan element a and mapping b to c. Consequently tp(b/a) = tp(c/a),and hence RA(a, y) is a principal formula. That the c1-isomorphismtypes of the tuples b and c coincide with the c1-isomorphism type Afollows from the denition of RA(x, y). ¤
The theory T1 Th(M) forM, which is constructed for provingTheorem 3.2.1, and the countable saturated model M are said tobe K∗
1-generic.Since every type over the empty set in T1 is determined by the
type of a corresponding c1-structure, and for every type q of a c1-
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structure not in the prime model, there exists a principal formula∃yl1 ...ylt RA(a, y) (where |= p∞(a)) for which
∃yl1 ...ylt RA(x, y)(a, y) ` q,
it follows that the model Mp∞ realizes all types of T1. Thus p∞(x)is a powerful type, and we have the following:
Theorem 3.2.2. There exists a complete theory with a non-principal powerful type expanding the theory T0.
We start with the requirements to be followed in order to con-struct the theory T2 ⊃ T0 in which all non-principal types are pow-erful. With this goal in mind, we redene relations RA by replacingcondition (1) by the following:
(1′) `(∃y RA(x, y) ↔ ∧
n<maxA¬Coln(x)
)∧
∃y (RA(x, y) ∧ ϕn(y)) ↔
∧
t<maxmaxA,n¬Colt(x)
, n ∈ ω,
and replacing (2) by (2′), which is obtained from the former byreplacing the formulas ϕn(y) by formulas ϕn(y) ∧ ¬ϕn+1(y) withthe formulas ρ entered into the latter.
Repeating the description of K∗1 subject to conditions (1′) and
(2′), again we obtain the class K∗2 of a countable language, which
(up to renaming of symbols) may be conceived of as coincident withthe language of K∗
1. Finite structures A with nite records formingthe class K∗
2 are called c2-structures. Denote by K2 the class ofall models of the language Col ∪ Q ∪ RAm
| m ∈ ω, everynite subset of which forms a c2-structure in K∗
2. Similarly to theconcept of a c1-embedding, the concept of a c2-embeddingf : A →c2 B for c2-structures A and B, under which an appropriaterecord WA (Wf(A) = WB ¹ f(A)) is preserved, is a generalization ofthe concept of a c-embedding. Thereby we also dene the conceptof a c2-embedding f : A →c2 N of a c2-structure A into a model Nin K2.
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Theorem 3.2.3. There exists a countable saturated modelM∈ K2 satisfying the following:
(a) if f : A →c2 M and g : A →c2 B are c2-embeddings, andB ∈ K∗
2, then there exists a c2-embedding h : B →c2 M such thatf = g h;
(b) if A and B are c2-isomorphic c2-structures in M, thentpM(A) = tpM(B);
(c) the restriction of M to the language Col∪Q is a K∗0-generic
model;(d) every formula RA(a, y), where |= p∞(a), is principal, and
a c2-c2-isomorphism type of every realization of RA(a, y) coincideswith the c2-isomorphism type A;
(e) every formula RA(x, a), where A is the c2-isomorphism typeof a tuple a, is principal, and every realization of the formulaRA(x, a) is a realization of the type p∞.
P r o o f of (a)-(d) repeats the argument for the respectiveitems in Theorem 3.2.1.
(e) Consider an arbitrary formula RA(x, a), where A is the c2-isomorphism type of a tuple a. We claim that RA(x, a) is principal.Let b and c be any elements for which |= RA(b, a) ∧ RA(c, a). Bythe denition of RA, then, we see that the c2structures B and C,consisting of the elements bˆa and cˆa are c2-isomorphic. SincetpM(B) and tpM(C) coincide, there exists an automorphism xinga and mapping b to c. Consequently tp(b/a) = tp(c/a), and henceRA(x, a) is a principal formula. The condition that RA(x, a) `p∞(x) follows from the denition of RA(x, y). ¤
The theory T2 Th(M) forM, which is constructed for provingTheorem 3.2.3, and the countable saturated model M are said tobe K∗
2-generic.From Theorem 3.2.3(e), it follows that for every non-principal
type q(y) of T2, the model Mq realizes p∞, and hence every non-principal type is powerful. Moreover, in view of Proposition 1.1.3,introduction of the predicates RA allows us, for every tuple a, hav-ing some c2-isomorphism type Am, to nd a realization a of p∞(x)such that |= RA(a, a), from which, in view of Proposition 1.1.3, itfollows that the model Ma coincides with Ma. Thus all prime mod-els over tuples realizing non-principal types are isomorphic to Mp∞ ,and we have the following:
Theorem 3.2.4. There exists a small theory T2 expanding thetheory T0 and satisfying the condition |RK(T2)| = 2.
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§ 3.3. A theory with three countable modelsFirst, we point out general principles for expanding T2, which
lead to the construction of a theory T with |RK(T )| = 2 and theproperty (CEP).
Let a0, a′0, a1, a
′1, . . . , an, a′n, a′′n, b0, b1 (where b0 6= b1) be realiza-
tions of a powerful type p∞(x) of a theory T2 such that Mai =Ma′i , i = 0, . . . , n, Man = Ma′′n , Mai ≺ Mai+1 , |= Q(ai+1, a
′i),
i = 0, . . . , n−1, Man ≺Mb0 , Man ≺Mb1 , |= Q(b0, a′n)∧Q(b1, a
′′n),
and the elements b0 and b1 are not linked via (b0, b1)- or (b1, b0)-routes. A corealization amalgam of models Mb0 and Mb1 over atype q of a tuple 〈a0, a
′0, a1, a
′1, . . . , an, a′n, a′′n, b0, b1〉 is a model M
(denoted by Mb0 ∗q Mb1), which is an expansion of Mb0 ∪Mb1 bybinary predicates Rq = (b0, b1) and R′
q = (b1, b0).5We expand T2 to T using all possible binary predicate sym-
bols Rq and R′q so that for any of the above-mentioned realiza-
tions a0, a′0, a1, a
′1, . . . , an, a′n, a′′n, b0, b1 of p∞, the model Mb0 ex-
tends to a prime model over b0 containing the corealization amalgamMb0 ∗q Mb1 .
Moreover, we require that a saturated model for the expandedtheory satises the conditions (1′) and (2′) specied in Section 3.2,and the following:
(3) Rq(b0, y) and R′q(b1, y) are principal formulas for any type q;
(4) the relation R∗ =⋃q(Rq ∪ R′
q) forms an acyclic undigraph
with shortest routes of unbounded lengths, which is composed ofmutually disjoint relations Rq∪R′
q, where R′q = (Rq)−1, Rq∩R′
q = ∅or Rq = R′
q, with innitely many images or preimages w.r.t. eachone of Rq, R′
q;(5) every connected component w.r.t. R∗ consists of one-color
elements, and for each color α ∈ ω ∪ ∞, there are innitely manyconnected components consisting of the elements in color α;
(6) the transitive closure of the relation R∗ is disjoint from a tran-sitive closure of the relation Q;
5Notice, that for c2-structures, in treating corealization amalgams, the el-ements ai and a′i coincide since every model Mp∞ has just one element overwhich all types realized in that model are principal. After introducing relationsRq and R′q, we obtain innitely many such elements, and in dealing with ele-mentary chains, we need to take into account a possible dierence between ai
and a′i.
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(7) the connected components w.r.t. R∗ form equivalence classeswhich are partially ordered under the transitive closure of Q ∪ id.
Subject to the above-specied conditions, models Mb0 will el-ementary include Mb1 and vice versa. Moreover, by construction,for any models Ma and Mb, where a, b |= p∞, the fact that Ma ≺Mb and Mb realizes some non-principal type over a implies thatthere exists a sequence a0, a
′0, . . . , an, a′n of realizations of type p∞
such that a0 = a, Mai = Ma′i , i = 0, . . . , n, Mai ≺ Mai+1 ,|= Q(ai+1, a
′i), i = 0, . . . , n − 1, and Man = Mb. Consequently,
any limit model M over p∞ is representable as the union of an el-ementary chain (Man)n∈ω over p∞, equal to an elementary chain(Ma′n)n∈ω, where |= Q(an+1, a
′n), |= p∞(a′n). So by construction,
any two models limit over p∞ will isomorphic. Thus the condi-tion (CEP) will hold, which, by Corollary 1.1.15 and in view of|RK(T )| = 2, will imply I(T, ω) = 3.
Prior to constructing T , we look at the theory Ta, describedunder items (4) and (5), of the language Σ = Col ∪ Rq | q ∈Q ∪ R′
q | q ∈ Q, where Q is some countable index set. Thistheory is a subtheory of T , and we call it a free acyclic theory withxed-color connected components, or, brie y, an facc-theory.
Let M be a model of Ta and A be a nite set in M. A ca-graph isa structure A of the language Σ consisting of the universe A coloredby the function Col, relations Rq and R′
q on A, and a nite recordW saying of the existence, lengths and tuples of names for arcs ofshortest routes, pairwise connecting the elements of A. By analogywith c-graphs, the ca-graphs A with corresponding universes A aredenoted by 〈A,Rq, R
′q,W 〉q∈Q.
For a ca-graph A = 〈A,Rq, R′q,W 〉q∈Q, cc(A) denotes the min-
imal graph Γ ⊇ 〈A,Rq, R′q〉q∈Q with marked arcs containing, for
every pair (a, b) ∈ A2, coupled via a route in accordance with theentry in W , a shortest (a, b)-route with the tuple of names of arcsspecied in W .
Dene a relation ⊆ca on the class of ca-graphs. We call the ca-graph A = 〈A,Rq,A, R′
q,A,WA〉q∈Q a ca-subgraph of the ca-graphB = 〈B, Rq,B, R′
q,B,WB〉q∈Q, and write A ⊆ca B if A ⊆ B, Rq,A =Rq,B ∩ A2, and WA is the record saying of shortest routes linkingthe elements of A in the graph cc(B).
We say that the ca-graph A = 〈A,Rq, R′q,W 〉q∈Q is closed if A
contains all routes specied in the record W , that is, A = cc(A).
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If A, B = 〈B,Rq,B, R′q,B,WB〉q∈Q, and C = 〈C,Rq,C , R′
q,C ,WC〉q∈Q are closed ca-graphs, and A = B∩C, then a free ca-amalgamof the ca-graphs B and C over A (denoted by B∗A C) is the ca-graph〈B ∪ C, Rq,B ∪Rq,C , Rq,B ∪Rq,C ,WB ∪WC〉q∈Q.
A one-to-one map f : A → B is called ca-embedding of the ca-graph A = 〈A,Rq,A, R′
q,A,WA〉q∈Q into the ca-graph B = 〈B, Rq,B,
R′q,B,WB〉q∈Q (written f : A →ca B) if f is an embedding of the
colored graph A = 〈A,Rq,A, R′q,A〉q∈Q into the colored graph B =
〈B,Rq,B, R′q,B〉q∈Q such that the record Wf(A) of a ca-subgraph of
the ca-graph B with universe f(A) coincides with a record obtainedfrom WA by replacing all elements a ∈ A by f(a).
Two ca-graphs, A and B, are said to be ca-isomorphic if thereexists a ca-embedding f : A →ca B with f(A) = B. The map f , inthis instance, is called a ca-isomorphism between A and B, and theca-graphs A and B are called ca-isomorphic copies.
Lemma 3.3.1. (ca-amalgamation lemma). The class of all clo-sed ca-graphs has the ca-amalgamation property (ca-AP), that is, forany ca-embeddings f0 : A →ca B and g0 : A →ca C, where A, B,and C are closed ca-graphs, there are a closed ca-graph D and ca-embeddings f1 : B →ca D and g1 : C →ca D such that f0f1 = g0g1.
P r o o f. There is no loss of generality in assuming thatA ⊆ca Band A ⊆ca C. As D we can take a closed ca-graph B ∗A C. ¤
Clearly, a saturated model of the facc-theory is representable asa generic model which is constructed from all possible closed ca-graphs using the ca-amalgamation lemma.
The next proposition gives a list of basic properties for facc-theories.
Proposition 3.3.2. For any facc-theory Ta, the following state-ments hold:
(1) a countable saturated model M of Ta consists of countablymany connected components for every color α ∈ ω ∪ ∞;
(2) if A and B are ca-isomorphic ca-subgraphs of the countablesaturated model M of Ta, then tpM(A) = tpM(B);
(3) a type tpM(A) of Ta is principal i all elements of A arenite in color, and all one-color elements belong to one connectedcomponent; moreover, the prime model M0 consists of elements ofnite colors each forming one connected component for every nitecolor;
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(4) a type tpM(A) of Ta is realized in Mp∞ i the subtype of allelements of nite colors in A is principal and the elements of innitecolors in A belong to one connected component.
P r o o f is obvious. ¤The construction of a theory T expanding both T2 and some
facc-theory proceeds by steps, similarly to how was T2.We dene the notion of a c3-graph subject to conditions given be-
low, assuming that the language of every c3-graph consists of unarypredicate symbols for the coloring Col, the binary predicate sym-bol Q, and a countable well-ordered set of binary predicate symbolsRq and R′
q, for some c3-isomorphism types corresponding to types q.I. If A = 〈A,Q, W 〉 is a c-graph in the class K∗
0, then the struc-ture 〈A, Q,Rq, R′
q, W ′〉q∈Q, obtained from A by adding the emptyrelations Rq and R′
q and the entries recording lengths of all shortestQ-routes, linking the elements of A, is a c3-graph.
II. Every ca-graph, to which the empty relation Q is added, isa c3-graph.
III. Let Γδ = 〈Aδ, Q, Rq,δ, R′q,δ, Wδ〉q∈Q, δ ∈ 0, 1, be c3-graphs
with universes Aδ = a0, a′0, a1, a
′1, . . . , an, a′n, a′′n, bδ, ai and a′i be
one-color elements linked via a (sole) shortest route in an undi-graph with R∗, i = 0, 1, . . . , n, and an and a′′n also be one-colorelements linked via a (sole) shortest route in an undigraph with R∗,|= Q(ai+1, a
′i), i = 0, . . . , n − 1, |= Q(b0, a
′n) ∧ Q(b1, a
′′n). Assume
that b0 and b1 are one-color elements which are not connected by(b0, b1)- or (b1, b0)-routes in a graph with the relation represented asthe union of Q and the (nitely many) relations Rq ∪ R′
q, involvedin the shortest routes, connecting the elements of A0 ∪ A1. Thenthe structure
Γ = 〈A0 ∪A1, Q, Rq, R′q,W0 ∪W1 ∪Wr(b0, b1)〉q∈Q,
is a c3-graph, which is called a corealization amalgam of Γ0 andΓ1, where the relations Q and Rq, dened on the c3-graphs Γ0 andΓ1, are joined, and to the empty relations Rr and R′
r (where ris a type describing quantier-free links between elements of A0 ∪A1, and also the interrelations between the elements of A0 ∪ A1
according to the record W0 ∪ W1) we add the pairs (b0, b1) and(b1, b0) respectively; Wr(b0, b1) is a record saying of the existenceof a set A0 ∪ A1 extending to Γ the graph 〈b0, b1, Q, Rq, R
′q〉q∈Q
with the empty relations Q, Rq, and R′q except Rr = (b0, b1) and
R′r = (b1, b0).
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IV. We dene an irre exive relation < on the set Q as follows:q1 < q2 ⇔ i Wq2(b0, b1) mentions Rq1 . Obviously, in dening Rq
and R′q by steps, the transitive closure of <, in view of item III, can
be extended to a well-order relation 6 on Q, consisting of all typesr described in III. Below, we assume that the set Σ2 Q ∪ Rq |q ∈ Q ∪ R′
q | q ∈ Q of binary signature symbols is well ordered,where Q is the least element, Rq1 does not exceed Rq2 if q1 6 q2, andR′
q1is sandwiched between Rq1 and its immediate successor Rq2 . 6
V. If Γ = 〈A, Q,Rq, R′q,W 〉q∈Q is a c3-graph then, for any set
A0 ⊆ A, a c3-graph (which is a c3-subgraph of the c3-graph Γ) isa structure like
〈A0, Q ∩A20, Rq ∩A2
0, R′q ∩A2
0,W0〉q∈Q,
where W0 consists of the following (induced by the c3-graph):(a) records Wq(b0, b1) for all pairs (b0, b1) belonging to relations
Rq ∩A20;
(b) entries describing lengths of shortest routes in the c-subgraphof a c-graph 〈A, Q,Wc〉 having universe A0, where Wc is an entryrecording shortest routes w.r.t. Q in the c3-graph Γ;
(c) entries describing lengths and tuples of names for arcs ofshortest routes in the ca-subgraph of a ca-graph 〈A,Rq, R
′q,Wca〉q∈Q
having universe A0, where Wca is an entry recording shortest lengthsand tuples of names for arcs of shortest routes w.r.t. relations Rq,R′
q, q ∈ Q, in the c3-graph Γ;(d) entries recording the existence and lengths n of shortest
(a, b)-routes (if any) via intermediate elements c, for which thereare (a, c)-routes w.r.t. Q, and (c, b) ∈ R0, where R0 is the least sig-nature symbol in Q; moreover, the formula θn(x, y), describing theshortest (a, b)-routes via intermediate elements c, is equivalent toevery formula describing the shortest (a, b)-routes via intermediateelements c′, where the (a, c′)-routes w.r.t. Q are of the same lengthn, while c′ and b are linked in the ca-graph by a given set of namesof arcs.
We require that the entries specied in (a)(d) cancel out for anyxed vertex pair (a, b), and that the record W itself, too, consists of
6Since the symbols of Σ2 are well ordered, every record Wr(b0, b1) in theconstruction of a generic model allows the c3-graph having universe b0, b1 andthe same record to be extended to a nite c3-graph in which all records likeWq(a, b) are realized.
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mutually exclusive entries for the pairs (a, b) ∈ A2 described in (a)(d). Furthermore, we insist on some formula θn(a, b) being involvedin the record W for any pair (a, b) coupled by some route in a graphwith Q ∪R∗ but not by any routes in a graph with Q, or with R∗.
VI. Let A, B = 〈B, QB, Rq,B, R′q,B,WB〉q∈Q and C = 〈C, QC , Rq,C ,
R′q,C , WC〉q∈Q be closed c3-graphs, that is, those which, together
with any two vertices belonging to one connected component w.r.t.R∗, contain a shortest R∗-route, and A = B ∩ C. Then the freec3-amalgam 〈B ∪C,QB ∪QC , Rq,B ∪Rq,C ,WB ∪WC ∪W 〉 of the c3-graphs B and C over A (denoted by B ∗A C) is a c3-graph , where Wis a record including all formulas θn(a, b) for vertices a, b ∈ B ∪ Cwhich both do not belong to B and C together and are connectedby a shortest route in which the least number n ≥ 1 of Q-arcs andnot less than one R∗-arc are involved.
VII. Let A = 〈A,Q, Rq, R′q,W 〉q∈Q be a c3-graph, (a, b) be a pair
of vertices for which some formula θn(a, b) belongs to the record W ,and α = (α1, . . . , αm) be a tuple of signature symbols in Σ2 inwhich Q occurs at least n times, and the symbols of Q occur atleast once. Then the structure, which is obtained by adding to A anexternal (a, b)-route whose tuple of arcs has the same set of namesas has α and the degrees of the new vertices are equal to two, isa c3-graph. The operation of adding the above-mentioned externalroutes is called a c3-tracing.
Denote by ∆0 the class consisting of all c3-graphs correspond-ing to some c- or ca-graph in accordance with items I and II, andincluding all possible c3-graphs of the forms:
Γ0,q = 〈0, 1, 2, (0, 1), (0, 2), Rq, R′q,W 〉q∈Q, Col(0) ≤ Col(1),
Col(1) = Col(2), Γ1,q = 〈0, 1, 2, (1, 0), (2, 0), Rq, R′q, W 〉q∈Q,
Col(0) ≥ Col(1), Col(1) = Col(2), where Rq = (1, 2) if Rq ∩R′q =
∅, and Rq = (1, 2), (2, 1) if Rq = R′q, W = Wq(1, 2);
Γ0,q1,δ1,...,qn,δn = 〈0, 1, 2, (0, 1), (0, 2), Rq, R′q,W 〉q∈Q,
Col(0) ≤ Col(1), Col(1) = Col(2), Γ1,q1,δ1,...,qn,δn = 〈0, 1, 2, (1, 0),(2, 0), Rq, R
′q, W 〉q∈Q, n ≥ 2, where the relations Rq and R′
q areempty and W contains information on there being a shortest R∗-route, connecting 1 and 2 via a sequence of arcs, having a tuple ofnames ((Rq1)
δ1 , . . . , (Rqn)δn), δ1, . . . , δn ∈ −1, 1;Γ0,n = 〈0, 1, 2, (0, 1), (0, 2), Rq, R′
q,W 〉q∈Q, Col(0) ≤Col(1) ≤ Col(2), Γ1,n = 〈0, 1, 2, (1, 0), (2, 0), Rq, R
′q,W 〉q∈Q,
Col(1) ≤ Col(2) ≤ Col(0), n ≥ 2, where Rq and R′q are empty
and W contains a formula θn(1, 2).
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Similarly to the concepts of a c1-embedding and of a c2-embedding, the notion of a c3-embedding f : A →c3 B for c3-graphsA and B under which an appropriate record WA (Wf(A) = WB ¹f(A)) is preserved generalizes the notions of a c-embedding and ofa ca-embedding.
Two c3-graphs, A and B, are said to be c3-isomorphic if thereexists a c3-embedding f : A →c3 B with f(A) = B. The map f ,in this event, is called a c3-isomorphism between A and B, and thec3-graphs A and B are called c3-isomorphic copies.
VIII. Any c3-graph is a nite structure and is obtained from thec3-graphs belonging to the class ∆0 by applying some nitely manyoperations of taking admissible recolorings, corealization amalgams,c3-subgraphs, free c3-amalgams, c3-tracings, and c3-isomorphiccopies in accordance with items IIIVII, an operation which, forany c3-graph A = 〈A,Q, Rq, R
′q,W 〉q∈Q and any pair of its vertices
(a, b) ∈ A2, Col(a) ≤ Col(b), not connected via Q- or R∗-routes inthe minimal graph including all the routes described in W , allows Wto be added one of arbitrarily chosen records saying of the exis-tence of a shortest (a, b)-route of length greater than the maximalof lengths of shortest sequences w.r.t. Q, and the inverse operationwhich, for any pair of elements, allows the record to be cleared ofpositive information on external shortest Q-routes and informationon the elements belonging to one R∗ ∪ id-equivalence class, or ontheir connectivity by formulas θn.
Denote by K∗3 the class of all c3-graphs, and by K3 the class
of all colored graphs whose every nite subgraph forms a c3-graphin K∗
3.A one-to-one map f : A → N is a c3-embedding of the c3-graph A
into the graph N ∈ K3 (written f : A →c3 N ) if f is a c3-embeddingof the c3-graph A into a c3-subgraph f(A) of N with universe f(A).
Combining the proofs of Lemmas 3.1.1 and 3.3.1 we arrive atthe following:
Lemma 3.3.3. (c3-amalgamation lemma). The class K∗3 has
the c3-amalgamation property (c3-AP), that is, for any c3-embeddingsf0 : A →c3 B and g0 : A →c3 C, where A, B, and C are closedc3-graphs in K∗
3, there exist a closed c3-graph D ∈ K∗3 and c3-
embeddings f1 : B →c3 D and g1 : C →c3 D such that f0f1 = g0g1.
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Theorem 3.3.4. There exists a countable, colored, saturatedgraph M∈ K3, satisfying the following:
(1) if f : A →c3 M and g : A →c3 B are c3-embeddings, andB ∈ K∗
3, then there exists a c3-embedding h : B →c3 M such thatf = g h;
(2) if A and B are c3-isomorphic closed c3-subgraphs of thegraph M, then tpM(A) = tpM(B);
(3) the coloring of the restriction M ¹ Q of the model M to thegraph language Σ = Q is inessential and Q-ordered;
(4) Q(x, y) is a principal formula in Th(M ¹ Σ2), and the for-mulas Rq(a, y) and R′
q(a, y) are principal for any element a ∈ M .
P r o o f is similar to the proof of Theorem 3.1.3, with thec-graphs replaced by closed c3-graphs and Lemma 3.3.3 applied tothese. To the descriptions of the formulas ϕn(X) and ψn(X, Y ), inthis instance, for every vertex pair (a, b), we add the following:
(a) positive information on tuples of names of shortest R∗-routes,if such (a, b)-routes exist;
(b) negations of R∗-routes of length at most n connecting a and band consisting of arcs whose names belong to the initial segment oflength n in the well-ordered set Q, if a and b are not linked by theR∗-routes;
(c) formulas θm in appropriate variables for all elements satisfy-ing these formulas;
(d) formulas ¬θ1 ∧ . . . ∧ ¬θn in appropriate variables for all ele-ments not satisfying any one of θm.
The description of c3-graphs implies that the coloring Col isinessential and Q-Q-ordered, and the property of being closed forc3-graphs consisting of one of Q- or R∗-arc implies being isolatedfor a formula Q(x, y) in Th(M ¹ Σ2), and for formulas Rq(a, y) andR′
q(a, y) for any element a ∈ M . ¤The theory T3 Th(M) of M, constructed for proving The-
orem 3.3.4, and the countable saturated model M are said to beK∗
3-generic.Similarly to Theorem 3.1.7, we prove the following:Theorem 3.3.5. (1) A type q of T3 is principal i q is extended
to a type r of T3 realized by closed c3-graphs every two distinct el-ements of which are connected by some Q- or R∗-route, and allelements of realizations of r are nite in color.
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(2) A type q of T3 is realized in the model Mp∞ i q is extendedto a type r of T3 realized by closed c3-graphs A in which any twodistinct elements ai and aj are connected by some Q- or R∗-route,and the following condition is met: if among elements of the set Aare elements of nite colors and elements of innite colors then:
(a) there exist elements af,1, . . . , af,k of nite colors which areconnected by R∗-routes and are such that for every element a ∈A \ af,1, . . . , af,k of nite color and every i ∈ 1, . . . , k, there isa Q-route leading from a to af,i;
(b) there exist elements a∞,1, . . . , a∞,l of innite color whichare connected by R∗-routes and are such that for every element a ∈A \ a∞,1, . . . , a∞,l of innite color and every j ∈ 1, . . . , l, thereis a Q-route leading from a∞,j to a;
(c) for any i ∈ 1, . . . , k and any j ∈ 1, . . . , l, there existQ-routes leading from af,i to a∞,j.
We dene the class K∗4 of nite structures equipped with nite
records saying of interrelations of the elements satisfying the condi-tions (1′) and (2′) specied in the previous Section, where instead ofc1-isomorphism types A, we treat all possible c3-isomorphism typeswhich are endowed with formulas ρ and are such that all primemodels over these types are not isomorphic either to a prime modelor to Mp∞ . We assume that every nite set A, which belongs toK∗
4 and is restricted to the graph language Σ2 with coloring Col,forms a c3-graph 〈A,Q,Rq, R
′q,W 〉q∈Q in K∗
3, and W is added posi-tive information on the interrelations of the elements w.r.t. projec-tions ∃yl1 ...ylt RAm
(x, y), and w.r.t. formulas ρ in accordance withitem (2′).
Finite structuresA with nite records WA forming K∗4, are called
c4-structures, and the c4-c4-structures which are closed c3-graphs arealso said to be closed.
Denote by K4 the class of all models in the language Col∪Σ2 ∪RAm
| m ∈ ω whose every nite subset forms a c4-structure in K∗4.
The notion of a c4-embedding f : A →c4 B for c4-structuresA and B, under which an appropriate record WA (Wf(A) = WB ¹f(A)) is preserved, is a natural generalization of the concepts of ci-embeddings introduced above. Thereby we also dene the conceptof a c4-embedding f : A →c4 N of a c4-structure A into a model Nin K4.
Two c4-structures, A and B, are c4-isomorphic if there is a c4-embedding f : A →c4 B with f(A) = B.
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Theorem 3.3.6. There exists a countable saturated modelM∈ K4 satisfying the following:
(a) if f : A →c4 M and g : A →c4 B are c4-embeddings, andB ∈ K∗
4, then there is a c4-embedding h : B →c4 M such thatf = g h;
(b) if A and B are c4-isomorphic closed c4-structures in M, thentpM(A) = tpM(B);
(c) the restriction of M to the language Col∪Σ2 is a K∗3-generic
model;(d) every formula RA(a, y), where |= p∞(a), is principal, and
the c4-isomorphism type of any realization of the formula RA(a, y)coincides with a c4-isomorphism type A;
(e) every formula RA(x, a), where A is the c4-isomorphism typeof a tuple a, is principal, and any realization of the formula RA(x, a)is a realization of the type p∞.
P r o o f is an obvious combination of the proofs of Theorems3.2.3 and 3.3.4. ¤
The theory T4 Th(M) of M, which is constructed for provingTheorem 3.3.7, and the countable saturated model M are said tobe K∗
4-generic.Following the argument of Theorem 3.2.4 and using Theorem
3.3.6, we state the following:Theorem 3.3.7. The theory T4 satises the condition
|RK(T4)| = 2.Theorem 3.3.8. There is a limit model of the theory T4 over
the type p∞ which is unique up to isomorphism.P r o o f. The existence of a limit model follows from Proposi-
tion 1.1.8 and Corollary 1.1.9, in view of the semi-isolation relationSIp∞ being nonsymmetric w.r.t. Q(x, y). In order to prove that ourlimit model is unique, it suces to show that any limit model Mover p∞ is saturated.
Let (Man)n∈ω be any elementary chain over p∞ whose unioncoincides with the limit model M. By Theorem 3.3.5, for everyelementary extension of the model Man to the model Man+1 , thereexists an R∗- or a Q-route leading from an+1 to an. Since the limitmodel is not isomorphic toMp∞ , and the modelMan is elementarilyextended to a prime model over an which coincides with Man+1
provided the R∗-route from an+1 to an exists, from (Man)n∈ω wecan remove all models Man for which R∗-routes from an to an−1
are available.
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With all the above-specied models removed, appealing to theconstruction of a generic model, we can extend the resulting elemen-tary chain (Ma′n)n∈ω by models Ma, where a are elements forming,for any n ∈ ω, one of the shortest Q-routes from a′n+1 to a′n, ifsuch Q-routes exist. Furthermore, the resulting elementary chainis extended to an elementary chain such that every model on oneside is neighbored by its coincident model, and on the other side |by a proper elementary substructure or superstructure.
Now, we consider an arbitrary 1-type q(x, b) ∈ S(b), where b isa tuple in M , and argue to show that q(x, b) is realized inM. Indeed,the tuple b belongs to some model Man , and, by denition of RA,for some c4-isomorphism type A, containing realizations of the typeq(x, y), and for some n′ ≥ n, there exists an element c ∈ Man′ suchthat some projection ∃zi1 , . . . , zim RA(c, z) is realized by b. Since theformula RA(c, z) is principal, there exists a tuple d ∈ M , realizingthat formula and extending b. By the choice of the c4-isomorphismtype A, some coordinate of d realizes the type q(x, b). Since thechosen type q is arbitrary, M is saturated. ¤
Appealing to Corollary 1.1.15 and Theorems 3.3.7 and 3.3.8, weconclude that the following theorem holds.
Theorem 3.3.9. There exists a complete generic theory T ex-panding the generic theory T3 and with I(T, ω) = 3.
§ 3.4. Realizations of basic characteristicsof complete theories with nitely manycountable models
Recall that Theorem 1.1.13 contains a characterization of theclass of complete theories with nitely many countable models. Weclaim that all situations described in that theorem are realized out.
Theorem 3.4.1. For any nite preordered set 〈X;≤〉 with theleast element x0 and the greatest class x1 in the ordered factor set〈X;≤〉/∼ w.r.t. ∼ (where x ∼ y ⇔ x ≤ y and y ≤ x), and for anyfunction f : X/∼→ ω satisfying the conditions f(x0) = 0, f(x1) > 0for |X| > 1, and f(y) > 0 for |y| > 1, there exist a complete theory Tand an isomorphism g : 〈X;≤〉 →RK(T ) such that IL(g(y)) = f(y)for any y ∈ X/∼.
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P r o o f. Let 〈X;≤〉 and f be as above. Without loss of gener-ality, we may assume that |X| > 1. We x a numbering ν : |X| → Xsuch that ν(m) < ν(n) and ν(m) 6∼ ν(n) imply m < n, and in cor-respondence with every ∼-class is an interval in |X|. Consider atheory T0 of unary predicates P1, . . . , P|X|−1 forming a partition on|X| − 1 innite classes with inessential coloring Col : M → ω ∪ ∞such that ` ∃≥ω (Pi(x) ∧ Coln(x)), i = 1, . . . , |X| − 1, n ∈ ω. Weclaim that there exists an expansion T of T0 with an isomorphismg : 〈X;≤〉 → RK(T ) such that:
(a) g(ν(i)) = Mpi , where Mpi is an isomorphism type of theprime model Mpi over a realization of the type pi(x) in S1(∅) forpi(x) being isolated by the set Pi(x)∧¬Coln(x) | n ∈ ω of formu-las, i = 1, . . . , |X| − 1, and p1(x), . . . , p|X|−1(x) are all non-principal1-types over ∅ in the variable x;
(b) IL(g(y)) = f(y) for any y ∈ X/∼.We construct T =
⋃i<|X|
Ti by induction so as to respect the
numbering ν. Assume T0, . . . , Tk−1 are already constructed andν(k), ν(k + 1), . . . , ν(k + l) form a ∼-class.
If f(ν(k)) = 0, then l = 0, and we dene Tk by expanding thelanguage of Tk−1 by new binary predicate symbols Rki (where theclass ν(k) covers ν(i), i 6= 0) so as to satisfy the following conditions:
(1) Rki(a, y) is a principal formula and Rki(a, y) ` pi(y) for anya |= pk;
(2) for any a, b |= pi, there are innitely many elements c |= pk
and innitely many d not realizing types p1(x), . . . , p|X|−1 such that|= Rki(c, a) ∧Rki(c, b) ∧Rki(d, a) ∧Rki(d, b);
moreover, c |= pk and |= Rki(c, a) imply that a does not semi-isolate c.
Clearly, (1) and (2) can be realized so that the model Mpkof
Tk has a unique realization of the type pk and hence IL(g(ν(k))) =0 = f(ν(k)). In addition, Mpk
, by induction, will realize all types pi
dominated by pk, that is, satisfying the relation ν(i) ≤ ν(k).Suppose f(ν(k)) = r > 0. We dene T 0
k by expanding thelanguage of Tk−1 by new binary predicate symbols Rki (where ν(k)covers ν(i), i 6= 0) satisfying (1) and (2), and by binary predicatesymbols R′
ij (where ν(i), ν(j) ∈ ν(k)) with the following conditions:
106
(3) R′ij(a, y) is a principal formula and R′
ij(a, y) ` pj(y) for anya |= pi;
(4) for any a, b |= pj , there are innitely many elements c |= pi
and innitely many d not realizing types p1(x), . . . , p|X|−1, with
|= R′ij(c, a) ∧R′
ij(c, b) ∧R′ij(d, a) ∧R′
ij(d, b);
moreover, c |= pi and |= R′ij(c, a) imply that a does not semi-
isolate c;(5) for any elements a and b not realizing types pk+l+1, . . . ,
p|X|−1, there exist innitely many c |= pj such that |= R′ij(a, c) ∧
R′ij(b, c);
(6) the relation R′k =
⋃ν(i),ν(j)∈ν(k)
R′ij forms a digraph isomorphic
to the digraph Γgen.As above, predicates Rki guarantee that types pi are dominated
by pk for ν(i) < ν(k) and ν(i) 6∼ ν(k), while relations R′ij ensure
that pi and pj are domination equivalent and that models Mpi andMpj are non-isomorphic, for i 6= j.
Now, similarly to the conditions (1′) and (2′) specied in Section3.2 with Q replaced by R′
k, we extend the language by predicatesymbols RA so that types pi, i = k, . . . , k + l, dominate all typesq(x) ∈ S(Tk) with pj(xi) 6⊆ q(x) for j > k+l, the types q in questionthat are not dominated by pi, i < k, dominate all types pk, . . . , pk+l,and the models Mq are isomorphic to Mpk
.For the condition IL(g(ν(k))) = r to be satised, on the struc-
ture of realizations of pk, we dene a graph structure with binaryrelations R′′
1 , . . . , R′′r such that:
(7) R′′i (a, y) is a principal formula and R′′
i (a, y) ` pk(y) for anya |= pk, i ≤ r;
(8) for any a, b |= pk, there are innitely many elements c |= pk
and innitely many d not realizing types p1(x), . . . , p|X|−1 for which
|= R′′i (c, a) ∧R′′
i (c, b) ∧R′′i (d, a) ∧R′′
i (d, b);
moreover, |= R′′i (c, a) and c |= pj imply that a does not semi-
isolate c;(9) the relation R′
k
⋃ r⋃i=1
R′′i forms a digraph isomorphic to the
digraph Γgen.
107
If Ma and Mb are prime models over realizations a and b of pk,respectively, such that |= R′′
i (a, b) and Mb ≺Ma, then we call Ma
an R′′i -extension of the model Mb. An elementary chain (Ms)s∈ω
over pk is called an R′′i -chain if Ms+1 is an R′′
i -extension of Ms forany s.
If Ma and Mb are prime models over realizations a and b of pi,respectively, with ν(i) ∼ ν(k), such that |= R′
ii(a, b) and Mb ≺Ma,then the model Ma is called an R′
ii-extension of Mb.Similarly to the conditions (3)-(7) given in Section 3.3, we extend
the language by symbols Rq so as to satisfy the following:(10) for any limit model M over a type pk+i, 0 ≤ i ≤ l, there
is a relation R′′j such that M is the union of an R′′
j -chain (Ms)s∈ω
over the type pk;(11) limit models M1 and M2 over pk are equivalent i there is
a predicate R′′i such that M1 and M2 are unions of R′′
i -chains butare not unions of R′′
j -chains, for j > i.Notice, that (10) and (11) are realized via predicates Rq \saying"
the following:(a) every R′
kk-extensionMa ofMb contains an R′′1-extension and
vice versa;(b) for any i, ν(i) ∼ ν(k), i 6= k, and for any nite elemen-
tary chain Ma1 , . . . ,Mas , a1, . . . , as |= pi, there exist realizationsb1, . . . , bs−1 of pk such that the sequenceMa1 ,Mb1 , . . . ,Mbs−1 ,Mas
is also an elementary chain;(c) if Mb0 and Mb1 are R′′
s -extensions of Man , q is the type of atuple 〈a0, a
′0 . . . , an, a′n, a′′n, b0, b1〉 of elements which realize type pk
and are such that Mai+1 is an R′′ti-extension of a model Ma′i , equal
to Mai , |= R′′ti(ai+1, a
′i), Mb0 and Mb1 are R′′
tn-extensions of themodel Man , equal to Ma′n and to Ma′′n , |= R′′
tn(b0, a′n)∧R′′
ti(b1, a′′n),
and elements b0 and b1 are not connected by (b0, b1)- or (b1, b0)-routes, then Mb0 contains a corealization amalgam Mb0 ∗q Mb1 ;
(d) if Ma1 , . . . ,Mas is a nite elementary chain such that themodel Maj+1 is an R′′
ij-extension of Maj , j = 1, . . . , s − 1, and
maxi1, . . . , is−2 < is−1, then Mas contains an R′′is−1
-extension ofMa1 and vice versa.
With the expansions exercised above we obtain a theory Tk =Tk+1 = . . . = Tk+l such that types pk, . . . , pk+l are domination equiv-alent, Mpk
, . . . ,Mpk+lare pairwise non-isomorphic, and the number
of limit models over pk, . . . , pk+l is equal to f(ν(k)).
108
Proceeding further with this process, at step |X| − 1, we ob-tain a theory T = T|X|−1 and an isomorphism g : 〈X;≤〉 → RK(T )such that g(ν(0)) is an isomorphism type of the prime model for T ,g(ν(m)) is an isomorphism type of Mpm , 1 ≤ m ≤ |X| − 1, andIL(g(y)) = f(y) for any y ∈ X/∼.
The possibility for the above-specied properties to be realizedis veried in essentially the same way as was sketched in construct-ing the theory with three countable models under Section 3.3 usingTheorem 2.5.1. ¤
§ 3.5. Theories with nite Rudin | Keisler preordersThe previous Sections have revealed a technique for construct-
ing all possible theories with nitely many pairwise nonisomorphiccountable models w.r.t. Rudin | Keisler preorders and the distri-bution functions of the number of limit models. The aim of thisSection is to generalize the main result of Section 3.4 from the clas-sication of theories with nitely many countable models to the caseof an arbitrary theory with a nite Rudin | Keisler preorder thathas at most countably many or continuum many limit models foreach equivalence class.
Theorem 3.5.1. For any nite preordered set 〈X;≤〉 with theleast element x0 and the greatest class x1 in the ordered factor set〈X;≤〉/∼ w.r.t ∼ (where x ∼ y ⇔ x ≤ y and y ≤ x), and for anyfunction f : X/∼→ ω ∪ ω, 2ω, satisfying the conditions f(x0) =0, f(x1) > 0 for |X| > 1, and f(y) > 0 for |y| > 1, there exista complete theory T and an isomorphism g : 〈X;≤〉 → RK(T ) suchthat IL(g(y)) = f(y) for any y ∈ X/∼.
Henceforth in this Section, we assume that the structure RK(T )is nite.
It was shown in Section 1.1 that each preordered set RK(T )contains the least element, which corresponds to the prime model,and the niteness of RK(T ) implies the existence of the largest∼RK-class corresponding to the prime models over the realizations ofpowerful types.
Repeating the proof of Proposition 1.1.7 and using the condition|RK(T )| < ω instead of the condition I(T, ω) < ω, we obtain
Proposition 3.5.2. If |RK(T )| < ω then each countable modelof T is prime over a realization of some type in S(T ) or limit oversome type in S(T ).
109
Notice, that by Morley Theorem (see M. Morley [127]), for eachclass M ∈ RK(T )/∼RK , the number of limit models IL(M) belongsto ω ∪ ω, ω1, 2ω.
Using this remark, by analogy with Theorem 1.1.13, we cometo the following theorem, which may be regarded as a syntacticcharacterization of the class of complete theories with nite Rudin |Keisler preorders.
Theorem 3.5.3. Any small theory T with a nite Rudin |Keisler preorder satises the following conditions:
(a) RK(T ) contains the least element M0 (the isomorphism typeof a prime model), and IL(M0) = 0;
(b) RK(T ) contains the greatest ∼RK-class M1 (the class of iso-morphism types of all prime models over realizations of powerfultypes), and |RK(T )| > 1 implies IL(M1) ≥ 1;
(c) if |M| > 1, then IL(M) ≥ 1.Moreover, we have the following decomposition formula:
I(T, ω) = |RK(T )|+|RK(T )/∼RK |−1∑
i=0
IL(Mi),
where M0, . . . , ˜M|RK(T )/∼RK |−1 are all elements of the partially or-dered set RK(T )/∼RK and IL(Mi) ∈ ω ∪ ω, ω1, 2ω for each i.
Therefore, by analogy with the theories with nitely many count-able models, the number of, and the relationship between, the count-able models of a theory with a nite Rudin | Keisler preorderis determined by the following two characteristics: the Rudin |Keisler preorder itself and the distribution function IL(·) of the num-ber of limit models that for each ∼RK-class may take a value inω ∪ ω, ω1, 2ω.
Show that all possibilities, described in Theorem 3.5.3 and sat-isfying the condition
rang(IL(·)) ⊂ ω ∪ ω, 2ωcan be realized. This will prove Theorem 3.5.1 conversing Theorem3.5.3 in assumption of Continuum Hypothesis.
110
We use the proof of Theorem 3.4.1. The construction of the the-ory T 0 with a given domination preorder 〈X;≤〉 repeats the con-struction of an analogous theory with nitely many countable mod-els step by step. The required expansion T of T 0, satisfying theconditions IL(g(y)) = f(y) for all y ∈ X/∼, will be constructed bya scheme similar to that of the proof of Theorem 3.4.1.
By the construction of T 0, it suces to consider the step k ofinduction for the case IL(g(ν(k))) = λ, λ ∈ (ω∪ω, 2ω)\0. Deneon the structure of realizations of the type pk a graph structure witharcs, colored by pairwise disjoint binary relations R′′
n, for n ∈ ω,satisfying the following conditions:
(1) R′′n(a, y) is a principal formula and R′′
n(a, y) ` pk(y) for anya |= pk, n ∈ ω;
(2) for any a, b |= pk, there exists innitely many elements c |= pk
and innitely many elements d that do not realize the types p1(x),. . . , p|X|−1(x) for which
|= R′′n(c, a) ∧R′′
n(c, b) ∧R′′n(d, a) ∧R′′
n(d, b),
n ∈ ω; moreover, |= R′′n(c, a) and c |= pi imply that a does not
semi-isolate c;3) the relation R′
k
⋃ r⋃i=1
R′′i forms a digraph isomorphic to the
digraph Γgen.Let Ma and Mb be prime models over realizations a and b of pk,
respectively, such that |= R′′n(a, b) and Mb ≺ Ma. Then Ma is
called an n-extension of Mb for n ∈ ω. Let w = 〈n1, . . . , nm〉 ∈ ωm
be an m-tuple, Mai , for i = 1, . . . , m+1, be models such that Mai+1
is an ni-extension of Mai , i = 1, . . . , m. Then the model Mam+1
is called a w-extension of Ma1 . An elementary chain (Ms)s∈ω issaid to be an f-chain (where f ∈ ωω) if Ms+1 is an f(s)-extensionof Ms for any s ∈ ω.
Likewise in the proof of Theorem 3.4.1, expanding the theory bythe predicates Rq and R′
q, we obtain a theory in which each limitmodel with the isomorphism type in g(ν(k)) can be represented asthe union of an f -chain for some sequence f ∈ ωω. Consequently,the number IL(g(ν(k))) is equal to the number of pairwise noni-somorphic unions of f -chains. The introduction of the additionalpredicates Rq and R′
q reduces the existence of the isomorphism
111
of the unions of f1- and f2-chains to the existence of the wordswm
1 , wm2 ∈ ω<ω, for m ∈ ω, satisfying the following conditions:
(a) the sequence fi is \similar" to the countable concatenationof the words w0
i , w1i , . . . , w
mi , . . ., i = 0, 1;
(b) any wm0 -extension is a wm
1 -extension, and vice versa, form ∈ ω.
Thus, the problem of constructing some extension of the theorywith the condition IL(g(ν(k))) = λ reduces to the problem of ndinga quotient of the set ωω by identifying words of the form wm
1 and wm2
that would contain exactly λ classes. We now formally dene andinvestigate the possibilities of taking the quotients of ωω.
Consider the semigroup S0 = 〈W ; ˆ 〉 of all nonempty words inthe alphabet ω under the operation ˆ of concatenation. If w1 and w2
are words in W , we call the formula w1 ≈ w2 an identity , as usual.Given some set I of identities wj
1 ≈ wj2 with j ∈ J including the
set I0 of all identities of view w ≈ w, we dene the set of identitiesdeducible from I. An identity w1 ≈ w2 is called deducible fromI if , there exists some nite sequence w1
1 ≈ w12, . . . , w
t1 ≈ wt
2 ofidentities such that wt
1 = w1, wt2 = w2, and each identity in the
sequence either belongs to I or follows from the previous identitiesby applying one of the following deduction rules:
(1) w1 ≈ w2
w2 ≈ w1, where w1, w2 ∈ W ;
(2) w1 ≈ w2; w2 ≈ w3
w1 ≈ w3, where w1, w2, w3 ∈ W ;
(3) w1 ≈ w2; w′1 ≈ w′2w1w′1 ≈ w2w′2
, where w1, w′1, w2, w
′2 ∈ W ;
(4) w1w2w3 ≈ w1w′2w3
w2 ≈ w′2, where w1, w2, w
′2, w3 ∈ W .
Henceforth we consider sets of identities I ⊇ I0 closed with re-spect to deducibility. Any set I of identities has a bijective corre-spondence with the semigroup SI = 〈W ; ˆ 〉/I that is the quotientof the semigroup S0 by the congruence ∼I :
w1 ∼I w2 ⇔ (w1 ≈ w2) ∈ I.
Let 〈L,≤〉 be the pair in which L is the set of all semigroups SI ,and SI1 ≤ SI2 is equivalent to I1 ⊆ I2. Clearly, the structure 〈L;≤〉
112
is a complete lattice, where sup
SIj
∣∣∣∣ j ∈ J
= SI for the set I of
all identities deducible from⋃
j∈J
Ij , and inf
SIj
∣∣∣∣ j ∈ J
= S ⋂
j∈JIj
.
The semigroup S0 is the bottom element in the lattice 〈L;≤〉, andthe one-element semigroup, in which all words are identied, is itstop element.
Dene the quotients of ωω corresponding to sets I of identities.Two sequences, f0 and f1, in ωω are called almost similar if thereexist l0, l1 ∈ ω such that f0(n + l0) = f1(n + l1) for all n ∈ ω.Two sequences, f0 and f1, are called strongly I-equivalent if fi isalmost similar to a nite concatenation of words wm
i ∈ W , m ∈ ω,i = 0, 1, where the identity wm
0 ≈ wm1 belongs to I for all m ∈ ω.
Two sequences, f and f ′, are called I-equivalent if there exists asequence f0, f1, . . . , fn ∈ ωω with f0 = f , fn = f ′, and fi and fi+1
are strongly I-equivalent for all i = 0, . . . , n− 1.Clearly, that after all pairwise identications of w1- and w2-
chains for all identities w1 ≈ w2 in I, the existence of an isomorphismbetween the unions of f - and f ′-chains becomes equivalent to theI-equivalence of f and f ′.
For any set I of identities, we denote by ωω/I the quotient ofωω by the relation of I-equivalence. Denote by T 0/I the theoryobtained from T 0 by pairwise identications of w1- and w2-chainsfor all identities w1 ≈ w2 in I. Clearly, we may assume that the setsωω/I are bijective with the theories T 0/I.
Denote by ILk(T 0/I) the number IL(M) for the class M of iso-morphism types of models of T 0/I, that includes the isomorphismtype of Mpk
.
Lemma 3.5.4. 1. If I1 ⊆ I2, then ILk(T 0/I1) ≥ ILk(T 0/I2).2. ILk(T 0/I0) = 2ω.3. ILk(T 0/w1 ≈ w2 | w1, w2 ∈ W) = 1.
P r o o f is obvious.The next lemma was implicitly proved in Section 3.4.
Lemma 3.5.5. For any n ∈ ω \ 0, there exists a set In ofidentities such that ILk(T 0/In) = n.
113
P r o o f. Fix some integer n ≥ 1 and consider the set In ofidentities deducible from the set of identities
n− 1 ≈ m,
where m ≥ n, andn1n2 . . . ns ≈ ns,
where maxn1, n2, . . . , ns−1 < ns. It is easy to see that each se-quence in ωω is In-In-equivalent to some constant sequence fr, thatis, fr(j) ≡ r, for all j ∈ ω, 0 ≤ r < n. Indeed, the identities writ-ten imply that each sequence f is In-equivalent to the sequence fr,where r is equal to the largest value below n− 1 of innite constantsubsequences of f , if existent, and r = n−1 if such subsequences failto exist. In addition, it is obvious that the sequences f0, . . . , fn−1
are pairwise In-nonequivalent. Therefore, ILk(T 0/In) = n. ¤
Lemma 3.5.6. There exists a set Iω of identities suchthat ILk(T 0/Iω) = ω.
P r o o f. Denote by Iω the set of identities deducible from theidentities
n1n2 . . . ns ≈ ns,
where maxn1, n2, . . . , ns−1 < ns, andn1n2 ≈ n1(n1 + 1)(n1 + 2) . . . (n2 − 1)n2,
where n1 < n2. We claim that ILk(T 0/Iω) = ω. Indeed, by therst system of identities each bounded subsequence f ∈ ωω is Iω-equivalent to the constant sequence fr ∈ ωω, fr(j) ≡ r, j ∈ ω,where r is the largest value of an innite constant subsequence of f .Moreover, the two systems of identities imply that each unboundedsequence is Iω-equivalent to the sequence fω ∈ ωω, where fω(j) = j,j ∈ ω. Therefore, each sequence f ∈ ωω is Iω-equivalent to somesequence fµ with µ ≤ ω. It is obvious that the sequences fµ arepairwise Iω-nonequivalent. Consequently, ILk(T 0/Iω) = ω. ¤
Theorem 3.5.1 is immediate from Lemmas 3.5.4 | 3.5.6.The question of the existence of a set Iω1 such that ILk(T 0/Iω1) =
ω1 remains open. Apparently, a positive answer could yield an alter-native for Knights approach (see R. W. Knight [107]) to construct-ing a counterexample to the Vaught conjecture on the nonexistenceof a theory T for which I(T, ω) = ω1.
114
§ 3.6. Rudin | Keisler preorders in small theories
The following modication of Theorem 3.4.1 represents a de-scription of preordered sets RK(T ) in small theories T .
Theorem 3.6.1. (1) For any small theory T , the preordered setRK(T ) is not more than countable, upward directed, and has a leastelement.
(2) For any nite or countable, preordered, upward directed set〈X;≤〉 having a least element, there exists a small theory T for whichRK(T ) ' 〈X;≤〉.
P r o o f. (1) That |RK(T )| ≤ ω follows from the propertyof T being small. The property for the preordered set RK(T ) =〈PM;≤RK〉 to be upward directed is implied by the following: ifM1 and M2 are isomorphism types of PM corresponding to mod-els Ma1 and Ma2 , then types tp(a1) and tp(a2) are dominated byq = tp(a1 ˆ a2); hence, M1 ≤RK M and M2 ≤RK M, where Mis the isomorphism type of Mq. The least element in RK(T ) isan isomorphism type of the prime model.
(2) In view of Theorem 3.4.1, there is no loss of generality inassuming that the set X is countable. A small theory T withRK(T ) ' 〈X;≤〉 is constructed similarly to how were the theo-ries constructed in proving Theorem 3.4.1, with the theory of unarypredicates P1, . . . , P|X|−1 replaced by a theory of pairwise disjointunary predicates Pi, i ∈ ω, each containing innitely many ele-ments. ¤
§ 3.7. Theories with non-dense structures of power-ful digraphs and theories with powerful typesand without powerful digraphs
In this Section, we dene a modication of construction in Sec-tion 3.2, producing a powerful digraph Γ = 〈X; Q〉 having transitiveclosure being a partial order with innitely many covering elementsfor any element in X.
115
The relation Q is a disjoint union of relations Q0 and Q1 suchthat Q0(x, y) and Q1(x, y) are principal formulas satisfying the fol-lowing:
` Qi(x, y) ↔ Q(x, y) ∧(∃z (Q(x, z) ∧ Q(y, z))
)1−i∧
(∃u (Q(x, u) ∧ Q(u, y))
)i,
i = 0, 1. Moreover, Q0(a, Γ) is the set of successors for a in TC(Γ).Denote by K∗ the class of all c-graphs A = 〈A, Q,W 〉, Q =
Q0∪Q1 such that |= Q0(a, b) implies an absence of (c, b)-routes forany c ∈ Q(a,A) \ b, and |= Q1(a, b) implies an absence of (b, c)-routes for any c ∈ Q(a,A) \ b. Here, if (a, b) ∈ Qi then the indexi is called an arc color (a, b).
Below, considering c-graphs in K0, we distinguish colors of arcsand use the language 〈Q0, Q1〉 instead of (or as an addition to) Qin digraphs and c-graphs.
The concepts of c-embedding and c-isomorphism are obviouslytransformed to the class of c-graphs of the language 〈Q0, Q1〉 pre-serving arc colors via mappings.
Denote by K∗0 the subclass of K∗ generated by stated below c-
graphs 〈A, Q0, Q1,W 〉 by taking of c-subgraphs, c-isomorphic copies,free c-amalgams, tracings (allowing for any chosen pair of vertices(a, b), that are not linked by routes, where Col(a) ≤ Col(b), toextend records W by an information on shortest (a, b)-routes of ar-bitrary length m, exceeding lengthes of all shortest routes being ingiven c-graph), and also detracings (allowing to remove aforesaidinformation):
(a) Γα,β,γ,0 = 〈0, 1, 2, (0, 1), (1, 2), (0, 2),W 〉, whereW = ∅;
(b) Γα,β,γ,1 = 〈0, 1, 2, (0, 1), (0, 2), (1, 2),W 〉, whereW = ∅;
(c) Γα,β,γ,s = 〈0, 1, 2, (0, 1), (0, 2),W 〉, where W =(1, 2, s), 2 ≤ s < ω, Col(0) = α, Col(1) = β, Col(2) = γ, α ≤ β ≤γ, γ ∈ ω ∪ ∞.
Lemma 3.7.1. ( -amalgamation lemma). The class K∗0 has the
-amalgamation property ( -AP), that is, for any c-embeddings f0 :A →c B and g0 : A →c C, where A,B, C ∈ K∗
0, there exist a c-graphD ∈ K∗
0 and c-embeddings f1 : B →c D and g1 : C →c D such thatf0 f1 = g0 g1.
P r o o f is obvious. ¤
116
Denote by K0 the class of colored acyclic digraphs in which anynite subgraph forms a c-graph in K∗
0.Theorem 3.7.2. There exists countable, colored, saturated di-
graph M ∈ K0 satisfying the following:(1) if f : A →c M and g : A →c B are c-embeddings and B ∈ K∗
0,then there exists a c-embedding h : B →c M such that f = g h;
(2) if A and B are c-isomorphic c-subgraphs of M, thentpM(A) = tpM(B);
(3) the coloring of restriction M ¹ Q of M to the graph languageΣ = Q is inessential and Q-ordered;
(4) the formula Q(x, y) is equivalent in Th(M ¹ Q) to the dis-junction of principal formulas Q0(x, y) and Q1(x, y).
P r o o f is similar to the proof of Theorem 3.1.3. The construc-tion of the saturated model M repeats steps of construction for themodel M in the proof of Theorem 3.1.3 with replacing of K∗
0 to theclass K∗
0 and of K0 to K0. ¤
Elements in Q0(a,M) are successors of a in the digraph TC(Γ),where Γ = M|Q, since, by construction, there are no elements b ∈Q(a,M) such that there exists a (b, c)-route with c ∈ Q0(a,M)\b.Thus, in TC(Γ), the partial order is not dense.
Similar Corollaries 3.1.43.1.6, we state that the relation SIp∞(x)
is non-symmetric, the digraph⟨p∞(M);Rp∞
Q(M)
⟩is powerful, and
the theory Th(M) is not simple.Notice, that constructions, described in Section 3.2, can be ap-
plied to M and also lead to creation of not ω-categorical, not simpletheory, for which any nonprincipal type is powerful. Thus, we obtainthe following theorem.
Theorem 3.7.3. There exists a generic theory T satisfying thefollowing conditions:
(1) |RK(T )| = 2;(2) the structure of realizations of some powerful type p ∈ S(T )
contains a powerful digraph Γ with unbounded lengths of shortestroutes and with innitely many covering elements for each elementin the transitive closure TC(Γ).
117
Now, we describe another modication od aforesaid construc-tion, allowing to create a small generic theory with a type havingthe local pairwise intersection property, but without the global pair-wise intersection property.
Let A = 〈A,Q, W 〉 be a c-graph. We replace the graph languageintroducing a sequence of pairwise disjoint binary predicates Qn,n ∈ ω, such that Q =
⋃n∈ω
Qn, and substitute in W , instead ofrecords on existence of external routes connecting elements in A, therecords on existence of these routes with an information of colors mi
for sequential arcs (bi, bi+1) ∈ Qmi forming these routes. Moreover,we allow, in obtained record W ′, for any two elements a, b ∈ A, towrite a formula-denable information of form
ϕk,m(a, b) ∃x(
k∧
i=0
¬Coli(x) ∧(
m∨
i=0
Qi(x, a)
)∧
(m∨
i=0
Qi(x, b)
))∧
¬∃y
k∧
i=0
¬Coli(y) ∧
m−1∨
i=0
k∧
j=0
Qji (y, a)
∧
m−1∨
i=0
k∧
j=0
Qji (y, b)
such that the set of formulas over A, correspondent to W ′, is con-sistent. The obtained structure A = 〈A,Qn,W ′〉n∈ω is called acv-graph.
The concepts of c-embedding and c-isomorphism, as above, aretransformed to the class of cv-graphs of the language 〈Q0, Q1, . . . ,Qn, . . .〉 with the preservation, via mappings, of arc colors and ofsatisability for ϕk,m(a, b) in the records.
A cv-graph A = 〈A,Qn,A,WA〉n∈ω is called a cv-subgraph of cv-graph B = 〈B, Qn,B,WB〉n∈ω (written A ⊆cv B) if A ⊆ B, Qn,A =Qn,B ∩ A2, n ∈ ω, and WA consists of all records in WB, in whichpairs of elements in A are used, and also of records, induced bythe cv-graph B, on existence or on absence of elements x and y inaccordance of ϕk,m(a, b).
Let A = 〈A,Qn,A,WA〉n∈ω, B = 〈B,Qn,B, WB〉n∈ω, and C =〈C, Qn,C , WC〉n∈ω be cv-graphs such that A = B∩C, Qn,A = Qn,B∩Qn,C , n ∈ ω, and WA consists of all records, included both in WBand in WC , in which pairs of elements in A take place, that inducedby cv-graphs B and C. We call the cv-graph 〈B ∪C,QB ∪QC ,WB ∪WC ∪ W 〉, a cv-amalgam of B and C over A if W consists of allpossible formulas, describing, for any elements b ∈ B \ A and c ∈C \ A, an existence and colors of elements d, for which there are
118
(d, b)- and (d, c)-routes (if their existence is implied by relationsQB ∪ QC and WB ∪ WC), and also an existence of elements e ofall nite colors m, not exceeding minCol(b), Col(c) and such that|=
m∨i=0
Qi(e, b) ∧m∨
i=0Qi(e, c). The record W also contains a nite set
of formulas of form ϕm−1,m(b, c), m ≤ minCol(b), Col(c), m ∈ ω,for the rest pairs (b, c), where b ∈ B \A, c ∈ C \A.
We denote by K0 the class of all cv-graphs that can be obtainedby all possible aforesaid transformations of c-graphs in K0. Denoteby K∗
0 the subclass of K0, generated by taking of cv-subgraphs, cv-isomorphic copies and free cv-amalgams, and also by tracings anddetracings starting by the following cv-graphs 〈A,Qn, W 〉n∈ω, wherethe index ·n in (a, b)n points out that (a, b) ∈ Qn:
Γα,β,γ,n1,n2,n3,W = 〈0, 1, 2, (0, 1)n1 , (1, 2)n2 , (0, 2)n3,W 〉,where Col(0) = α, Col(1) = β, Col(2) = γ, α ≤ β ≤ γ, γ ∈ω∪∞, W consists of nonempty nite set of formulas ϕm−1,m(0, 1),m ≤ minCol(0), Col(1), m ∈ ω, and also of nonempty nite set offormulas ϕm−1,m(0, 2), m ≤ minCol(0), Col(2), m ∈ ω.
Lemma 3.7.4. ( -amalgamation lemma). The class K∗0 has the
-amalgamation property ( -AP), that is, for any cv-embeddings f0 :A →cv B and g0 : A →cv C, where A,B, C ∈ K∗
0, there exist a cv-graph D ∈ K∗
0 and cv-embeddings f1 : B →cv D and g1 : C →cv Dsuch that f0 f1 = g0 g1.
P r o o f is obvious. ¤Denote by K the class of colored acyclic digraphs in which each
nite subgraph forms a cv-graph in K∗0.
Theorem 3.7.5. There exists countable, colored, saturated di-graph M ∈ K satisfying the following:
(1) if f : A →cv M and g : A →cv B are cv-embeddings andB ∈ K∗
0, then there exists a cv-embedding h : B →cv M such thatf = g h;
(2) if A and B are cv-isomorphic cv-subgraphs in M,then tpM(A) = tpM(B);
(3) the coloring of restriction M ¹ Q of M to the graph lan-guage Σ = 〈Q0, Q1, . . . , Qn, . . .〉 is inessential and Qn-ordered foreach n ∈ ω;
(4) p∞(x) has the (LPIP) but doesn't have the (GPIP).
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P r o o f is similar to the proof of Theorem 3.1.3. The construc-tion of the saturated model M repeats steps of construction for themodel M in the proof of Theorem 3.1.3 with replacing of K∗
0 to K∗0
and of K to K. The property (LPIP) for p∞(x) holds, since, by thegeneric construction, for any two elements, a and b, having the samecolor at least n + 1, there exists an element c of color n such that|=
n∨i=0
Qi(c, a)∧n∨
i=0Qi(c, b). An absence of (GPIP) is followed, since,
by Compactness Theorem, for any n ∈ ω there exist realizations aand b of p∞ for which there are no realizations c of p∞ such that(c, a), (c, b) ∈ ∪
n⋃
i=0Qj
i | j ∈ ω
. ¤
Using the construction of M, we have that for any two realiza-tions, a and b, of p∞(x), there exist numbers m,n ∈ ω and a realiza-tion c of p∞ such that |= Qm(c, a)∧Qn(c, b). Moreover, for any ele-ments d1 and d2 and for any color α ≥ maxCol(d1), Col(d2), thereexists an element e of color α, for which |= Q0(d1, e) ∧ Q0(d2, e).Therefore, an obvious modication of construction in Section 3.2,applied for the generic model M, states the following theorem.
Theorem 3.7.6. There exists a generic theory T satisfying thefollowing conditions:
(1) |RK(T )| = 2;(2) some powerful type p ∈ S(T ) has the (LPIP), but doesn't
have the (GPIP).Constructions, similar to aforesaid in Sections 3.33.6, show that
the generic models M and M can be extended to models with the-ories having given domination preorder and given distribution func-tion of numbers of limit models.
C h a p t e r 4STABLE GENERIC EHRENFEUCHTTHEORIES (A SOLUTION OF THELACHLAN PROBLEM)
§ 4.1. Small stable generic graphs with innite weight.Bipartite digraphs
B. Herwig [81] described a generic construction that was a mod-ication of the Hrushovski construction [89]. Herwigs inventionmade it possible to construct a small stable theory of a graph withcolored edges which possesses a unique 1-type such that its ownweight is innite. Thus, the realization was given for one of theessential properties shared by all stable Ehrenfeucht theories.
In the present Section, grounding on Herwig construction [81],we describe a generic construction that yields a family of small stablebipartite edgeless digraphs with colored arcs such that all 1-typeshave innite weight. We thus solve the existence problem for a smallstable acyclic digraph with colored arcs which has types with inniteweight.1. Denitions and properties. The language Σbp of a genericbipartite graph Γbp consists of binary symbols Ip, p ∈ ω, and unarysymbols J0 and J1. The relations corresponding to J0 and J1 aredisjoint and divide the universe in two parts. The relations corre-sponding to Ip are assumed irre exive, antisymmetric, and pairwisedisjoint, i.e., each pair of vertices (a, b) belongs to at most one re-lation Ip, so that if (a, b) ∈ Ip then a ∈ J0, b ∈ J1. In what followsin this Section, a graph is a bipartite digraph of language Σbp sat-
121
isfying the above conditions. The structure consisting of the emptyset and empty relations is also a graph, and we denote it by ∅. Theclass of all graphs under consideration is denoted by Γbp.
We x a positive real β. Let lnmn
stand for the best rationalapproximation of β such that ln
mn≤ β and mn ≤ n. The index of β
is Ind(β) =∑i>0
(β − limi
). In [81], B. Herwig observed that the set
β ∈ R+ | Ind(β) > N is dense and open for each N ∈ ω. Thus, theset β | Ind(β) = ∞ is dense in R+. Furthermore, if Ind(β) = ∞and q ∈ Q then Ind(βq) = ∞.
Choose a positive real α1 < 12 with Ind( 1
α1) = ∞. Below we will
dene a sequence (αk)k∈ω\0 of weights of Ik-arcs, where αk+1 = αkNk
with Nk suciently large naturals.Dene a prerank function y that assigns a real number to every
nite graph A by the rule
y(A) = |A| −∞∑
k=1
αk · ek(A),
where ek(A) is the number of Ik-arcs in A.Notice that, in the denition of a prerank function, the summa-
tion is always nite due to the niteness of A.A p-approximation of a prerank function y is a function yp as-
signing a real to every nite graph A by the rule
yp(A) = |A| −p∑
k=1
αk · ek(A).
Given a graph A, we consider its representation as the points1A = (|A|; y1(A)) in the lattice1
L1 = (n; n− α1 ·m) | n,m ∈ ω.Dene a monotonically increasing unbounded sequence (b1
n)n≥1: b11
1, b1n+1 b1
n+(1−α1· lnmn
), where lnmn
is the best rational approxima-tion of 1
α1satisfying mn ≤ n. The unboundedness of the sequence
follows from the condition Ind( 1α1
) = ∞.1As in [81], here and below, a lattice is some discrete set of points on the
co-ordinate plane.
122
Given a graphA, we writeA ∈ Kbp1 iA is nite and y1(A′) ≥ b1
nfor every nonempty graph A′ ⊆ A, where n = |A′|. We put k1 1and thus conclude the denition of step 1.
Suppose that at step p, we have dened the number αp, thelattice Lp of the possible yp-values, and the monotonically increasingsequence (bp
n)n≥kp . Choose a natural kp+1 > kp subject to bpkp+1
>
p+2 and a suciently small real εp+1 > 0 (εp+1 < εp for p > 1) suchthat, for all (p1, y1), (p2, y2) ∈ Lp with p1, p2 ≤ kp+1 and y1 6= y2,we have |y1− y2| > εp+1. Choose a number αp+1 = αp
Npwith Np ∈ ω
and Np > 2 so that αp+1 · kp+1(kp+1−1)2 < εp+1. The lattice
Lp+1 (n; n− αp+1 ·m) | n,m ∈ ωrenes Lp.
The sequence (bp+1n )n≥kp+1 is constructed according to the fol-
lowing relations: bp+1kp+1
= bpkp+1
−2εp+1, bp+1n+1 = bp+1
n +(1−αp+1 · lnmn
),where ln
mnis the best rational approximation of 1
αp+1satisfying mn ≤
n. The condition Ind( 1αp+1
) = ∞ implies the unboundedness of themonotonically increasing sequence (bp+1
n )n≥kp+1 . Moreover, the leastelement bp+1
kp+1of this sequence satises the inequality
bp+1kp+1
> p + 1 + ε2, (4.1)
in view of ε2 < 13 .
Lemma 4.1.1. For every positive p ∈ ω and arbitrary nitegraphs A and B, where A is a proper subgraph of B, |B| = n < kp+1,the following assertions hold:
(1) We have yp(B)− y(B) < εp+1.(2) The inequality yp(A) < yp(B) holds i y(A) < y(B). The
inequality yp(A) < yp(B) holds i yq(A) < yq(B) for every q ≥ p.(3) The least possible positive slope2 sl((p1, y1), (p2, y2)) between
the points (p1, y1) and (p2, y2) (p1 < p2 ≤ n, n ≥ kp) in the latticeLp coincides with the slope sl((n, bp
n), (n + 1, bpn+1)) between (n, bp
n)and (n + 1, bp
n+1).(4) If q < p and m ≥ kp then bp
m ≤ bqm − 2εq+1.
2that is, the tangent of slope angle for the line segment, connecting points.
123
P r o o f. (1) Only the unordered pairs of distinct elements ofB are used in counting the values ek(B) Each pair contributes atmost one value ek(B) to calculation. The number of such pairs doesnot exceed kp+1(kp+1−1)
2 . Therefore, yp(B) can dier from y(B) by atmost αp+1 · kp+1(kp+1−1)
2 < εp+1.(2) Since y(A) = yq(A) and y(B) = yq(B) starting from some q,
it is sucient to show the equivalence yp(A) < yp(B) ⇔ yq(A) <yq(B) for every q ≥ p. Choose q ≥ p arbitrarily and suppose thatyp(A) < yp(B). Then yp(A) < yp(B)−εp+1 by the denition of εp+1.In view of assertion (1), we have
yq(A) ≤ yp(A) < yp(B)− εp+1 < y(B) ≤ yq(B).
Therefore, yq(A) < yq(B).To show the converse, suppose that yp(A) ≥ yp(B) and, without
loss of generality, assume that A 6= B. We claim that yp(A) > yp(B).Indeed, by hypothesis, the positive part |A| of yp(A) is less than thepositive part |B| of yp(B). Since yp(A) and yp(B) have the forms|A| −MA · αp and |B| −MB · αp respectively, from the irrationalityof αp, it follows that yp(A) 6= yp(B).
We now obtain the inequality yq(A) > yq(B) from the followingrelations:
yq(A) ≥ y(A) > yp(A)− εp+1 > yp(B) ≥ yq(B).
(3) The minimal possible positive slope equals
min
p1 − αp ·m1
p1
∣∣∣∣p1
m1> αp, p1 ≤ n
which, in turn, equals 1 − αp · lnmn
, where lnmn
is the best rationalapproximation of 1
αpsatisfying mn ≤ n.
(4) It is sucient to show that bq+1m ≤ bq
m − 2εq+1. This can beestablished by induction on m on using assertion (3) and the factthat Lq+1 is a renement of Lq. ¤
2. Generic class and generic theory. Given a nite graph A,we write A ∈ Kbp
p+1 i A ∈ Kbpp and yp(A′) ≥ bp
n for any graphA′ ⊆ A, where n = |A′|, kp ≤ n < kp+1.
124
Put Kbp0
∞⋂p=1
Kbpp and denote by Kbp the class of all graphs
whose every nite subgraph belongs to Kbp0 .
Let A be a subgraph of M, where M belongs to Kbp. Wesay that A is a self-sucient subgraph of M and write A 6 M ify(A) ≤ y(B) for every nite graph B, where A ⊆ B ⊆M. If A 6 Mand M is a nite graph then A is called a strong subgraph of M.
We have to show that the class Tbp0 of all quantier-free types
corresponding to the graphs in Kbp0 with the relation 6′ (where
Φ(A) 6′ Ψ(B) ⇔ A 6 B) is a self-sucient generic class that(upon adding the necessary formulas to the types describing the self-sucient closures) satises the property of uniform t-amalgamation.According to Theorem 2.5.1, this fact will yield the ω-saturation ofthe (Tbp
0 ; 6′)-generic model.We start with the following:
Remark 4.1.2. (1) The conditions A ∈ Kbp0 and kp ≤ |A| =
n < kp+1 do not imply that y(A) ≥ bpn. Nevertheless, y(A) cannot
be much less than bpn: since yp(A) − y(A) < εp+1, we have y(A) >
bpn − εp+1.
Moreover, in view of (4.1), given a graph A ∈ Kbp0 of cardinality
|A| ≥ max2, kp, the inequality y(A) > p holds.(2) Let A be a graph in Kbp
0 , |A| = p, M be a graph in Kbp, andA ⊆ M. Then A 6 M i yp(A) ≤ yp(B) for all nite graphs B,where A ⊆ B ⊆M.
Indeed, if |B| < kp+1 then yp(A) ≤ yp(B) is equivalent to y(A) ≤y(B) by Lemma 4.1.1(2). If |B| ≥ kp then
yp(B) ≥ y(B) > p ≥ yp(A).
Moreover, in order to verify that A is self-sucient in M, itsuces to choose nA = kp and check the validity of the relationyp(A) ≤ yp(B) only for the graphs B, A ⊆ B ⊆ M, satisfying|B| < nA.
Therefore, the condition A 6 M is denable by a formula thatdescribes the absence of n < nA new elements in M \ A such thatn < α1 · e1 + . . . + αp · ep, where p = |A|, es is the number of newIs-arcs.
The following lemma is a direct consequence of the denition:
125
Lemma 4.1.3. (1) If A 6 Bthen A ⊆ B.(2) If A 6 C, B ∈ Kbp
0 , and A ⊆ B ⊆ C then A 6 B.(3) The empty graph ∅ is the least element of the structure
(Kbp0 ;6).Lemma 4.1.4. If A,B, C ∈ Kbp
0 , A 6 B, and C ⊆ B thenA ∩ C 6 C.
P r o o f. Assume the contrary. Then there exist n new elementsin C \ A such that n < α1 · e1 + . . . αp · ep, where es is the numberof new Is-arcs. Since all new elements lie in B \A, they violate thecondition A 6 B. ¤
Lemma 4.1.5. The relation 6 is a partial order on Kbp0 .
P r o o f. The re exivity and antisymmetry of 6 are evident.We show that 6 transitive. Assume the contrary and consider
nite graphs A,B, C ∈ Kbp0 such that A 6 B, B 6 C, and A 66 C.
Denote by p the cardinality of A. Choose a graph A′, A ⊆ A′ ⊆ C,with the least possible value yp(A′). This value is attainable by agraph of cardinality less than nA, since the set yp(D) | A ⊆ D ⊆C, yp(D) ≤ yp(A) is a subset of the nite set n − αp · m | n <nA,m < nA
αp.
We claim thatA′ is a minimal strong subgraph of C containingA.Indeed, we have A′ 6 C, since, for any graph D with A′ ⊆ D ⊆ C, if|D| < kp+1 then y(D) ≥ y(A′) in view of yp(D) ≥ yp(A′) by Lemma1.1(2); and if |D| ≥ kp+1 then y(D) > y(A) ≥ y(A′). Moreover,every subgraph D with A ⊆ D A′ is not a strong subgraph ofC, since the minimality of yp(A′) implies yp(D) ≥ yp(A′), and therelation |A′| 6= |D| yields y(D) > y(A′) by Lemma 4.1.1(2).
The minimality of A′ and the existence of a strong subgraphB of C imply that A′ C. Now, from the inequality |A′| 6= |C|and from the irrationality of αs, we conclude that yp(A′) 6= yp(C),and Lemma 4.1.1(2) yields y(A′) < y(C). By Lemma 4.1.4, we haveB ∩ A′ 6 A′. Therefore, B ∩ A′ = A′ and A′ ⊆ B. The last relationcontradicts A 6 B. ¤
Let A, B = 〈B; Ip,B, J0,B, J1,B〉, and C = 〈C; Ip,C , J0,C , J1,C〉 begraphs, with A = B ∩ C. The free amalgam of B and C over A(written B∗A C) is the structure 〈B∪C; Ip,B ∪ Ip,C , J0,B ∪J0,C , J1,B ∪J1,B〉.
Observe that, in constructing a free amalgam, the initials (end-points) of arcs can be joined only with the initials (endpoints) of
126
arcs. Therefore, the structure B ∗A C is a graph containing A, B,and C as subgraphs. This graph is bipartite with irre exive andantisymmetric pairwise disjoint relations Ip and unary relations J0
and J1 that have no common elements and form a partition of theuniverse in two parts so that if (a, b) ∈ Ip then a ∈ J0 and b ∈ J1.
An embedding f of a graph A in a graph B (f : A → B) is saidto be strong if f(A) 6 B.
Lemma 4.1.6. (amalgamation lemma). The class Kbp0 has the
amalgamation property (AP), that is, for any strong embeddings f0 :A → B and g0 : A → C, where A,B, C ∈ Kbp
0 , there exist a graphD ∈ Kbp
0 and strong embeddings f1 : B → D and g1 : C → D suchthat f0 f1 = g0 g1.
P r o o f. Without loss of generality we may assume that A 6 B,A 6 C, and A = B ∩ C. We claim that the graph D B ∗A C is asrequired. To this end, by the symmetry of the denition of a freeamalgam, it is sucient to show that B 6 D and D ∈ Kbp
0 .Assume that B 66 D. Then there exist n new elements in D \B
such that n < α1 ·e1 + . . . αp ·ep, where p = |B| and es s the numberof new Is-arcs. Since all new elements lie in C \A, they violate thecondition A 6 C.
Since each subgraph of D has the form B0 ∗A0 C0, where A0 6 B0
and A0 6 C0, in order to show that D ∈ Kbp0 , it suces to verify
that yp(D) ≥ bpn, where n = |D|, kp ≤ n < kp+1. Suppose that
|B| ≤ |C| = m, kq ≤ m < kq+1, and A ( B. By Lemma 4.1.1(3)and by condition A 6 B, it follows that
sl((|A|, yq(A)), (|B|, yq(B))) ≥ sl((l, bql ), (l + 1, bq
l+1))
for any l ≥ m. Then the equality
sl((|C|, yq(C)), (|D|, yq(D))) = sl((|A|, yq(A)), (|B|, yq(B)))
yields
sl((|C|, yq(C)), (|D|, yq(D))) ≥ sl((m, bqm), (|D|, bq
|D|)). (4.2)
The fact that C belongs to Kbp0 implies that the point (|C|, yq(C)) is
above the point (m, bqm). Then, by (4.2), we conclude that
(|D|, yq(D)) is above (|D|, bq|D|), i.e., yq(D) ≥ bq
|D|.
127
If p = q then the required inequality yp(D) ≥ bpn holds. Oth-
erwise, we have q < p and then yp(D) diers from yq(D) by lessthan αq+1 · kq+1 · (kq+1 − 1) < 2εq+1, since D contains less thankq+1 · (kq+1 − 1) arcs. By Lemma 4.1.1(4), we conclude thatyp(D) ≥ bp
n. ¤Lemmas 4.1.34.1.6 implyCorollary 4.1.7. The class (Tbp
0 ; 6′) is self-sucient.Denote the (Tbp
0 ; 6′)-generic theory by T bp.We claim that, upon adding a formula χΦ(A) such that
(T bp, A) ` χΦ(A) to every quantier-free type Φ(A) ∈ Tbp0 , we ob-
tain a self-sucient class (T0; 6′′) satisfying the uniform t-amalga-mation property.
Indeed, in view of Remark 4.1.2 for every graph A ∈ Kbp0 of
cardinality p and every graph M |= 1T bp, where A ⊆M, we have:
A 6 M⇔ yp(A) ≤ yp(B) for any graph Bwith A ⊆ B ⊆M and |B| ≤ kp.
Since the cardinalities of the graphs B are bounded only in termsof the cardinality of A and since checking the condition yp(A) ≤yp(B) involves only counting the links over the relations I1, . . . , Ip,the self-suciency condition A 6 M is expressible by a universalformula χA(X) of the language I1, . . . , Ip, where the set of vari-ables X is bijective with A.
Let A and B be graphs in Kbp0 , M be a generic model of the
theory T bp with A 6 B 6 M. Denote by ψA,s(X) ( by ψB,s(X, Y ),respectively) a quantier-free formula that describes the quantier-free I1, . . . , Is, J0, J1-type of A (of B), where X and Y are disjointsets of variables bijective with A and B \A, respectively. Then, forevery s ≥ |B|, the following formula is true in M:
∀X ((χA(X) ∧ ψA,s(X)) → ∃Y (χB(X, Y ) ∧ ψB,s(X, Y ))) .
The last relation implies the uniform t-amalgamation property forthe class (T0; 6′′) that we obtain from (Tbp
0 ; 6′) by adding the for-mulas to the types which establish the cardinality bounds and theI1, . . . , Ip, J0, J1-structures of the self-sucient closures, as well asthe formulas χA(A) to the types of self-sucient sets A.
128
The addition of the above-mentioned formulas ensures the exis-tence of nite closures for arbitrary nite sets of models of T bp.
In view of Theorem 2.5.1, the following holds:Theorem 4.1.8. A (Tbp
0 ; 6′)-generic model M is saturated. Inaddition, each nite set A ⊆ M can be extended to its self-sucientclosure A ⊆ M , and the type tpX(A) is deducible from the setχA(X) ∪ ψA,s(X) | s ∈ ω \ 0.
Let N be an ω-saturated model of T bp.Proposition 4.1.9. For each nite set A in N , we have
acl(A) = A.P r o o f. The inclusion A ⊆ acl(A) follows from the uniqueness
of A in N . We now take an arbitrary element b ∈ N \ A and putB A ∪ b. By the construction of a generic model, there existinnitely many isomorphic pairwise disjoint copies of the set B \ Aover A, that is, b 6∈ acl(A). ¤
3. Stability of the generic theory. We claim that the generictheory T bp is stable. Let N be a suciently saturated model of T bp.The rank function in N is the function
rN : A | A is a nite subgraph of N → R+,
that is dened by rN (A) = infy(B) | A ⊆ B ⊆fin N (here and else-where the notation B ⊆fin N means that B is a nite substructureof N , and B ⊆fin N means that B is a nite subset of N).
Observe that, in view of the self-suciency of the class (Tbp0 ; 6′),
the inmum in the above equality is always attainable and coincideswith y(A), i.e., rN (A) = y(A).
In what follows, we x a model N , denote the function rN by rfor brevity, and replace the graphs A by their universes in N . Wealso assume that all sets under consideration are subsets of N .
The relative prerank function y(A/B) of a graph A over a graphB is given by the relation
y(A/B) = y(A ∪B)− y(B)
and the relative rank function r(A/B) of A over B is given by
r(A/B) = r(A ∪B)− r(B).
129
We establish some intermediate properties of the relative rankfunction.
Clearly, r(A/B) ≥ 0.Lemma 4.1.10. For any nite sets B and C, we have
y(B) + y(C) ≥ y(B ∪ C) + y(B ∩ C).
The inequality becomes an equality i B ∪ C = B ∗B∩C C.P r o o f. By denition, the value
y(B) + y(C)− y(B ∪ C)− y(B ∩ C)
equals∞∑
k=1
αk · ek, where ek is the number of Ik-arcs that connect
the elements of B \ C with the elements of C \ B. Therefore, therequired inequality holds and becomes an equality precisely in theabsence of the specied Ik-arcs, i.e., when the graph with universeB ∪ C is the free amalgam B ∗B∩C C. ¤
Lemma 4.1.11. If B1 ⊆ B2 then r(A/B1) ≥ r(A/B2).P r o o f. Put C A ∪B1, D B1, E B2. By Lemma
4.1.10 and in view of D ⊆ C, D ⊆ E, and D ≤ C ∩ E, it followsthat
y(C) + y(E) ≥ y(C ∪ E) + y(C ∩ E) ≥ y(C ∪ E) + y(D).
Then y(C)− y(D) ≥ y(C ∪ E)− y(E). This implies
r(A/B1) = y(C)− y(D) ≥ y(C ∪ E)− y(E) ≥ y(C ∪ E)− y(E) =
r(A ∪B2)− r(B2) = r(A/B2). ¤If A is a nite set and X is some (not necessarily nite) set then
r(A/X) infr(A/B) | B ⊆fin X.
Lemma 4.1.12. Let A1 and A2 be nite sets and let X be a setsuch that A1 \X = A2 \X. Then r(A1/X) = r(A2/X).
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P r o o f. Assume the contrary. Without loss of generality wemay assume that r(A1/X) = r(A2/X) + ε for some ε > 0. Choosea nite set B2 ⊆ X such that r(A2/B2) < r(A2/X)+ ε. Denote theset (A1 ∩X) ∪ (A2 ∩X) ∪B2 by C. Then
r(A1/X) ≤ r(A1/C) = r(A2/C) ≤ r(A2/B2) < r(A2/X) + ε.
We have r(A1/X) < r(A2/X) + ε, which contradicts the choiceof ε. ¤
A set X ⊆ N is called closed (in N ) (written X 6 N) if A ⊆ Xfor any nite set A ⊆ X. The closeness is equivalent to the absencein N \X of any new n elements such that n <
∞∑k=1
αk · ek, where ek
is the number of new Ik-arcs.
Lemma 4.1.13. Let X and Y be sets in N . Then the followinghold:
(1) if X 6 N and Y 6 N then X ∩ Y 6 N ;(2) there exists a least closed set X ⊇ X; moreover, X =
⋃A |A is a nite subset of X, and X = acl(X);
(3) if X ⊂ Y then X ⊆ Y .The set X is called the intrinsic closure of X (in N ).P r o o f. (1) Suppose that X 6 N and Y 6 N . Let A be a nite
subset of X ∩ Y . Since A ⊂ X, A ⊂ Y , and the sets X and Y areclosed, we have A ⊆ X and A ⊆ Y . Consequently, A ⊆ X ∩ Y .Therefore, X ∩ Y is closed.
(2) Denote the set⋃A | A is a nite subset of X by Z. Clearly,
each closed superset of X includes Z and X ⊆ Z. On the otherhand, if A is a nite subset of Z then there exists a nite set B ⊆ Xsuch that A ⊆ B. So A ⊆ B ⊆ Z. Therefore, Z is the leastclosed set containing X. The equality X = acl(X) now follows fromProposition 4.1.9.
(3) It is sucient to observe that, for arbitrary nite graphsA and B, if A ⊆ B then A ⊆ B. Indeed, A ⊆ A ∩ B 6 N .Consequently, A ⊆ A ∩B and A ⊆ B. ¤
By analogy with Denition 3.30 in the work by J. T. Baldwin andN. Shi [37], we say that nite sets A and B are independent over Zand write A ↓r
Z B if r(A/Z) = r(A/(Z∪B)) and A ∪ Z∩B ∪ Z ⊆ Z.
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We say that sets X and Y are independent over Z and writeX ↓r
Z Y if A ↓rZ B for any nite sets A ⊆ X and B ⊆ Y .
The following assertion is an analog of Lemma 3.31 in the workby J. T. Baldwin and N. Shi [37]:
Lemma 4.1.14. If X and Y are closed sets, Z = X ∩ Y , andX ↓r
Z Y , then X ∪ Y is closed.P r o o f. First, observe that, by Lemma 4.1.13(1), the set Z is
closed.Assume now, that X ∪ Y is not closed. Then there exist nite
sets A ⊆ X, B ⊆ Y , and C such that (A ∪ B, C) is a minimalpair and C is not contained in A ∪ B. Put ε y(A ∪ B) − y(C).By the choice of A, B, and C, for any nite sets A′ and B′ withA ⊆ A′ ⊆ X and B ⊆ B′ ⊆ Y , we have y(C/(A′ ∪B′)) ≤ −ε. Thenr(A′ ∪B′ ∪C) ≤ y(A′ ∪B′ ∪C) ≤ y(A′ ∪B′)− ε. In order to arriveat a contradiction, using the independence of X and Y over Z, wechoose a pair (A1, B1) such that A ⊆ A1 ⊆ X, B ⊆ B1 ⊆ Y , andr(A1 ∪B1 ∪ C) > y(A1 ∪B1)− ε.
Since X ↓rZ Y , we have r(A/Z) = r(A/(Z ∪B)). Choose a nite
set Z0 ⊆ Z such that r(A/Z0) < r(A/Z) + ε2 . Put A0 = A ∪ Z0,
B0 = B∪Z0 and D0 = A0∩B0. By Lemma 4.1.12, we have r(A/Z) =r(A0/Z). Moreover, from A ∪ Z0 = A0 it follows that r(A/Z0) =r(A0/Z0) ≥ r(A0/D0). Therefore, r(A0/D0) < r(A0/Z) + ε
2 .Put D1 D0, A1 A0, and B1 B0. Clearly, y(A1/D1) =
r(A1/D1) and |y(A1/D1) − r(A0/Z0)| < ε2 . In addition, D1 ⊆
A1∩ B1.We claim that A1 and B1 are the required sets. Since r(A1 ∪
B1 ∪ C) ≥ r(A0 ∪ B0), it is sucient to show that r(A0 ∪ B0) >y(A1 ∪B1)− ε. We have
r(A0 ∪B0) = r(A0/B0) + r(B0) ≥ r(A0/B0 ∪ Z) + r(B0) =
r(A0/Z) + r(B0) > r(A0/Z0)− ε
2+ r(B0) >
(y(A1/D1)− ε
2
)− ε
2+ y(B1) = y(A1/D1) + y(B1)− ε.
In view of the self-suciency of D1, we have y((A1∩B1)/D1) ≥ 0.Then
y(A1/D1) + y(B1)− ε =
y(A1/(A1 ∩B1)) + y((A1 ∩B1)/D1) + y(B1)− ε ≥132
y(A1/(A1 ∩B1) + y(B1)− ε ≥ y(A1 ∪B1)− ε.
Comparing the rst and last expressions, we obtain
r(A0 ∪B0) > y(A1 ∪B1)− ε. ¤
The following denition naturally generalizes the above notionof a free amalgam of graphs. A set U (in N ) is called a free amalgamof X and Y over Z and denoted by X ∗Z Y if X∪Y = U , X∩Y = Z,and there are no arcs that connect elements in X \Y with elementsin Y \X.
Lemma 4.1.15. If X is a self-sucient set, Y is a closed set,and Z = X ∪ Z ∩Y , then X ↓r
Z Y i X ∪ Z ∪Y is a closed set thatcoincides with the free amalgam X ∪ Z ∗Z Y .
P r o o f. Denote X ∪ Z by X ′.Suppose that X ↓r
Z Y . The fact that X ′ ∪ Y is closed wasshown in Lemma 4.1.14. We prove that X ′ ∪ Y = X ′ ∗Z Y . Assumeto the contrary that X ′ ∪ Y 6= X ′ ∗Z Y , i.e., there exists an Ip-arc that connects x ∈ X ′ \ Z with y ∈ Y \ Z. Choose a niteset Z0 ⊆ Z such that x ∈ X ∪ Z0. Then, for every nite self-sucient set Y0 ⊇ Z0 ∪ y, we have r(X/Z ∩ Y0) ≥ r(X/Y0) + αp
and, therefore, r(X/Z) ≥ r(X/Y ) + αp. The latter contradicts theequality r(X/Z) = r(X/Y ) which is a consequence X ↓r
Z Y .Suppose now, that X ′ ∪Y is a closed set that coincides with the
free amalgam X ∪ Z ∗Z Y . We show that X ↓rZ Y . It is sucient to
establish that
infr(X/A) | A ⊆fin Z ≤ infr(X/B) | B ⊆fin Y .Choose an arbitrary nite set B ⊆ Y . Denote by C the self-sucientset B ∪X which is contained in X ′ ∪ Y by hypothesis. Since C =(C ∩ X ′) ∗C∩Z (C ∩ Y ) and the set C = (C ∩ X ′) ∪ (C ∩ Y ) isself-sucient, it follows that X ∪ (C ∩ Z) ∪ (C ∩ Y ) is also self-sucient and coincides with X ∪ (C ∩ Z) ∗C∩Z C ∩ Y . Since B ⊆C ∩ Y , we have r(X/B) ≥ r(X/C ∩ Y ). It remains to observethat r(X/C ∩ Z) = r(X/C ∩ Y ). Indeed, denoting by X ′′ the setX ∪ (C ∩ Z), we obtain
r(X/C ∩ Z) = y(X ′′)− y(C ∩ Z) ≥ y(X ′′)− y(X ′′ ∩ (C ∩ Y )) ≥y(X ′′ ∪ (C ∩ Y ))− y(C ∩ Y ) ≥ r(X ′′/C ∩ Y ) = r(X/C ∩ Y ).
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The rst inequality is an equality due to C ∩ Z = X ′′ ∩ (C ∩ Y ).The second inequality is an equality by Lemma 4.1.10 in view ofX ′′ ∪ (C ∩ Y ) = X ′′ ∗C∩Z (C ∩ Y ). The third inequality is also anequality, since the set X ′′ ∪ (C ∩ Y ) is self-sucient. ¤
Lemma 4.1.15 readily impliesCorollary 4.1.16. If X is a self-sucient set, Y is a closed
set, Z = X ∪ Z ∩Y , and X ↓rZ Y then the type tp(X/Y ) is uniquely
dened by the type tp(X/Z), a description of X ∪ Z∪Y to be closed,and a quantier-free type that describes the coincidence of X ∪ Z∪Ywith the free amalgam X ∗Z Y .
Lemma 4.1.17. If X is a closed set and a is an element in Nnot belonging to X, then there exists at most countable, closed subsetX ′ ⊆ X such that a ↓r
X′ X.P r o o f is similar to that of Lemma 3.32 in the work by
J. T. Baldwin and N. Shi [37]. We choose a sequence Xn, n ∈ ω, of -nite subsets of X such that Xn−1 ⊆ Xn and r(a/Xn)−r(a/X) <1n , n ∈ ω\0. Put U
⋃n∈ω
Xn and X ′ X∩a ∪ U . The set X ′
is closed and at most countable in view of Lemma 4.1.13. ApplyingLemma 4.1.11, we infer r(a/X ′) = r(a/X) = r(a/(X ′ ∪X)).In addition, (a ∪X ′) ∩X = X ′. Consequently, a ↓r
X′ X. ¤Recall the notion of a weight (see S. Shelah [22]). Let p be a type
of a theory T , λ be a cardinal. The weight w(p) of type p is greaterthan or equal to λ if there exists a nonforking extension tp(a/A)of p and a sequence (ai)i∈λ independent over A such that every ai
is dependent on a over A. We assign w(p) λ if w(p) ≥ λ andw(p) λ+.
Theorem 4.1.18. (1) The theory T bp is stable, small, and hasprecisely two 1-types (of elements of color J0 and those of color J1).
(2) Each 1-type of T bp has innite weight; namely, for everyelement a of color Ji, i = 0, 1, there are innitely many independentelements bn, n ∈ ω, of color J1−i that depend on a.
P r o o f. (1) Theorem 4.1.8 implies that T bp is small. In orderto prove the stability of T bp, we estimate the number of 1-types inS(N), where N is a model of T bp. Consider an arbitrary elementa. By Lemma 4.1.17, there exists at most countable, closed setX ⊆ N such that a ∪X ∩ N = X and a ↓r
X N . By Corollary
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4.1.16, the type tp(a/N) is dened by tp(a/X), by a descriptionof the closeness of a ∪X ∪N , and by a quantier-free type thatdescribes the coincidence of a ∪X ∪ N with the free amalgama ∪X ∗X N . Therefore, in order to count the number of types inS(N), it is sucient to count the number of types in S(X) and thenumber of ways of choosing the countable set X in N . Then
|S(N)| ≤ 2ω · |N |ω = |N |ω.
Consequently, T bp is stable.By construction, every singleton a is self-sucient. In view
of Theorem 4.1.8, this means that there exist precisely two 1-typesof T bp; moreover, one of them contains the formula J0(x) and theother contains J1(x).
(2) Let A be a graph with universe a, and let Ap = a, bpbe the universe of a two-element graph Ap that contains the Ip-arc between a and bp. Put B1 A1, Bp+1 Bp ∗A Ap+1, B ⋃p∈ω\0
Bp. By construction, B is a closed subgraph of some generic
model M of T bp. Since α1 + . . . + αp+1 < 1 and b1, . . . , bp+1 6a, b1, . . . , bp+1 6 B 6 M , we have bp+1 ↓r
∅ b1, . . . , bp. There-fore, (bp)p∈ω\0 is an innite independent sequence of elements ofthe same type, where each bp is dependent on a. ¤
Let M be a model, ϕ(x), ψ(x, y) be formulas of Th(M), and Xbe a set in M, which is dened by ϕ(x): X = ϕ(M). We say thatX has the pairwise ψ-intersection property if
M |= ∀x, y (ϕ(x) ∧ ϕ(y) → ∃z (ψ(z, x) ∧ ψ(z, y))).
Since αp < 12 for every natural p ≥ 1, it follows that arbitrary
elements a and b of color J0 are connected via an element c of colorJ1 so that aIpc and bIpc. In the same way, arbitrary elements a andb of color J1 are connected via an element c of color J1 so that cIpaand cIpb. Furthermore, in either case, there exist innitely manysuch elements c. Thus, we have
Proposition 4.1.19. For every natural p ≥ 1, the set of el-ements of color Ji has the pairwise I
(−1)1−i
p -intersection property,i = 0, 1.
The following two assertions clarify the structure of the primemodels MA over nite sets A.
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Lemma 4.1.20. If A and B are self-sucient sets in N andA 6 B then the type tp(B/A) is isolated i B is a complete bipartitegraph over A, i.e., arbitrary two elements a ∈ B and b ∈ B \ A,dierent in J-color, are connected by an Ip-arc.
P r o o f. Let Y be a set of variables bijective with B\A. If B is acomplete bipartite graph overA then Theorem 4.1.8 implies that thetype tp((B\A)/A) is isolated by the formula χB(A∪Y )∧ψB,s(A∪Y ),where s is a number that exceeds all color numbers for the arcs inB. If B is not a complete bipartite graph over A , then Theorem4.1.8 implies that tp((B \ A)/A) is isolated by the set of formulasχB(A ∪ Y ) ∪ ψB,s(A ∪ Y ) | s ∈ ω, but is not isolated by niteparts of this set. ¤
Lemma 4.1.20 implies
Corollary 4.1.21. Let A be a self-sucient set in N . Themodel MA is a complete bipartite graph over A. The set of iso-morphism types of prime models over nite sets coincides with theset of isomorphism types of the models of form MA, where A is aself-sucient set such that MA is a complete bipartite graph over A.
§ 4.2. Small stable generic graphs with innite weight.Digraphs without furcations
In the present Section, we introduce the concept of a digraphwithout furcations and, on the basis of Section 4.1, describe a genericconstruction that yields a family of small stable digraphs withoutfurcations such that all 1-types of non-intermediate elements haveinnite weight. We thus solve the existence problem for small stabletheories of nite (graph) language that possess types with inniteweight.1. Denitions and properties. Let A = 〈A,Q, W 〉 be a c-graph. A vertex a ∈ A is an upper (lower) furcation in A if there arevertices b, c, d ∈ A, c 6= d, such that the following conditions hold:
(a) (b, a) ∈ Q or (b, a, n) ∈ W (respectively, (a, b) ∈ Q or(a, b, n) ∈ W ) for some n;
(b) (a, c) ∈ Q or (a, c, n) ∈ W (respectively, (c, a) ∈ Q or(c, a, n) ∈ W ) for some n;
(c) (a, d) ∈ Q or (a, d, n) ∈ W (respectively, (d, a) ∈ Q or(d, a, n) ∈ W ) for some n.
136
Upper (lower) furcations are often called just furcations.A c-graph or acyclic digraph is called a graph without furcations.
The class of all graphs without furcations is denoted by Γnf .We denote by Γbp
c the class of all graphs with language Qthat can be obtained from graphs of the class Γbp by replacing allIp-arcs (a, b) by some number of (a, b)-Q-routes of length at least p,including the shortest route of length p, such that all intermediatevertices in these routes are of degree 2. Clearly, any graph in Γbp
c hasno furcations. Furthermore, the acyclic digraphs in which degreesof vertices are at most 2 are graphs without furcations.We denotethe class of all such graphs by Γ≤2.
Obviously, every graph in Γ≤2 consists of a number of connectedcomponents K, and each component is a nite or innite route withor without initial element, as well as with or without nal element.As above, we mark initial elements by color J0 and nal elementsby color J1. The class Γ≤2 is closed under adding isolated vertices,i.e., vertices of degree 0. We assume that the set of isolated verticesis disjoint with J0 ∪ J1.
Let Γi = 〈Xi, Qi〉, i ∈ I, be graphs with the unique commonelement a of color Jj , j ∈ 0, 1. The graph 〈⋃
i∈I
(Xi),⋃i∈I
(Qi)〉 issaid to be the free amalgam of Γi, i ∈ I, over a and denoted by∗i∈IΓi.
We denote by Γnf0 the closure of Γbp
c ∪Γ≤2 under disjoint unionsand free amalgams over vertices.
It is easy to observe that any graph in Γnf has no furcations.Furthermore, each graph without furcations can be represented asa disjoint union of amalgams over vertices of graphs in Γbp
c ∪ Γ≤2.Therefore, Γnf = Γnf
0 .Denoting the operation of taking amalgams over vertices by A
and the operation of taking disjoint unions by D, we obtain thefollowing formula:
Γnf = D(A(Γbpc ∪ Γ≤2)).
In what follows, we consider graphs without furcations and theirc-subgraphs A = 〈A,Q,W 〉 in which the functions Col are one-color: ρCol = 0, and each vertex is either initial (of color J0), ornal (of color J1), or intermediate (of degree 2 in some extension ofA, and of color distinct from both J0 and J1). Moreover, for each
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vertex a ∈ A not belonging to J0 ∪ J1, the record W is assumedto contain information on the length of the shortest (b, a)-Q-route,where b ∈ J0 (if there exists such a vertex b in a c-graph or someits extension), as well as information on the length of the shortest(a, c)-Q-route, where c ∈ J1 (if there exists such a vertex c in c-graphor some its extension). c-Subgraphs considered independently arecalled c-graphs as usual.
Let A = 〈A,Q, W 〉 be a c-graph.We denote by e1(A) the number|Q|; and by ek(A), k ∈ ω\0, 1, the number of pairs (a, b) of verticesin A such that (a, b, k) ∈ W .
We dene a prerank function y, that assigns a real to each c-graph A by the rule
y(A) = |A| −∞∑
k=1
αk · ek(A),
where αk are dened in Section 4.1. Notice that, in the denitionof a prerank function, the summation is always nite due to theniteness of c-graphs A.
A p-approximation of a prerank function y is a function yp thatassigns a real to each c-graph A by the rule
yp(A) = |A| −p∑
k=1
αk · ek(A).
Given a c-graph A, we write A ∈ Knf1 i y1(A′) ≥ b1
n for everynonempty c-graph A′ ⊆c A, where n = |A′|.
Lemma 4.2.1. For every positive p ∈ ω and arbitrary c-graphsA and B, where A is a proper c-subgraph of B, |B| = n < kp+1, thefollowing assertions hold:
(1) We have yp(B)− y(B) < εp+1.(2) The inequality yp(A) < yp(B) holds i y(A) < y(B). The
inequality yp(A) < yp(B) holds i yq(A) < yq(B) for every q ≥ p.P r o o f is similar to that of Lemma 4.1.1. ¤
2. Generic class and generic theory. Given a c-graph A, wewrite A ∈ Knf
p+1 i A has no vertices that are forbidden for addingarcs to, A ∈ Knf
p , and yp(A′) ≥ bpn for every c-graph A′ ⊆c A, where
n = |A′|, kp ≤ n < kp+1.
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Put Knf0
∞⋂p=1
Knfp and denote by Knf the class of all graphs
(without furcations) whose all c-subgraphs belong to Knf0 .
Let A be a c-subgraph of a graph (of c-graph) M (of a graph)in Knf . We say that A is a self-sucient c-subgraph of the graph(respectively, c-graph) M and write A 6c M if y(A) ≤ y(B) forevery c-graph B, where A ⊆c B ⊆c M. If A 6c M and M is ac-graph then A is called a strong c-subgraph of M.
Notice that, for each c-graph A = 〈A,Q, W 〉 ∈ Knf0 and each
intermediate vertex a ∈ A, the set of all elements, that form a Q-route Sa including the vertex a, is contained in the denable closureof the set a: Sa ⊆ dcla. This means that every extension of ato a maximal Q-route Sa is uniquely dened. The set
A ∪ ∪Sa | a is an intermediate vertex in Ais called the route closure of A (inside a given graph that includesA as a c-subgraph) and denoted by ccl(A).
Clearly, the set ccl(A) ∩ (J0 ∪ J1) is nite for every c-graph A.Below, we assume that the record W contains information on
interconnection between the elements of ccl(A) by only externalshortest Q-routes. Thus, the record W of the expanded c-graphA = 〈A,Q, W 〉 contains the structure description of the set ccl(A)(possibly, including innite information on the routes of innitelength whose intermediate vertices are of degree 2), together withnite information on lengths of the external (w.r.t. ccl(A)) shortestQ-routes that connect the elements of ccl(A)∩J0 with the elementsof ccl(A) ∩ J1 only by external shortest Q-routes.
We are to prove that the class T of all types corresponding tothe expanded c-graphs in Knf
0 with the relation 6′c (where Φ(A) 6′
c
Ψ(B) ⇔ A 6c B) includes a self-sucient generic subclass Tnf0
that dominates T and satises (after adding the necessary formulasdescribing the self-sucient closures to the types) the uniform t-amalgamation property. This fact yields the ω-saturation of the(Tnf
0 ; 6′c)-generic model that realizes all types Φ(X) corresponding
to the types Φ(A) in T.Remark 4.2.2. (1) The conditions A ∈ Knf
0 and kp ≤ |A| =n < kp+1 do not imply that y(A) ≥ bp
n. Nevertheless, y(A) cannotbe much less than bp
n: since yp(A) − y(A) < εp+1, we have y(A) >bpn − εp+1.
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Moreover, in view of the inequality (4.1), for a graph A ∈ Knf0
of cardinality |A| ≥ max2, kp, the inequality y(A) > p holds.(2) Let A be a c-graph in Knf
0 , |A| = p, let M be a graph in Knf ,and A ⊆c M. Then A 6c M i yp(A) ≤ yp(B) for each c-graph B,where A ⊆c B ⊆c M.
Indeed, if |B| < kp+1, then yp(A) ≤ yp(B) is equivalent toy(A) ≤ y(B) by Lemma 4.2.1(2), and if |B| ≥ kp then
yp(B) ≥ y(B) > p ≥ yp(A).
Furthermore, in order to verify that A is self-sucient in M,it suces to choose nA = kp and check the validity of the relationyp(A) ≤ yp(B) only for c-graphs B, A ⊆c B ⊆c M, satisfying |B| <nA.
Thus, the condition A 6c M is denable by a formula thatdescribes the absence of n < nA new elements of M \A such that
n < α1 · e1 + α2 · (e2 − e′2) + . . . + αp · (ep − e′p),
where p = |A|, e1 is the number of new arcs, es is the number of newpairs of vertices (a, b) that are connected only by external shortestroutes of length s, and e′s is the number of pairs of vertices (a, b) thatare no longer connected only by external shortest routes of length sin the extended c-graph.
Moreover, each triple (a, b, s) that enters the calculation of e′sturns into a sequence of triples (a0, a1, s1), . . . , (ak−1, ak, sk) suchthat a0 = a, ak = b, the elements a1, . . . , ak−1 are of degree 2,the elements ai−1 and ai are connected by an arc (for si = 1) orthe unique external shortest route of length si (for si > 1) in theextended c-graph, and s1 + . . .+ sk = s. Therefore, when we extenda c-graph and remove a triple (a, b, s) that enters the calculation ofy(·), the value −αs is replaced by the positive (in view of αi < 1
2 ,i ≥ 1) value (k − 1)− αs1 − . . .− αsk
in the new calculation of y(·).Thus, the value y(·) can be diminished only by adding vertices ofcolor J0 or J1 to a c-graph; and the condition A 6c M is denableby a formula that describes the absence of n < nA new elements ofM \ A such that n < α1 · e1 + . . . + αp · ep, where p = |A| and es isthe number of new pairs of vertices (a, b) connected only by externalshortest routes of length s.
The following lemma is a direct consequence of the denition:
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Lemma 4.2.3. (1) If A 6c B then A ⊆c B.(2) If A 6c C, B ∈ Knf
0 , and A ⊆c B ⊆c C then A 6c B.(3) The empty graph ∅ is the least element of the structure
(Knf0 ; 6c).Lemma 4.2.4. If A,B, C ∈ Knf
0 , A 6c B, and C ⊆c B thenA ∩ C 6c C.
P r o o f. Assume the contrary. Then there exist n new elementsin C \ A such that n < α1 · e1 + . . . + αp · ep, where p = |A|, es isthe number of new pairs of vertices (a, b) connected only by externalshortest routes of length s. Since all new elements lie in B \A, theyviolate the condition A 6c B. ¤
Lemma 4.2.5. The relation 6c is a partial order on Knf0 .
P r o o f is similar to that of Lemma 4.1.5 upon replacing thegraphs of Kbp
0 by the c-graphs in Knf0 , and the relations ⊆ and 6,
by the relations ⊆c and 6c, respectively. ¤A c-graph A is J-closed if A contains, together with each inter-
mediate vertex a, the initial vertex (if it exists) and the nal vertex(if it exists) of the route Sa.
Let A, B = 〈B, QB,WB〉, and C = 〈C, QC ,WC〉 be expanded J-closed c-graphs, A = B ∩ C and ccl(A) = ccl(B) ∩ ccl(C) (the lastequality indicates that, together with each common Q-route S oflength ≥ 2 which is described at once in B and C, the set A containsan intermediate vertex common for B and C which belongs to S).The free c-amalgam of c-graphs B and C overA (in symbols, B∗AC) isthe c-graph 〈B∪C,QB∪QC∪Q,WB∪WC∪W 〉, where Q (respectivelyW ) is the set of the arcs (of the records on external shortest routesover B∪C) with endpoints in B∪C that are described in the graphwith universe ccl(A) .
By denition, every free c-amalgam is J-closed. Moreover, therecords on route closures ensure that the c-graph B ∗A C is a graphwithout furcations.
A c-embedding f of a c-graph A into a c-graph B (written f :A →c B) is said to be strong if f(A) 6c B.
Lemma 4.2.6. (amalgamation lemma). The class Knf0 has the
c-amalgamation property c-(AP), i.e., for arbitrary strong c-embed-dings f0 : A →c B and g0 : A →c C such that A,B, C ∈ Knf
0 areJ-closed c-graphs, there exist a J-closed c-graph D ∈ Knf
0 and strongc-embeddings f1 : B →c D and g1 : C →c D such that f0f1 = g0g1.
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P r o o f. Without loss of generality, we may assume that A 6c
B, A 6c C, A = B ∩ C, and ccl(A) = ccl(B) ∩ ccl(C). We claimthat c-graph D B ∗A C is the one sought. To this end, in view ofthe symmetry of the denition of free c-amalgam, it suces to showthat B 6c D and D ∈ Knf
0 .Suppose that B 66c D. Then there exist n new elements in D \B
such that n < α1 · e1 + . . . + αp · ep, where p = |B| and es is thenumber of new pairs of vertices (a, b) that are connected only byexternal shortest routes of length s. Since all new elements lie inC \A, they violate the condition A 6c C.
Since each intermediate vertex is of degree 2 and all numbersαp are less than 1
2 , the property of c-graph to belong to the classKnf
0 is preserved under adding nitely many intermediate vertices.Therefore, it suces to establish that the c-graph D belongs to Knf
0in the case when D has no intermediate vertices. In this case, eachJ-closed c-subgraph of D has the form B0 ∗A0 C0, where A0 6c B0
and A0 6c C0. Therefore, to verify that D ∈ Knf0 , it is sucient to
ascertain that yp(D) ≥ bpn, where n = |D|, kp ≤ n < kp+1.
Suppose that |B| ≤ |C| = m, kq ≤ m < kq+1, and A ( B. ByLemma 4.1.1(3) and the condition A 6c B, we have
sl((|A|, yq(A)), (|B|, yq(B))) ≥ sl((l, bql ), (l + 1, bq
l+1))
for any l ≥ m. Then fromsl((|C|, yq(C)), (|D|, yq(D))) = sl((|A|, yq(A)), (|B|, yq(B)))
we derivesl((|C|, yq(C)), (|D|, yq(D))) ≥ sl((m, bq
m), (|D|, bq|D|)). (4.3)
Since the c-graph C belongs to Knf0 , the point (|C|, yq(C)) is above
the point (m, bqm). Hence, the inequality (4.3) implies that the point
(|D|, yq(D)) is above the point (|D|, bq|D|), i.e., yq(D) ≥ bq
|D|.If p = q, then we establish the required inequality yp(D) ≥ bp
n
Otherwise, i.e., if q < p then the value yp(D) diers from yq(D) byless than αq+1 · kq+1 · (kq+1− 1) < 2εq+1, since D contains less thankq+1 · (kq+1 − 1) arcs together with records on Q-routes. In view ofLemma 4.1.1(4), we conclude that yp(D) ≥ bp
n. ¤Denote by Tnf
0 the class of all types of T that correspond toJ-closed c-graphs; and by 6′′
c , the self-suciency relation on Tnf0
induced by 6′c. Lemmas 4.2.34.2.6 imply
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Corollary 4.2.7. The class (Tnf0 ; 6′′
c ) is self-sucient.Denote the (Tnf
0 ; 6′′c )-generic theory by T nf .
We claim that the self-sucient class (T0;6′′′c ) possessing the
uniform t-amalgamation property arises after adding a formula χΦ(A)such that (T nf , A) ` χΦ(A) to each self-sucient type Φ(A) ∈ Tnf
0 .Indeed, each intermediate vertex generates at most two vertices
in J0 ∪ J1; therefore, the J-closure of p-element c-graph contains atmost 3p elements. Hence, by Remark 4.2.2, for an arbitrary c-graphA ∈ Knf
0 of cardinality p and an arbitrary graph M |= T nf , whereA ⊆c M, the following relation holds:
A 6c M⇔ y3p(A) ≤ y3p(B) for any c-graph Bsuch that A ⊆c B ⊆c M and |B| ≤ k3p.
Since the cardinalities of c-graphs B are bounded in terms of thecardinality of A, while the verication of y3p(A) ≤ y3p(B) involvesonly calculation of connections by the relations Q1, . . . , Q3p, the self-suciency condition A 6c M is expressed by a formula χA(X) ofgraph language Q, where the set X of variables is bijective withthe set A.
Let A and B be c-graphs in Knf0 , M be a generic model of
T nf , and A 6c B 6c M. Denote by ψA,s(X) (respectively, byψB,s(X, Y )) the formula that describes the Q1, . . . , Qs, J0, J1-typeof A (respectively, of B), where X and Y are disjoint sets of variableswhich are bijective with A and B \ A. Then for every s ≥ |B|, thefollowing formula is true in M:
∀X ((χA(X) ∧ ψA,s(X)) → ∃Y (χB(X, Y ) ∧ ψB,s(X, Y ))) .
The last relation implies the uniform t-amalgamation property forthe class (T0; 6) obtained from (Tnf
0 ;6′′′c ) by adding the formulas
that dene cardinality bounds and Q1, . . . , Q3p, J0, J1-structuresof self-sucient closures to the types as well as the formulas χA(A)to the types of self-sucient sets A.
By adding the above-stated formulas, we ensure that every niteset of models of the theory T nf possesses a nite closure.
For a nite set A in a model N of T nf , we denote the self-sucient closure of the J-closure of A by A and say that A is theJ-self-sucient closure of A. The sets A considered independentlyare called J-self-sucient sets.
Theorem 2.5.1 implies
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Theorem 4.2.8. The (Tnf0 ; 6′′′
c )-generic model M is saturated.Moreover, every nite set A ⊆ M can be extended to its J-self-sucient closure A ⊆ M , and the type tpX(A) is deduced from theset χA(X) ∪ ψA,s(X) | s ∈ ω \ 0.
Let N be an ω-saturated model of T nf .Proposition 4.2.9. For each nite set A in N , the equality
acl(A) =⋃B | B ⊆fin ccl(A) is valid. Moreover, the sets A \ A
and acl(A) \ ccl(A) do not contain intermediate elements.P r o o f. The set A\A does not contain intermediate elements,
since, after removing an intermediate element a from a nite set B,the value y(·) diminishes at least by a positive value 1−αs1−αs2+αs,where s1 and s2 | are the lengths of the external shortest routesover B that connect a with the remaining elements b and c, ands = s1 + s2 is the length of the shortest route, external over B \a,that connects b and c.
Denote the set⋃B | B ⊆fin ccl(A) by X. The inclusion X ⊆
acl(A) follows from the fact that the route Sa is uniquely determinedfor every intermediate vertex a ∈ A, and the sets B are unique in N ,where B ⊆fin ccl(A). Given an element b ∈ N \X, put C A ∪ b.Since C \ (A ∪ b) does not contain intermediate elements, by theconstruction of generic model, there are innitely many c-isomorphicpairwise disjoint copies of the set C \ X over X, i.e., b 6∈ acl(A).Thus, acl(A) ⊆ X.
Now, since B \ B does not contain intermediate elements, theequality acl(A) = X implies that acl(A) \ ccl(A) does not containintermediate elements either. ¤
3. Stability of the generic theory. We claim that the generictheory T nf is stable. Let N be a suciently saturated model of T nf .The rank function in N is the function
rN : A | A is a c-subgraph of N → R+,
that is dened by the equality rN (A) = infy(B) | A ⊆c B ⊆c N.Observe that, since the class (Tnf
0 ; 6′′c ) is self-sucient, the in-
mum in the above equality is always attainable and coincides withy(A): rN (A) = y(A).
In what follows, we x a model N , denote the function rN by rfor brevity, and replace c-graphs A by their universes in N . We alsoassume that all sets under consideration are subsets of N .
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The relative prerank function y(A/B) of a c-graph A over a c-graph B is given by the relation
y(A/B) = y(A ∪B)− y(B),
and the relative rank function r(A/B) of a c-graph A over a c-graph B is given by the relation
r(A/B) = r(A ∪B)− r(B).
We now establish some necessary properties of the relative rankfunction.
Clearly, r(A/B) ≥ 0.
Lemma 4.2.10. For any nite sets B and C, we have
y(B) + y(C) ≥ y(B ∪ C) + y(B ∩ C).
The inequality becomes an equality i there are no elements b ∈ B\Cand c ∈ C\ B connected by external shortest routes over B ∪C andnone of external shortest routes over B ∩ C contains intermediatevertices of B ∪ C.
P r o o f. By hypothesis, the value
V y(B) + y(C)− y(B ∪ C)− y(B ∩ C)
is equal to∞∑
k=1
αk · (ek(B ∩ C) + ek(B ∪ C)− ek(B)− ek(C)).
All summands in the last expression, and therefore the expressionitself, are equal to zero if there are no elements b ∈ B \ C andc ∈ C \ B connected by external shortest routes over B ∪ C andnone of external shortest routes over B ∩ C contains intermediatevertices of B ∪ C.
If there are elements b ∈ B \ C and c ∈ C \ B connected byexternal shortest routes over B ∪ C and none of external shortestroutes over either B ∩ C or B or C contains intermediate vertices
145
of B ∪ C then the positive value V equals∞∑
k=1
αk · ek, where e1 is
the number of Q-arcs, ek, k > 1, is the number of pairs (b, c) ∈(B \ C) × (C \ B) of elements that are connected only by externalshortest Q-routes over B ∪ C of length k.
Suppose that some of external shortest route over either B ∩ Cor B or C contains intermediate vertices of B ∪ C. Consider allexternal shortest routes over B ∩ C, as well as over B and C, suchthat their lengths do not change if we consider B∪C. Observe that,in calculation of V , the quantity for each of these routes equalszero while, for the pairs (b, c) ∈ (B \ C) × (C \ B) of elementsconnected by arcs or only by the external shortest routes over B∪Cof length k, that are not proper subroutes of external shortest routesover B ∩ C or B or C, the value V includes the nonnegative value∞∑
k=1
αk ·ek, where e1 is the number of Q-arcs, ek, k > 1, is the number
of pairs (b, c) ∈ (B \ C) × (C \ B) of elements connected only byexternal shortest Q-routes over B ∪ C of length k. Therefore, itsuces to establish that the positive values in V form the elementscorresponding to each external shortest route over B ∩C or B or Cthat contains intermediate vertices of B ∪ C.
Let (a1, a2) ∈ B2 ∪ C2 be a pair of vertices that are connectedover B ∩ C or B or C only by external shortest (a1, a2)-routes oflength s, and let a′0, . . . , a
′n, n ≥ 2, be all elements of B ∪ C that
are intermediate vertices or endpoints of (a1, a2)-routes. For anarbitrary pair (a′i, a
′j) of elements connected by a (unique) external
shortest (a′i, a′j)-route Si over B∪C, denote its length by si. By the
choice of vertices, we have si < s.Consider the case when there exist pairs (b1, b2) ∈ B2 and
(c1, c2) ∈ C2 of vertices that are connected only by external shortest(b1, b2)- and (c1, c2)-routes SB and SC over B and C, respectively,including the route Si. Denote the lengths of these routes by sB
and sC . By the choice of vertices, we have si < sB or si < sC . Thefollowing cases are possible:
(a) si = sB and si < sC , i.e., (a′i, a′j) = (b1, b2) and (a′i, a
′j) 6=
(c1, c2);(b) si < sB and si = sC , i.e., (a′i, a
′j) 6= (b1, b2) and (a′i, a
′j) =
(c1, c2);(c) si < sB and si < sC , i.e., (a′i, a
′j) 6= (b1, b2) and (a′i, a
′j) 6=
(c1, c2).
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In the rst case, the information on Si and SB is presentedin V by the dierence αsi − αsB equal 0. In the second case, theinformation on Si and SC is presented in V by the dierence αsi −αsC equal 0. Furthermore, the route SC (in the rst case), or theroute SB (in the second case), includes an external shortest (a′i′ , a
′j′)-
route over B∪C such that (a′i′ , a′j′) 6∈ B2∪C2. Thus, we reduce the
calculation of V to examining the third case, when the informationon Si, SB, and SC is presented in V by the expression
αsi − αsB − αsC . (4.4)
which is positive in view of αsB <αsi2 and αsC <
αsi2 . The sum of
values of the positive expressions (4.4) gives a lower bound for V .Suppose that the above pair (b1, b2) ∈ B2 exists while the pair
(c1, c2) ∈ C2 does not exist. Then, by examining the cases si = sB
and si < sB in a similar way, we come to the expression αsi − αsB
which is zero in the rst case and positive in the second case; andagain the positive expressions give a lower bound for V .
Arguing similarly in the case when the pair (c1, c2) ∈ C2 existswhile the pair (b1, b2) ∈ B2 does not exist, we obtain a positive lowerbound for V as well. ¤
Lemma 4.2.11. If B1 ⊆ B2 then r(A/B1) ≥ r(A/B2).P r o o f is a word-for-word repetition of the proof of Lemma
4.1.11. ¤If A is a nite set and X is some set (not necessarily nite) then
r(A/X) infr(A/B) | B ⊆fin X.
Lemma 4.2.12. Let A1 and A2 be nite sets and let X be a setsuch that A1 \X = A2 \X. Then r(A1/X) = r(A2/X).
P r o o f is similar to that of Lemma 4.1.12. ¤A set X ⊆ N is called closed (in N ) (written X 6 N) if ccl(X) =
X and A ⊆ X for every nite set A ⊆ X. The last condition isequivalent to the fact that N \X does not contain n new elementssuch that n <
∞∑k=1
αk · ek, where e1 is the number of new Q-arcs, ek,k > 1, is the number of new pairs of elements connected only byexternal shortest Q-routes of length k.
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Lemma 4.2.13. Let X and Y be sets in N . Then the followinghold:
(1) if X 6 N and Y 6 N then X ∩ Y 6 N ;(2) there exists a smallest closed set X
c ⊇ X; moreover, Xc =⋃A | A ⊆fin ccl(X) and X
c = acl(X);(3) if X ⊂ Y then X
c ⊆ Yc.
The set Xc is called an intrinsic closure of X (in N ).
P r o o f. (1) Suppose that X 6 N and Y 6 N . Clearly, ccl(X∩Y ) = ccl(X) ∩ ccl(Y ) = X ∩ Y , i.e., the set X ∩ Y is route-closed.Let A be a nite subset of X ∩ Y . Since A ⊆ X, A ⊆ Y , andboth X and Y are closed, we have A ⊆ X and A ⊆ Y . Therefore,A ⊆ X ∩ Y . Thus, X ∩ Y is closed.
(2) Denote the set⋃A | A ⊆fin ccl(X) by Z. Obviously, each
closed superset of X includes Z, and X ⊆ Z. On the other hand,in view of Proposition 4.2.9, we obtain ccl(Z) = Z, and if A is anite subset of Z then there exists a nite set B ⊆ ccl(X) such thatA ⊆ B. We claim that A ⊆ B. Indeed, A ⊆ A∩B 6c N . Therefore,A ⊆ A ∩ B and A ⊆ B. The inclusions A ⊆ B and B ⊆ Z implyA ⊆ Z. Thus, Z is the least closed set that includes X.
(3) Clearly, X ⊂ Y implies ccl(X) ⊆ ccl(Y ). It remains toobserve that, for arbitrary c-graphs A and B, if A ⊆ B then A ⊆ B.Indeed, A ⊆ A ∩ B 6c N , as above. Therefore, A ⊆ A ∩ B andA ⊆ B. ¤
By analogy with Section 4.1, we say that nite sets A and B areindependent over Z and write A ↓r
Z B if r(A/Z) = r(A/(Z ∪ B))and A ∪ Z
c ∩B ∪ Zc ⊆ Z.
We say that sets X and Y are independent over Z and writeX ↓r
Z Y if A ↓rZ B for all nite sets A ⊆ X and B ⊆ Y .
Lemma 4.2.14. If X and Y are closed sets, Z = X ∩ Y , andX ↓r
Z Y then X ∪ Y is closed.P r o o f. Since ccl(X ∪ Y ) = ccl(X) ∪ ccl(Y ) and both X and
Y are route-closed, the set X ∪ Y . is route-closed. The implication
A ⊆fin (X ∪ Y ) ⇒ A ⊆fin (X ∪ Y )
is veried similarly to the proof of Lemma 4.1.14 on using Lemma4.2.11. ¤
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Lemma 4.2.15. If X is a J-self-sucient set, Y is a closedset, and Z = X ∪ Z
c∩Y then X ↓rZ Y i X ∪ Z
c∪Y is a closed setsuch that there are no elements x ∈ X ∪ Z
c \ Y and y ∈ Y \X ∪ Zc
connected by external shortest routes over X ∪ Zc ∪ Y , and none
of external shortest routes over Z contains intermediate vertices ofX ∪ Z
c ∪ Y .P r o o f is similar to that of Lemma 4.1.15 on using Lemmas
4.2.11 and 4.2.13. ¤Lemma 4.2.15 readily impliesCorollary 4.2.16. If X is a J-self-sucient set, Y is a closed
set, Z = X ∪ Zc∩Y and X ↓r
Z Y then the type tp(X/Y ) is uniquelydened by the type tp(X/Z), a description of the propertyfor X ∪ Z
c∪Y to be closed, and the type that describes the absence ofelements x ∈ X ∪ Z
c \Y and y ∈ Y \X ∪ Zc , connected by external
shortest routes over X ∪ Zc ∪ Y and the absence of intermediate
vertices of X ∪ Zc ∪ Y for external shortest routes over Z.
Lemma 4.2.17. If X is a closed set and a is an element ofthe model N that does not belong to X then there exists at mostcountable closed subset X ′ ⊆ X such that a ↓r
X′ X.P r o o f is a word-for-word repetition of that of Lemma 4.1.17
on using Lemmas 4.2.11 and 4.2.13. ¤Theorem 4.2.18. (1) The theory T nf is stable, small and has
countably many 1-types: of elements of color J0, of elements of colorJ1, of intermediate elements that are at some nite distance fromelements of color J0 and (or) elements of color J1, and of intermedi-ate elements that are not connected by routes with elements of colorJ0 ∪ J1.
(2) Any 1-type of non-intermediate elements of T nf has inniteweight; namely, for every element a of color Ji, i = 0, 1, there areinnitely many independent elements bn, n ∈ ω, of color J1−i thatdepend on a.
P r o o f. (1) Theorem 4.2.8 implies that T nf is small. The proofof stability for T nf is similar to that for T bp in view of Corollary4.2.16 and Lemma 4.2.17.
By construction, every singleton a is self-sucient. In view ofTheorem 4.2.8, this means that there exist countably many 1-typesof T nf , and each of these types is dened by one of the followingsets:
149
(a) J0(x);(b) J1(x);(c) the set of formulas describing the fact that an intermediate
element x s at some xed nite distance from an element of J0 andat some xed nite distance from an element of J1;
(d) the set of formulas describing the fact that an intermediateelement x is at some xed nite distance from an element of Ji andthat there are no connections between x and elements of J1−i bya Q-route, i = 0, 1;
(e)the set of formulas describing the absence of connections be-tween an element x and elements of J0 ∪ J1 by Q-routes.
(2) Let A be a c-graph with universe a, Ap = a, bp be theuniverse of two-element c-graph Ap that contains the Qp-arc be-tween a and bp, where a ∈ Ji and bp ∈ J1−i. Put B1 A1,Bp+1 Bp ∗A Ap+1, B
⋃p∈ω\0
Bp. By construction, B is a self-
sucient subgraph of some generic model M of T nf . Since α1 +. . . + αp+1 < 1 and b1, . . . , bp+1 6c a, b1, . . . , bp+1 6c B 6c M ,we have bp+1 ↓r
∅ b1, . . . , bp. Thus, (bp)p∈ω\0 is an innite inde-pendent sequence of elements of the type J1−i(x) such that eachelement bp depends on a. ¤
Since αp < 12 for every natural p ≥ 1, it follows that every two
elements a and b of color J0 are connected via an element c of colorJ1 so that aQc and bQc. In the same way, every two elements a andb of color J1 are connected via an element c of color J0 so that cQaand cQb. Furthermore, in either case, there are innitely many suchelements c. Thus, we obtain
Proposition 4.2.19. The set of elements of color Ji has thepairwise Q(−1)1−iintersection property, i = 0, 1.
The following two assertions clarify the structure of the primemodels MA over nite sets A.
Lemma 4.2.20. If A and B are J-self-sucient sets in N andA 6c B then the type tp(B/A) is isolated i the following conditionshold:
(1) each intermediate element of B \A that belongs to an innitechain is connected with an intermediate element of A by a Q-route;
(2) every two dierently J-colored elements a ∈ B ∩ (J0 ∪ J1)and b ∈ (B \A) ∩ (J0 ∪ J1) are connected by Q-routes.
150
P r o o f. Let Y be the set of variables bijective in B \A. If theconditions (1) and (2) are satised then, by Theorem 4.2.8, the typetp((B \ A)/A) is isolated by the formula χB(A ∪ Y ) ∧ ψB,s(A ∪ Y ),where s is a positive number that exceeds all lengths of externalshortest routes connecting elements of B. If any of the conditions (1)and (2) is violated then, by Theorem 4.2.8, the type tp((B\A)/A) isisolated by the set of formulas χB(A∪Y )∪ψB,s(A∪Y ) | s ∈ ω,but is not isolated by any nite part of this set. ¤
Lemma 4.2.20 impliesCorollary 4.2.21. Let A be a J-self-sucient set in N . In the
model MA, each intermediate element of MA \ A that belongs toan innite chain is connected with an intermediate element of A bya Q-route. Furthermore, every two dierently J-colored elementsa ∈ M ∩ (J0 ∪ J1) and b ∈ (M \ A) ∩ (J0 ∪ J1) are connected byQ-routes. The set of isomorphism types of prime models over nitesets coincides with the set of isomorphism types of the models MA,where A are J-self-sucient sets.
§ 4.3. Small stable generic graphs with innite weight.Powerful digraphs
In this Section, on the base of generic constructions in Sections4.1 and 4.2, we describe a generic construction producing a family ofstable powerful digraphs with almost inessential ordered colorings.1. Tandem c-graphs without furcations. A furcation-tandemin a c-graph A (in a graph M, respectively) is a pair (ab, at) of ver-tices such that the following conditions hold in cc(A) (in M):
(a) ab has zero outcoming semi-degree deg+ab and non-zero inputsemi-degree deg−ab;
(b) at has zero input semi-degree deg−at and non-zero outcomingsemi-degree deg+at;
(c) deg−ab + deg+at ≥ 3.Clearly that, for any c-graph (respectively, for acyclic digraph),
an identication of vertices of furcation-tandem (ab, at) forms a fur-cation. Conversely, any furcation-tandem is a result of replace-ment of a furcation a by a pair (ab, at) of dierent vertices, whereendpoints of routes, that input in a, are replaced by ab, and ini-tials of routes, outcoming from a, are replaced by at. Further-more, a c-graph (a digraph) without furcations is formed as result
151
of all possible replacements of furcations by furcation-tandems ingiven c-graph (acyclic digraph). If resulted c-graph (digraph) isequipped by an equivalence E with singleton classes for old ver-tices and with two-element classes, that contain furcation-tandemsw.r.t. old vertices, then it is called a tandem c-graph without fur-cations T (A) = 〈T (A), Q, W,E〉 (tandem graph without furcationsT (M) = 〈T (M), Q, E〉), correspondent to c-graph A (digraph M).Moreover, we consider, for the deniteness, that any vertex a of de-gree 0, lying in T (A), belongs to J1.
Clearly, for any c-graph A = 〈A, Q,W 〉, the number of arcs inT (A) is equal to the number of arcs in Q, and the number of recordson its external shortest routes is the number of triples (a, b, n) ∈ W .
For any c-graph A (respectively, for acyclic digraph M), wedenote by Af (respectively, by Mf ) the set of all furcations in A(in M), and we denote by Anf (Mnf) the set of all vertices in A(in M) that are not furcations in A (in M): Anf = A \ Af (Mnf =M \ Mf ). We shall use notations Af , Anf , Mf , and Mnf insteadof Af , Anf , Mf , and Mnf , respectively, if a c-graph (a digraph) isxed.
Clearly, if A ⊆c B then Af ⊆ Bf .Constructing c-graph T (B) w.r.t. c-graph A, where A ⊆c B, we
shall assume that any vertex a ∈ A ∩ (Bf \ Af ) coincides with ab
if deg+Aa = 0, and is equal to at, otherwise. Each vertex a in T (B)
w.r.t. A will be denoted by aT (A).Let f0 : A →c B be a c-embedding. A c-embedding g : T (A) →c
T (B) is called a canonical c-embedding of the c-graph T (A) into thec-graph T (B) w.r.t. f0 (written g : T (A) →c,f0 T (B)) if g(a) =f0(a)T (f0(A)) for any vertex a ∈ Anf , and g(ab) = f0(a)b and g(at) =f0(a)t for any vertex a ∈ Af .
Clearly, for any c-embedding f0 : A →c B, a canonical c-embed-ding g : T (A) →c,f0 T (B) exists.
Two c-graphs, T (A) and T (B), are called canonically c-isomorphicif there exist c-isomorphisms f0 : A →c B and g : T (A) →c,f0 T (B).Furthermore, the map g is said to be a canonical c-isomorphismbetween T (A) and T (B) w.r.t. c-isomorphism f0, and c-graphsT (A) and T (B) are called canonically c-isomorphic copies (w.r.t.c-isomorphism f0).
For any c-graphs A, B, and C, the following assertions hold, con-necting, by canonical c-isomorphisms, set-theoretic operations overc-graphs with correspondent operations over their tandem images.
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Proposition 4.3.1. Two c-graphs, T (A∩B) and T (A)∩T (B),are canonically c-isomorphic i (A ∩B)f = Af ∩Bf .
P r o o f. Suppose, that c-graphs T (A ∩ B) and T (A)∩ T (B)are canonically c-isomorphic. Then each furcation-tandem (ab, at)in T (A ∩ B) is a furcation-tandem both in T (A) and in T (B), andconversely, each common furcation-tandem of T (A) and of T (B) isa furcation-tandem in T (A ∩ B). The rst conclusion means that(A∩B)f ⊆ Af ∩Bf , and the second that Af ∩Bf ⊆ (A∩B)f . Thus,(A ∩B)f = Af ∩Bf .
Now, suppose that (A ∩ B)f = Af ∩ Bf . Then constructing c-graph T (A∩B), we observe that vertices in A∩B become furcation-tandems i these vertices become furcation-tandems both in T (A)and T (B). Thus, having simultaneous transformations of furcationsto furcation-tandems, c-graphs T (A ∩ B) and T (A) ∩ T (B) equallypreserve route connections between elements in A ∩ B, i.e., the c-graphs T (A ∩ B) and T (A) ∩ T (B) are canonically c-isomorphic. ¤
Proposition 4.3.2. Let M be an acyclic digraph containing c-subgraphs A and B so that Mf ∩ A = Af and Mf ∩ B = Bf .Then c-graphs T (A ∪M B) and T (A) ∪T (M) T (B) are canonicallyc-isomorphic i (A ∪B)f = Af ∪Bf .
P r o o f is similar the proof of Proposition 4.3.1 with replace-ments of symbols ∩ by ∪. ¤
Proposition 4.3.3. If A = B ∩ C then the c-graphs T (B ∗A C)and T (B)∗T (A) T (C) are canonically c-isomorphic i Af = Bf ∩Cf .
P r o o f is analogous to the proof of previous Propositions.It should only be noticed that, by denition of free amalgam, theequality Af = Bf ∩ Cf implies (B ∪ C)f = Bf ∪ Cf . ¤
If A ⊆c B then the c-graph A will be denoted by B ¹ A.For a c-graph A, we denote the set A ∪ ∪ab, at | a ∈ A by
Υ(A). Then for all possible c-graphs B such that A ⊆c B and theset B, together with each vertex a ∈ A, doesn't contain vertices ab
and at distinguished from a, and together with each vertex aj ∈ A,doesn't contain the vertex a distinguished from ab and at,3 restric-tions T (B) ¹ Υ(A) correspond all possibilities of replacements ofvertices in Anf by furcation-tandems w.r.t. c-extensions of A. A
3For c-graphs, the condition of inconsistency for presence of dierent elementsa and aj is stated to avoid collisions, and it will be always supposed for nitesets of c-graphs.
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c-graph T (B) ¹ Υ(A), in which all vertices of A are replaced byfurcation-tandems, is denoted by T ∗(A).
A c-graph T (A) is called a canonical c-subgraph of c-graph T (B)if A ⊆c B and T (A) = T (B) ¹ Υ(A).
Let A be a c-subgraph of c-graph B. The c-graph B is calleda hereditary extension in furcations of A (written A ⊆f B) if Af =Bf ∩Ain, where Ain is the set of vertices in A having nonzero inputand outcoming semi-degrees.
Clearly, the relation ⊆f is re exive, antisymmetric and transi-tive.
Proposition 4.3.4. Let A be a c-subgraph of c-graph B, and theset B, together with each vertex a ∈ A, doesn't contain vertices ab
and at distinguished from a, and together with each vertex aj ∈ A,doesn't contain the vertex a distinguished from ab and at. ThenT (A) is a canonical c-subgraph of T (B) i A ⊆f B.
P r o o f. Suppose, that T (A) is a canonical c-subgraph of T (B).Then the furcation-tandem (ab, at) of each vertex a in Bf coincideswith the furcation-tandem of a in A or contains the vertex a in A.For the rst case, we have a ∈ Af , and for the second case, a cannot be intermediate in A, i.e., have non-zero input and outcomingsemi-degrees, since deg+ab = 0 and deg−at = 0 for the vertex abeing in B. Thus, Af = Bf ∩Ain, that is, A ⊆f B.
Now suppose, that A ⊆f B. Then constructing T (A) and T (B),each intermediate vertex of A is (not) replaced by furcation-tandemboth for A and for B, and each vertex a ∈ A, transforming to thefurcation-tandem for B, is equal to ab if deg+
Aa = 0, and is equalto at otherwise. Since arcs and an information on external shortestroutes is preserved by constructions of T (A) and T (B), we have thatT (A) is a canonical c-subgraph of T (B). ¤
LetA be a c-subgraph of c-graph B, and letA′ be a c-subgraph ofc-graph T ∗(A), in which all furcations a in Bf ∩A are represented byfurcation-tandems (ab, at), and the rest vertices in A by themselvesor by furcation-tandems. Then the tandem c-graph without furca-tions, corespondent to c-graph that obtained from A′ by adding ofall vertices in B \A, all arcs, and the information on external short-est routes, connecting (in B) elements of B \A with elements of B,is said to be a tandem c-graph without furcations correspondent to Bw.r.t. A′, and denoted by T (BA′).
Clearly, T (B) = T (B∅) for any c-graph B, and if A′ 6= ∅ thenT (B) is naturally embeddable in T (BA′).
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2. Prerank functions. We dene a preprerank function y0 thatassigns a real to every c-graph A by the rule
y0(A) = 2 · |Af |+ |Anf | −∞∑
k=1
αk · ek(A).
Notice that, in the denition of a preprerank function, the sum-mation is always nite due to the niteness of c-graphs. More-over, for any c-graph A, the value y0(A) is equal to y(T (A)), where2 · |Af |+ |Anf | = |T (A)|. Thus,
y0(A) = |T (A)| −∞∑
k=1
αk · ek(A).
A p-approximation of preprerank function y0 is a function y0p
assigning a real to every c-graph A by the rule
y0p(A) = 2 · |Af |+ |Anf | −
p∑
k=1
αk · ek(A).
Clearly, y0p(A) = yp(T (A)).
Below we describe constructions simultaneously for c-graphs Aand correspondent c-graphs T (A) without furcations.
The function y0(·), dened as prerank functions above, has thefollowing essential lack. If one take c-graphs with non-negative val-ues y0(·), then c-subgraphs, obtained by removing of furcations, canhave values less than any given negative number.
Indeed, if c-graph An consists of furcation a, vertices b1, . . . , bn,c1, . . . , cn, set Q = (b1, a), . . . , (bn, a), (a, c1), . . . , (a, cn) of arcs,and empty set W , then the c-subgraph Bn of An, that formedfrom An by removing of a, has the following value of preprerankfunction: y0(Bn) = 2n− α2 · n2. Therefore, lim
n→∞ y(Bn) = −∞.For the control of real balance, in c-graphs A, between the num-
ber of elements and the number of links, by prerank function, wehave to take into consideration a virtual presence (as an addition torecords W ) of nitely many so-called \compulsory" furcations, i.e.,furcations a such that there are no super uous shortest routes con-necting dierent pairs (a′, a′′) ∈ ⋃
n∈ω\0Qn(a,A)× ⋃
n∈ω\0Qn(A, a)
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and avoiding these furcations. Moreover, before adding of all com-pulsory furcations, belonging to shortest (a′, a′′)-routes, the pairs(a′, a′′) don't take into consideration for the calculation of valueof prerank function, and after adding of all compulsory furcations,we take into consideration all pairs connected by external shortestroutes only. Thus, the prerank function will have only non-negativevalues and it will allow to include all compulsory furcations in self-sucient closures of c-graphs.
Thus, a furcation a of c-graph A is called A-compulsory if
y(T ∗(B)) > y(T ((A ¹ (B ∪ a))T ∗(B)))
for some c-subgraph B ⊆c A.4 Furthermore, the furcation a is alsocalled B-compulsory.5
An A-compulsory furcation is called B-external (where B ⊆c
A) if it doesn't belong to B. An A-compulsory furcation is calledcompulsory if A is xed. A B-external compulsory furcation is saidto be external if c-graph B is xed.
By denition of Knf0 , for any xed number n ∈ ω and any c-graph
T ∗(A) ∈ Knf0 , there are cardinality bounds, depending on cardinal-
ity |T ∗(A)|, for number of new external shortest routes of lengths atmost k, connecting vertices in T ∗(A) with some new vertex a so thatobtained c-graph B with universe T ∗(A) ∪ a or T ∗(A) ∪ ab, atbelongs to Knf
0 . Therefore, a sequential choice of numbers αs, sat-isfying all aforesaid condition, allows to produce the property thatsome vertex, ab or at, for each compulsory furcation a, belongs toa self-sucient closure for the set of its predecessors or for the setof its successors in T ∗(A).
Indeed, removing any furcation a connected with b1, . . . , bl, c1,. . . , cm only by external shortest (b1, a)-, . . ., (bl, a)-, (a, c1)- and(a, cm)-routes of lengths s1, . . . , sl, s
′1, . . . , s
′m, respectively, such that
for any pair (bi, cj) the length sij of shortest (bi, cj)-route equalssi + s′j , we obtain only external shortest routes between bi and cj .
4By denition of y(·), for checking that the furcation a is compulsory, it suf-
ces to consider c-subgraphs B ⊆c A ¹(
⋃n∈ω\0
Qn(a,A) ∪ ⋃n∈ω\0
Qn(A, a)
).
5Here, the dierence of A-compulsion and B-compulsion is that the furcationa belongs to A and doesn't belong to B. Moreover, belonging to a self-sucientclosure B, the vertex a is an element integrated in B.
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Moreover, in prerank function y(·), sums∑i,j
(αsi + αs′j ) are replaced
by∑i,j
αsij , where αsij < minαsi , αsj. Considering all possible
congurations of c-graphs with universes a, b1, . . . , bl, c1, . . . , cmand having only external shortest routes of lengths at most k − 1,we can, on each sequential step of denition for the number αk,choose this number so small that the following conditions (∗) aresatised:
(1) if for c-graphs B and C with universes a, b1, . . . , bl anda, c1, . . . , cm respectively, where deg+
B (a) = deg−C (a) = 0, the rela-tions
y(T (BT ∗(B¹b1,...,bl))) ≥ y(T ∗(B ¹ b1, . . . , bl)) (4.5)and
y(T (CT ∗(C¹c1,...,cm))) ≥ y(T ∗(C ¹ c1, . . . , cm)) (4.6)hold then, for c-graph D with the universe a, b1, . . . , bl, c1, . . . , cm,the relation QB ∪ QC , the record WB ∪ WC , and for the c-graphE D ¹ b1, . . . , bl, c1, . . . , cm, the inequality
y(T (DT ∗(E))) ≥ y(T ∗(E)) (4.7)
is satised;(2) if a vertex a is not an E-compulsory furcation then, for the
c-graphs B′, C′, and E ′ that obtained from B, C, and E , respectively,by removing of all (bi, bj)- and (ci, cj)-routes, we have
∞∑
k=1
αk · ek(E ′) <12·min
∞∑
k=1
αk · ek(B′),∞∑
k=1
αk · ek(C′)
. (4.8)
Considering two-element c-graph, B and C, and using the condi-tion (2), we conclude that
∞∑
k=1
αk · ek(E ′) <12· αl,
where l is the greatest index, till which, inclusive, all values ek(E ′)equal zero. Moreover, taking all possible c-graphs satisfying (∗),
157
we may rewrite the inequality (4.8) by the following sequence ofinequalities for all natural p ≥ 1:
p∑
k=1
αk · ek(E ′) <12·min
p∑
k=1
αk · ek(B′),p∑
k=1
αk · ek(C′)
.
Below, we suppose that (∗) is satised and this property is takeninto consideration in the denition of αk's. It means, in particular,that if (4.7) is violated then (4.5) or (4.6) is not true, i.e., if a D-compulsory furcation a belongs to the self-sucient closure of Ethen a belongs to the self-sucient closure of B ¹ (B \ a) or ofC ¹ (C \ a).
The conditions (∗) also imply that, for any non-compulsory fur-cation a of c-graph A, all connections between elements in A \ ashortest routes by external over A \ a, containing a, can be re-placed by links being external shortest routes over A\a such thatsets of intermediate elements are pairwise disjoint.
The following modication of the preprerank function allows toadd all A-compulsory furcations to the self-sucient closure of A,that will be guaranteed by extension of the record WA.
We allow for each c-graph B ⊆c A to have less than k|T (B)| com-pulsory furcations (these furcations mat be not only B-compulsory,but also compulsory w.r.t. c-graphs that obtained by extensions of Bby other, of less than k|T (B)|, compulsory furcations).6 Now, we addto each record WB a type describing the structure of c-graph B to-gether with all its compulsory furcations, and also describing anabsence of any another compulsory furcations. Obtained expandedc-graphs are called cc-graphs.
We denote by Acf the set of all A-compulsory furcations of thecc-graph A belonging A. The cc-graph A is called cf-closed if Acf
contains all A-compulsory furcations.The operation of adding to a cc-graph A of all its compulsory
furcations with the structure, described in W (A), is called the op-eration of cf-closure of cc-graph A. The cc-graph, being the resultof cf-closure of cc-graph A, is denoted by cfc(A).
6In view of constructions in Section 4.2, the stated condition is satised forany c-graph A such that T (A) ¹ B ∈ Knf
0 for any c-subgraph B ⊆c A.
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A cc-graph A is a cc-subgraph of cc-graph B (written A ⊆cc B)if the following conditions hold:
(1) the c-graph 〈A,QA, W1〉 with the record W1 of the structureof c-graph in A is a c-subgraph of the c-graph 〈B,QB, W2〉 with therecord W2 of the structure of c-graph in B;
(2) the record WA coincides with the restriction of the recordWB to the set A.
A cc-graph A is called a cc-subgraph of graph M if the triple〈A,QA,W 〉, with the record W of structure of c-graph in A, is a c-subgraph of M and the record WA of the cc-graph A is co-ordinatedwith tpM(A). We write A ⊆cc M if A is a cc-subgraph of M.
Clearly, ifA ⊆cc B then anyA-compulsory furcation is B-compul-sory and, in particular, Acf ⊆ Bcf .
We dene a prerank function y1 that assigns a real to every cc-graph A by the rule
y1(A) = 2 · |Af |+ |Anf | −∞∑
k=1
αk · e1k(A),
where e11(A) is the number of arcs in A, e1
k(A), k ≥ 2, is the numberof pairs (a, a′) ∈ A2, connected only by external shortest (a, a′)-routes of length k and such that there are no (a, a′)-routes of length kcontaining A-external A-compulsory furcations.
The construction of tandem cc-graph Tc(A) (T ∗c (A)) withoutfurcations, correspondent to cc-graph A, consists of removing fromc-graph T (A) (T ∗(A)) of all tuples (a1, a2, n), for which there ex-ist external shortest (a1, a2)-routes of length n containing externalcompulsory furcations.
By denition, we have |T ∗c (A)| = 2 · |A|, |Tc(A)| = |T (A)|and y1(A) = y(Tc(A)) = y(T ∗c (A))− |Anf |.
A p-approximation of prerank function y1 is a function y1p as-
signing a real to every cc-graph A by the rule
y1p(A) = 2 · |Af |+ |Anf | −
p∑
k=1
αk · e1k(A).
Clearly, y1p(A) = yp(Tc(A)).
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3. Generic class and generic theory. We are going to denea class of cc-graphs, in which like the classes of c-graphs from Sec-tions 4.1 and 4.2, on the one hand, there is a balance between num-bers of elements and number of connections by lower estimates bp
n forvalues of prerank function, and, on the other hand, a decreasing ofvalues of prerank function with extensions of universes of cc-graphsdetermines that elements of extensions belong to self-sucient clo-sures of given cc-graphs. At the same time, doubling weights ofvertices for elements a ∈ Anf being in Bf , where A ⊆cc B, we cannot observe the self-suciency of cc-graphs A (meaning an exceed-ing of number of new element, not of virtual pairs of elements, overthe numbers of new links) by inequalities y1(A) ≤ y1(B). The samecause, even for the following denition of self-suciency, doesn'tallow to obtain the nite closure property for models of generic the-ory and, as corollary, to have saturated generic model taking, asabove, the class of cc-graphs A for which all cc-subgraphs A′ satisfyy1
p(A′) ≥ bpn, where n = |T (A′)| and kp ≤ n < kp+1.
For the realization of required properties, we restrict the class ofconsidered cc-graphs to a subclass, allowing to bound the numberof iterations for self-sucient closures of cc-subgraphs dependenton cardinalities of given cc-graphs. The self-suciency conditionwill be dened as nondecreasing of values of prerank balance be-tween number of new vertices and number of new links in tandem c-graphs without furcations w.r.t. tandem c-graph without furcationsof given self-sucient cc-graph for which each vertex is representedby a furcation-tandem.
LetA be a cc-subgraph of cc-graph B, andA′ be a cc-subgraph ofT ∗c (A), in which all furcation vertices a in Bf ∩A are represented byfurcation-tandems (ab, at), and the rest vertices of A by themselvesor by furcation-tandems. Then the tandem cc-graph without furca-tions, correspondent to cc-graph that obtained from A′ by adding allvertices of B \A and of correspondent arcs and the information oncompulsory furcations and on external shortest routes, connecting(in B) elements of B \ A with elements of B, is said to be a tan-dem cc-graph without furcations correspondent to B w.r.t. A′, andis denoted by Tc(BA′).
Recall [25], that a graph A is a part of graph B if A ⊆ B andB contains all arcs of A. Similarly, a cc-graph A is called a part ofcc-graph B (written A v B) if A ⊆ B, B contains all arcs of A, andthe record WB includes all records in WA. We also write A vM ifA is a part of some cc-graph B being a cc-subgraph of graph M.
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Notice, that replacing a cc-graph A by its parts with the uni-verse A, we can consider all subsets of A, consisting of furcations, asnon-furcations and, by observation of parts, to test cc-subgraphs A′of A being self-sucient in A. Moreover, by (∗), the non-self-suciency in A satises the following hereditary property: if A con-tains a subset B ⊆ A \A′ that by adding to A′ produces, w.r.t. A′,a negative dierence between number of new vertices and weightednumber of new links in a part of A with the universe A′ ∪ B, thenfor any cc-graph A′′, where A′ ⊆cc A′′ ⊆cc A and B ⊆ A \ A′′, wealso have a negative dierence between number of new vertices andweighted number of new links, w.r.t. A′′, in some part of A withthe universe A′′ ∪B.
Lemma 4.3.5. For every positive p ∈ ω and arbitrary cc-graphsA and B, where A is a proper part of B, |T (B)| = n < kp+1, thefollowing assertions hold:
(1) We have y1p(B)− y1(B) < εp+1.
(2) The inequality y1p(A) < y1
p(B) holds i y1(A) < y1(B). Theinequality y1
p(A) < y1p(B) holds i y1
q (A) < y1q (B) for every q ≥ p.
P r o o f is similar to that of Lemma 4.1.1 with replacementsof y by y1, and graphs A and B by cc-graphs Tc(A) and Tc(B),respectively. ¤
Given a cc-graph A, we write A ∈ Kf1 i y1
1(A′) ≥ b1n for any
nonempty cc-graph A′ v cfc(A), where n = |T (A′)|.For a cc-graph A, we write A ∈ Kf
p+1 i A ∈ Kfp and y∆
p (A′) ≥bpn for any cc-graph A′ v cfc(A), where y∆
p (A′) is a minimal valueyp(Tc(A0))+∆1+ . . .+∆m, ∆i = yp(Tc((Ai+1)T ∗c (Ai)))−yp(T ∗c (Ai)),A0 ⊆cc A1 ⊆cc . . . ⊆cc Am = A′, n is a natural number, for whichy∆
p (A′) = n − s · α1 with some s ∈ ω, kp ≤ n < kp+1. Notice, thatthe denition is correct, since the value n in n−s·α1 doesn't dependon p.
Put Kf0
∞⋂p=1
Kfp and denote by Kf the class of all acyclic
digraphs whose cc-subgraphs belong to Kf0 .
Let A be a cc-subgraph of a graph (respectively, of a cf-closed cc-subgraph) M (of a graph) in Kf . We say that A is a self-sucientcc-subgraph of the graph (respectively, cc-graph) M and write A 6cc
M if for any cc-graphs A′ v A, B′ v M, A′ = A0 ⊆cc A1 ⊆cc
. . . ⊆cc Am = B′, the inequality ∆1 + . . . + ∆m < 0, where ∆i =
161
y(Tc((Ai+1)T ∗c (Ai))) − y(T ∗c (Ai)), implies B′ ⊆ A. If A 6cc M andM is a cc-graph then A is called a strong cc-subgraph of M.
In particular, if a cc-graph A is self-sucient then A is cf-closed.We have to show that the class Tf
0 of all types corresponding tothe cc-graphs of Kf
0 with the relation 6′cc (where Φ(A) 6′
cc Ψ(B) ⇔A 6cc B) is self-sucient and satises (after adding the necessaryformulas describing the self-sucient closures to the types) the uni-form t-amalgamation property. This fact yields the ω-saturation of(Tf
0 ; 6′cc)-generic model that realizes all types Φ(X) corresponding
to the types Φ(A) in Tf0 .
Remark 4.3.6. (1) The conditions A ∈ Kf0 and kp ≤ |T (A)| =
n < kp+1 do not imply that y∆(A) infp
y∆p (A) ≥ bp
n. Nevertheless,y∆(A) cannot be much less than bp
n: since y1p(A) − y1(A) < εp+1
then y∆(A) > bpn − εp+1.
Moreover, in view of the inequality (4.1), for a cc-graph A ∈ Kf0
with |T (A)| ≥ max2, kp, we have y∆(A) > p.(2) Suppose that A is a cc-graph of Kf
0 , |T ∗(A)| = p, M isa graph in Kf , A ⊆cc M. Then A 6cc M i, for any cc-graphs A′ vA, B′ v M, A′ = A0 ⊆cc A1 ⊆cc . . . ⊆cc Am = B′, the inequality∆p
1 + . . . + ∆pm < 0, where ∆p
i = yp(Tc((Ai+1)T ∗c (Ai)))− yp(T ∗c (Ai)),implies B′ ⊆ A.
Indeed, if |Tc((B′)T ∗c (A′))| < kp+1 then ∆p1 + . . . + ∆p
m < 0is equivalent to ∆1 + . . . + ∆m < 0 by Lemma 4.3.5(2), and if|Tc((B′)T ∗c (A′))| ≥ kp then ∆p
1 + . . . + ∆pm ≥ ∆1 + . . . + ∆m >
p− yp(T ∗c (A′)) ≥ 0.Moreover, in order to verify that cc-graph A is self-sucient
in M, it suces to choose nA = kp and check the validity of therelations B′ ⊆ A only for cc-graphs A′ v A and B′ with A′ ⊆cc B′,B′ v B ⊆cc M, ∆p
1 + . . . + ∆pm < 0, |T ∗(B′)| < nA.
Thus, the condition A 6cc M is denable by a formula thatdescribes the absence of ni new, w.r.t. T ∗(Ai−1) (where A0 ⊆ccA1 ⊆cc . . . ⊆cc Am vM, A0 v A), vertices of Am ∪Υ(Am), wheren1+. . .+nm < nA, such that the set of these vertices is not containedin A ∪Υ(A) and
n < α1 · e1 + α2 · (e2 − e′2) + . . . + αp · (ep − e′p), (4.9)
where p = |T ∗(A)|; e1 is the number of new arcs; es, s > 1, is
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the number of new pairs of vertices (a, b) that are connected onlyby external shortest routes of length s, and such that there are noshortest (a, b)-routes containing Am-external Am-compulsory furca-tions; e′s is the number of pairs of vertices (a, b) used in calculationof ∆i and that stop to be linked only by external shortest routes oflength s in extended, w.r.t. Ai, cc-subgraph of the cc-graph Am.
Below, we suppose that the record WA of each cc-graph A is ex-tended by an information on (im)possibility of an extension for A,by less than nA new vertices, to obtain the value y(·) that is non-decreasing for extensions. Thus, any type, describing a place of Aw.r.t. external elements, is expanded by an information on minimalself-sucient extension of A, including an information on compul-sory furcations.
The following lemma is a direct consequence of the denition:Lemma 4.3.7. (1) If A 6cc B then A ⊆cc B.(2) If A 6cc C, B ∈ Kf
0 , and A ⊆cc B ⊆cc C, then A 6cc B.(3) The empty graph ∅ is the least element of the structure
(Kf0 ; 6cc).Lemma 4.3.8. If A,B, C ∈ Kf
0 , A 6cc B, and C ⊆cc B, thenA ∩ C 6cc C.
P r o o f. Assume the contrary. Then there exist n1, . . . , nm newelements in Υ(C) such that
n1 + . . . + nm < α1 · e1 + α2 · (e2 − e′2) + . . . + αp · (ep − e′p),
where p = |T ∗(A)|; es and e′s are values as in (4.9). Since all newelements belong to Υ(B), they violate the condition A 6cc B. ¤
Lemma 4.3.9. The relation 6cc is a partial order on Kf0 .
P r o o f. The re exivity and antisymmetry of 6cc are obvious.We claim that 6cc is transitive. Assume the contrary and con-
sider cc-graphs A,B, C ∈ Kf0 such that A 6cc B, B 6cc C and
A 66cc C. By assumption, there exists a part A′ of A and its ex-tension C′, being a part of C and not being a part of B, such thatA′ = A0 ⊆cc A1 ⊆cc . . . ⊆cc Am = C′ and ∆1 + . . . + ∆m < 0. SinceA′ is a part of B, the last inequality contradicts to the self-suciencyof B in C. ¤
Lemma 4.3.10. For any cc-subgraph A of graph M in Kf ,there exists the least cc-graph A that contains A is a self-sucientcc-subgraph of M. Moreover, |A| < k|T ∗(A)|, each vertex in T (A) \Υ(A) has degree not less than two and belongs to J0 ∪ J1.
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P r o o f. An existence of self-sucient extensions of the cc-graph A in the graphM is implied by denition of Kf , namely, frominequalities y∆
p (A′) ≥ bpn, where A′ are parts of A with universes A.
The same inequalities implies that there exists an upper cardinal-ity estimate k|T ∗(A)| for minimal self-sucient extensions of A. ByLemma 4.3.8, an intersection of any two such self-sucient exten-sions is also a self-sucient cc-subgraph of M containing A. Thismeans, that there exists the least self-sucient extension of A, andit is equal to the self-sucient closure A.
Each vertex in T (A)\Υ(A) has degree not less than two, since theself-sucient extension A is minimal. The condition T (A)\Υ(A) ⊆J0 ∪ J1 is implied by the fact that, removing an intermediate vertexa of degree 2, the value of the prerank function becomes less onpositive quantity 1−αs + αs1 + αs2 , where s = s1 + s2 is the lengthof external (when a is removed) shortest route containing a. Thus,there are no new intermediate vertices that belong to a superset of Awith the least value y(·). ¤
By Lemma 4.3.10, we have a right, for each cc-graph A in Kf0 ,
to add an information on self-sucient closureA in a graphM∈Kf .Clearly, such addition is not unique (depend on choice of M) andgenerate, for each cc-graph, nitely many possibilities of c-isomor-phism types of self-sucient closures. This number is dened by pos-sibilities for distributions of lengths of shortest routes between lessthan k|T ∗(A)| elements, forming universes for c-isomorphism typesof self-sucient closures. If a cc-graph A coincides with its self-sucient closure, A is called a self-sucient cc-graph.
Below, we suppose that the record W of each cc-graph containsan information on its self-sucient closure, and this information isin accordance with extensions and restrictions of cc-graphs.
An injective map f : A → B is called cc-embedding of cc-graph A = 〈A,QA,WA〉 into a cc-graph B = 〈B, QB,WB〉 (writtenf : A →cc B) if f is an embedding of the c-graph, that formedfrom A by restriction of the record WA to c-graph one, into the c-graph, that formed from B by restriction of the record WB to c-graphone, and such that the restriction of WB to f(A) coincides with thesubstitution to WA of elements in f(A) instead of correspondentelements of A.
Two cc-graphs, A and B, are called cc-isomorphic if there existsa cc-embedding f : A →cc B with f(A) = B. Furthermore, the
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map f is called a cc-isomorphism between A and B, and cc-graphsA and B are cc-isomorphic copies.
An injection f : A → N is called a cc-embedding of cc-graph Ainto a digraph N (written f : A →cc N ) if f is a cc-embedding of Ainto the cc-subgraph f(A) of N with the universe f(A).
A cc-embedding f of cc-graph A into a cc-graph B is strong iff(A) 6cc B.
Let A, B = 〈B,QB,WB〉, and C = 〈C,QC ,WC〉 be self-sucientcc-graphs, and A = B ∩ C. The free cc-amalgam of cc-graphs Band C over A (written B ∗A C) is the self-sucient cc-graph 〈B ∪C, QB∪QC ,WB∪WC ∪W 〉, where W is the record on self-suciencyof resulted cc-graph.
Notice, that, by denition, any free cc-amalgam is cf -closed.
Lemma 4.3.11. (amalgamation lemma). The class Kf0 has the
cc-amalgamation property cc-(AP), i.e., for any strong cc-embeddingsf0 : A →cc B and g0 : A →cc C, where A,B, and C are self-sucientcc-graphs in Kf
0 , there exist a self-sucient cc-graph D ∈ Kf0 and
strong cc-embeddings f1 : B →cc D and g1 : C →cc D such thatf0 f1 = g0 g1.
P r o o f. Without loss of generality, we may assume that A 6cc
B, A 6cc C, A = B ∩ C. We claim that cc-graph D B ∗A C is asrequired. To this end, in view of the symmetry of the denition offree cc-amalgam, it suces to show that B 6cc D and D ∈ Kf
0 .Assume that B 66cc D. Then there exist some n1, . . . , nm new
elements in Υ(D), not all in Υ(B), such that
n1 + . . . + nm < α1 · e1 + α2 · (e2 − e′2) + . . . + αp · (ep − e′p),
where p = |T ∗(B)|; es and e′s are values as in (4.9). Since elements,that contradict to self-suciency of B in D, belong to Υ(C), theseelements, for an extension of A by elements, via which elementsof B \ A are linked with new elements in C, violate the conditionA 6cc C in view of (∗).
Indeed, if routes, connecting elements in B with elements inC \ A, contain compulsory furcations, then correspondent links arenot used in the calculation of ∆i. So, w.l.o.g., shortest routes S,connecting a ∈ B \ A and b ∈ C \ A, doesn't contain compulsoryfurcations. Suppose now, that some value ∆1 + . . .+∆m is negativeafter adding of elements of C \A to some B′ ⊆ B.
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By denition of free amalgam, for any pair (a, a′) ∈ B×(C\A) ofvertices, connected only by external shortest routes over B′∪(C\A),one of the following conditions holds:
(1) a belongs to A;(2) a doesn't belong to A, and some external shortest (a, a′)- or
(a′, a)-route over B′ ∪ (C \A) contains an intermediate vertex in A.Denote by A′ the set of all intermediate vertices of A described
in (2). After adding the set A′ and correspondent links to B′, thenegative part
∞∑k=1
αk·ek of value ∆1+. . .+∆m increases in view of (∗),and positive part is the same, since adding these minimal links withelements in C \A to parts in D, we preserve non-furcation vertices.Then for the set (B′ ∩ A) ∪ A′, some value ∆1 + . . . + ∆m is alsonegative after adding of elements in C \ A. The last contradicts tothe self-suciency of A in C.
Now, we claim that any cc-graph D belongs to Kf0 . It means
that y∆p (E) ≥ bp
n, where E v D, n is a natural number, for whichy∆
p (E) = n− s · α1 with some s ∈ ω, kp ≤ n < kp+1.Denote by E1 the cc-graph E ¹ B, by E2 the cc-graph E ¹ C, and
by E3 the cc-graph E ¹ A. Suppose for deniteness, that for valuesy∆
r (E1) = l1 − s1 · α1 and y∆q (E2) = l2 − s2 · α1, kr ≤ l1 < kr+1,
kq ≤ l2 < kq+1, correspondent to value y∆p (E), the inequality l1 ≤
l2 holds, and w.l.o.g., E1 * A and E2 * A. Notice, that values∆1 + . . .+∆m are not negative (i.e., are positive) after adding to E2
of elements in E1\E3. By Lemma 4.3.5(2), the correspondent values∆q
1 + . . . + ∆qm are also positive. By Lemma 4.1.1(3), we conclude
that
sl((l2, y∆q (E2)), (n, y∆
q (E))) ≥ sl((l2, bql2), (n, bq
n)). (4.10)
Since E2 belongs to Kf0 , the point (l2, y∆
q (E2)) is above the point(l2, b
ql2). Then, by inequality (4.10), we have that the point (n, y∆
q (E))is above the point (n, bq
n), i.e., y∆q (E) ≥ bq
n.If p = q, then the required inequality y∆
p (E) ≥ bpn holds. Oth-
erwise, we have q < p and then y∆p (E) diers from y∆
q (E) by lessthan αq+1 · kq+1 · (kq+1 − 1) < 2εq+1, since E contains less thankq+1 · (kq+1 − 1) arcs and records on shortest Q-routes. By Lemma4.1.1(4), it follows that y∆
p (E) ≥ bpn. ¤
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Lemmas 4.3.74.3.11 imply
Corollary 4.3.12. The class (Tf0 ; 6′
cc) is self-sucient.
Denote the (Tf0 ;6′
cc)-generic theory by T f .We claim that, upon adding a formula χΦ(A) such that (T f , A) `
χΦ(A) to every type Φ(A) ∈ Tf0 , we obtain a self-sucient class
(T0;6′′cc) having the uniform t-amalgamation property.
Indeed, in view of Remark 4.3.6, for every cc-graph A ∈ Kf0 of
cardinality p and every graph M |= T f , A ⊆cc M, the conditionA 6cc M is equivalent to inequalities ∆p
1+ . . .+∆pm > 0 after exten-
sions of subsets of A by elements such that not all of them belongto A. Since we have to check these conditions only for extensionsof cardinalities bounded by nA7 and it is sucient to count linksonly by relations Q1, . . . , Qp, then the property of self-suciencyA 6cc M is denable by a formula χA(X) of the graph languageQ, where the set X of variables is bijective with the set A.
Let A and B be cc-graphs in Kf0 , M be a generic model of T f ,
and A 6cc B 6cc M. Denote by ψA,s(X) (respectively, ψB,s(X, Y ))a formula that describes the Q1, . . . , Qs-type of A (of B), whereX and Y are disjoint sets of variables bijective with A and B \ A.Then, for every s ≥ |T (B)|, the following formula is true in themodel M:
∀X ((χA(X) ∧ ψA,s(X)) → ∃Y (χB(X, Y ) ∧ ψB,s(X, Y ))) .
The last relation implies the uniform t-amalgamation property forthe class (Tf
0 ; 6′′cc) that we obtain from (Tf
0 ;6′cc) by adding the
formulas to the types which establish the cardinality bounds andthe Q1, . . . , Qp-structures of the self-sucient closures, as well asthe formulas χA(A) to the types of self-sucient sets A.
Since, by Lemma 4.3.10, the theory T f has the nite closureproperty, then, in view of Theorem 2.5.1, the following holds:
Theorem 4.3.13. A (Tf0 ; 6′′
cc)-generic model M is saturated.In addition, each nite set A ⊆ M can be extended to its self-sucient closure A ⊆ M , and the type tpX(A) is deduced from theset [Φ(A)]AX , where Φ(A) is the type in Tf
0 such that M |= Φ(A).7The upper estimate for cardinalities of extensions depends only on doubled
cardinality of A.
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Let N be an ω-saturated model of T f .Proposition 4.3.14. For each nite set A in N , we have
acl(A) = A.P r o o f. The inclusion acl(A) ⊇ A follows from the uniqueness
of A in any elementary extension of N .We now take an arbitrary element b ∈ N \ A and show that
b 6∈ acl(A). Denote by B some self-sucient cc-subgraph of N suchthat A∪ b ⊆ B and, for each vertex in Bf ∩A, a vertex of degree1 is added to A so that, for extended self-sucient cc-graph A′, theconditions A′f = Bf ∩ A and b 6∈ A′ hold. In view of constructionof generic model, there exist innitely many cc-isomorphic pairwisedisjoint copies of B\A′ over A′, i.e., b 6∈ acl(A). Thus, acl(A) ⊆ A. ¤
4. Stability of generic theory. The scheme of proof for stabil-ity of T f will be rather dierent from the proof of stability of generictheories in Sections 4.1 and 4.2, since the specicity of generic con-struction, represented in previous Subsection, generates the struc-ture with properties that don't allow to use standard arguments asin the work by J. T. Baldwin and N. Shi [37]. In particular, wedon't dene analogues of rank function and don't use them as a toolfor the proof (these analogues, for instance, don't have the mono-tonicity property), and lead a direct analysis of saturated model,allowing to state countable separability (countable baseness) of -nite sets from given closed sets and, as corollary, to nd an upperestimate for number of types over a set that guarantees the stabilityof T f .
Let N be a suciently saturated model of T f . Below, all con-sidered cc-graphs A are supposed to be parts of cc-subgraphs in N ,and all considered sets are subsets of N . Furthermore, self-sucient(in N ) cc-graphs A are denoted by their universes A, and the uni-verses A are called self-sucient sets.
A set X ⊆ N is called closed (in N ) (denoted by X 6 N) if, forany nite set A ⊆ X, an inclusion A ⊆ X holds, i.e., in T (N)\Υ(X),there are no any n new elements for which the inequality (4.9) istrue.
Lemma 4.3.15. Let X and Y be sets in N . Then the followingholds:
(1) if X 6 N and Y 6 N then X ∩ Y 6 N ;
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(2) there exists the least closed set X ⊇ X; moreover, X =⋃A |
A ⊆fin X, and X = acl(X);(3) if X ⊂ Y then X ⊆ Y .The set X is called the intrinsic closure of X (in N ).P r o o f repeats the proof of Lemma 4.1.13. ¤A set V (in N ) is called a free cc-amalgam of closed sets X and
Y over a set Z (written X ∗Z Y ) if X ∪ Y = V , X ∩ Y = Z andthere are no pairs (a, b) ∈ V 2 \ (X2 ∪ Y 2) of vertices connected byarcs or by only external shortest routes over V .
We say that sets X and Y are independent over Z and writeX ↓Z Y if X ′∩Y ′ = Z, X ′∪Y ′ = X ′ ∗Z Y ′, and X ′∪Y ′ 6 N , whereX ′ = X ∪ Z and Y ′ = Y ∪ Z.
Lemma 4.3.16. If X is a self-sucient set, Y is a closed set,Z = (X ∪ Z) ∩ Y and X ↓Z Y , then the type tp(X/Y ) is uniquelydetermined by the type tp(X/Z), a description of (X ∪ Z)∪Y to beclosed, and a type, describing the coincidence of (X ∪ Z) ∪ Y withthe free cc-amalgam X ∗Z Y .
P r o o f. By the construction of generic model, clearly, takingtwo complete types, q1(X) and q2(X), over Y such that these typescontain tp(X/Z), a description of (X ∪ Z) ∪ Y to be closed, anda type, describing the coincidence of (X ∪ Z) ∪ Y with the freeamalgam X ∗Z Y , there exists an automorphism of |Y |+-saturatedmodel, mapping a realization of q1(X) to a realization of q2(X). ¤
Lemma 4.3.17. If X is a closed set and a is an element in Nnot belonging to X then there exists at most countable closed subsetX ′ ⊆ X such that a ↓X′ X.
P r o o f. Notice, that the element a is connected by arcs oronly external shortest routes over X with at most countably manyelements of X. Indeed, assuming that there are uncountably manysuch elements, there exist uncountably many elements in X, con-nected with a by arcs or by only external shortest routes of the samelength, and moreover, a is a common origin or a common endpointof these routes. But an exceeding of number of routes of a certainlength over their correspondent weight αk means that a belongs tothe closure of chosen elements in X. It is impossible, since X isclosed and a 6∈ X. Aforesaid arguments also show that each at mostcountable set, that disjoint with X, also has at most countably manylinks with X by arcs or external shortest routes.
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Assume, that the required set X ′ doesn't exist. Then there is anuncountable set of pairwise dierent nite closed subsets Xi ⊆ X,for which self-sucient closures Xi ∪ a don't contained in X∪aand some elements in Xi ∪ a\X are connected with Xi by pairwisedierent (w.r.t. i) arcs or only external shortest routes. Moreover,w.l.o.g. sets Xi ∪ a form cc-isomorphic cc-graphs, for which allcorrespondent (w.r.t. i) elements either coincide (xed) or pairwisedierent (mobile). By denition of self-sucient closure, the sets Xi
have minimal, by inclusion, subsets X ′i, for which some extensions of
X ′i ∪ a in Xi ∪ a have exceedings of numbers of new links over
numbers of new elements in accordance with the inequality (4.9),i.e., some sums ∆1 + . . . + ∆m, obtained via extensions of X ′
i ∪ aby elements, not lying in X, are negative. Again, since the familyof sets X ′
i is uncountable, we may assume, that the sets X ′i ∪ a
are pairwise dierent and form cc-isomorphic cc-graphs. Thus, thesums ∆ ∆1 + . . . + ∆m coincide for all X ′
i.If sums ∆ can be formed by xed elements a1, . . . , ak in Xi ∪ a\
X only, then, in view of Xi ∪ a being cc-isomorphic copies, we canchoose n sets X ′
i with an exceeding of weighted number of links be-tween elements in X ′
i and xed elements more than 2 · k + 1. Thischoice guarantees that the elements a1, . . . , ak and a belong to theself-sucient closure of the union of these sets Xi. It contradictsthe condition a 6∈ X.
If negative sums ∆ are produced only by adding of correspondentmobile elements of Xi ∪ a \ X, then we chose n sets Xi, wheren · α > 2, α is the least weight of shortest routes that used incalculation of ∆ for mobile vertices. Adding xed elements bij ofXi ∪ a \ X to the union of sets Xi and mobile elements cik ofXi ∪ a \X, connecting elements of Xi ∪ a correspondent to thesum ∆ by arcs or only external shortest routes, and not connectingmobile elements of dierent sets Xi ∪ a \X by arcs and externalshortest routes, we obtain a cc-graph (being a free amalgam over aset of some elements of X and xed elements including a), which hasa negative sum ∆′
1 + . . .+∆′m′ after adding to the union of chosen n
sets Xi elements bij , cik, and a. The last again contradicts the factthat a doesn't belong to the closed set X.
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Thus, aforesaid uncountable family of nite sets Xi doesn't ex-ist, and a required set X ′ can be chosen as the intersection of theset X and the closure Y of the at most countable union of at mostcountable sets Xi containing the element a, and also all possible ele-ments in X, for which there are connections with elements in Y \Xby arcs or only external shortest routes. ¤
Theorem 4.3.18. The theory T f is stable, small and has theunique 1-type. The weight of this type is innite.
P r o o f. Theorem 4.3.13 implies that T f is small. In order toprove the stability of T f , we estimate the number of 1-types in S(N),where N is a model of T f . Consider an arbitrary element a. ByLemma 4.3.17, there exists at most countable closed set X ⊆ N suchthat a ∪X ∩N = X and a ↓X N . By Lemma 4.3.16, the typetp(a/N) is determined by tp(a/X), a description of a ∪X ∪N tobe closed, and a type describing that a ∪X ∪ N coincides withthe free amalgam a ∪X ∗X N . Therefore, in order to count thenumber of types in S(N), it is sucient to count the number oftypes in S(X) and the number of ways of choosing the countableset X in N . Then
|S(N)| ≤ 2ω · |N |ω = |N |ω.
Consequently, T f is stable.By construction, every singleton a is self-sucient. In view
of Theorem 4.3.13, this means that there exist the unique 1-typeof T f . Let A be the cc-graph with the universe a, Ap = a, bpbe an universe of two-element self-sucient cc-graph Ap contain-ing the Qp-arc between a and bp. Put B1 A1, Bp+1 Bp ∗AAp+1, B
⋃p∈ω\0
Bp. By construction, B forms a closed set in
some generic model M of T f . Since α1 + . . . + αp+1 < 1 andb1, . . . , bp+1 6cc a, b1, . . . , bp+1 6cc B 6 M , we have bp+1 ↓∅b1, . . . , bp. Therefore, (bp)p∈ω\0 is an innite independent se-quence of elements, where each bp is dependent on a. ¤
Since α1 < 12 , any two elements, a and b, are connected via an
element c so that aQc and bQc, and also via an element d so thatdQa and dQb. Thus, any model of T f has the pairwise intersectionproperty.
The uniqueness of 1-type implies the transitivity of automor-phism group for any homogeneous model of T f .
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Since any two-element set a, b, where aQb, is self-sucient,then, by Theorem 4.3.13, the formula Q(x, y) is principal in T f .
Since, by Proposition 4.3.14, acl(a) = a for any element ain a model M of T f , we have
acl(a) ∩⋃n∈ω
Qn(M, a) = a.
Comparing aforesaid properties with the denition of powerfuldigraph and using Theorem 4.3.18, we have the following:
Theorem 4.3.19. Any generic model of Tf is a small, stable,powerful digraph.
The following two assertions clarify the structure of the primemodels MA over nite sets A.
Lemma 4.3.20. If A and B are self-sucient sets in N andA 6cc B then the type tp(B/A) is isolated i B is a complete
∞⋃k=1
Qk-
graph over A, i.e., any two elements, a ∈ B and b ∈ B \ A, areconnected by a Qk-arc.
P r o o f. Let Y be a set of variables bijective with B \A. If B isa complete
∞⋃k=1
Qk-graph over A, then Theorem 4.3.13 implies that
the type tp((B \A)/A) is isolated by a principal formula of Φ(A, Y ),where Φ(B) is a type in Tf
0 such that N |= Φ(B). This formulaexists in view of formula denability of self-suciency and the factthat records on existence of shortest routes between elements arenite. If B is not complete
∞⋃k=1
Qk-graph over A then Theorem
4.3.13 implies that tp((B \ A)/A) is isolated by the type Φ(A, Y ),but is isolated by no nite part of this set. ¤
Lemma 4.3.20 impliesCorollary 4.3.21. Let A be a self-sucient set in N . The
model MA is a complete∞⋃
k=1
Qk-graph over A. The set of isomor-phism types of prime models over nite sets coincides with the setof isomorphism types of the models MA, where A is a self-sucientset .
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5. Powerful digraph with almost inessential ordered col-orings. The aforesaid construction of stable generic powerful di-graphs admits the following modication. Consider the class Kf
0
and expand each cc-graph A in Kf0 by all possible colorings Col :
A → ω ∪ ∞ such that if vertices a, a′ ∈ A are linked by (a, a′)-route then Col(a) ≤ Col(a′). We denote the expanded class by Kf
0 .Notice, that countably many possibilities of colorings of vertices cor-respond to each nonempty cc-graph.
Since cc-graphs are contained in acyclic digraphs, the free amal-gams of self-sucient cc-graphs of Kf
0 are also self-sucient cc-graphs in Kf
0 , and by Lemma 4.3.11, the following holds:
Lemma 4.3.22. (amalgamation lemma). The class Kf0 has the
cc-amalgamation property cc-(AP), i.e., for any strong cc-embeddings f0 : A →cc B and g0 : A →cc C, where A,B, and C areself-sucient cc-graphs in Kf
0 , there exist a self-sucient cc-graphD ∈ Kf
0 and strong cc-embeddings f1 : B →cc D and g1 : C →cc Dsuch that f0 f1 = g0 g1.
Therefore, the generic construction produces a generic stablesaturated powerful digraph Γcpg, in which the coloring of elementsis Q-ordered. Moreover, the type of any self-sucient set A is de-ned by a formula describing its self-suciency, a set of formulasdescribing existence or absence of routes between elements. Sincefor graph restrictions, a preservation of information on interconnec-tions of elements of self-sucient sets means that complete typescoincide, then the coloring of Γcpg is almost inessential.
Thus, the following holds:
Theorem 4.3.23. There exists a small, stable, generic, power-ful digraph Γ = 〈X,Q〉 with an almost inessential Q-ordered color-ing.
Denote the theory of generic powerful digraph Γcpg with thealmost inessential ordered coloring by T cpg.
Combining the proofs of Theorem 3.1.7 and Lemma 4.3.20, weobtain the following theorem, describing the types realized in a primemodel of T cpg, and in a prime model Mp∞ over a realization of typep∞(x) of innite elements in color.
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Theorem 4.3.24. (1) A type q of T cpg is principal i any twodierent elements, ai and aj, of any realization a of q are linked bysome (ai, aj)- or (aj , ai)-route, and all elements of realizations of qhave nite colors.
(2) A type q of T cpg is realized in Mp∞ i, for any realizationa of q, any two dierent elements, ai and aj, are linked by some(ai, aj)- or (aj , ai)-route and the following condition holds: if someelements of a are nite in color, af is an element of nite color,being a common endpoint of routes connecting all elements of nitecolors with af , and some elements of a are innite in color, a∞ isan element of innite color, being a common origin of routes, con-necting all elements of innite color with a∞, then there exists an(af , a∞)-route.
§ 4.4. On expansions of powerful digraphsIn this Section, we investigate a possibility of expansion of struc-
ture of a stable powerful graph to a structure of stable Ehrenfeuchttheory.
We claim, that the simplest form of expansion, proposed in Sec-tion 3 (an expansion by 1-inessential ordered coloring and locallygraph ∃-denable multiplace relations guaranteing the realization-equivalence of all nonprincipal types), can not preserve the structurein the class of stable structures: a presence on 1-inessential orderedcoloring with locally graph ∃-denable relations, that guaranteingthe realization-equivalence of all nonprincipal types, implies an ex-istence of formula in two free variables and with the strict orderproperty.
1. Type unstable theories. The following concepts generalizethe correspondent notions of Classication Theory [3], [22].
Let q(x, y) be some (not necessary complete) type of theory T ,where x and y be disjoint tuples of variables; M be a countablysaturated model of T .
The type q(x, y) is called unstable or having the order propertyif there exist tuples an, bn, n ∈ ω, for which |= q(ai, bj) ⇔ i ≤ j.
We say that the type q(x, y) has the independence property ifthere exist tuples an, n ∈ ω, such that, for any set w ⊆ ω, thereexist a tuple bw for which |= q(an, bw) ⇔ n ∈ w.
The type q(x, y) has the strict order property if there exist tuplesan, n ∈ ω, such that q(an,M) % q(an+1,M) for every n ∈ ω.
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The theory T is called type stable if T doesn't have unstabletypes q(x, y). The theory T is type unstable or has the type orderproperty, the type independence property, or the type strict orderproperty if some type q(x, y) of T has the correspondent property.
Clearly, the order property, the independence property, and thestrict order property for types generalize correspondent concepts forformulas (see [3], p. 342), and the type order property is implied bythe type independence property and the type strict order property.
Let Γ = 〈X, Q〉 be a graph without loops, a be a vertex in Γ. Theset 5Q(a)
⋃n∈ω
Qn(a, Γ) (respectively, 4Q(a) ⋃
n∈ωQn(Γ, a)) is
called the upper (lower) Q-cone of a. We call the Q-cones 5Q(a)and 4Q(a) by cones and denote by 5(a) and 4(a), respectively, ifa relation Q is xed.
We have the following criterium for an inclusion of cones in an-other cones.
Proposition 4.4.1. Fore any vertices a and b of graph Γ, thefollowing conditions are equivalent:
(1) 5(a) ⊇ 5(b);(2) 4(a) ⊆ 4(b);(3) a = b or the vertex a is reachable from b.
Now, we obtain two immediate corollaries.Corollary 4.4.2. For any sequence an, n ∈ ω, of vertices in
a graph Γ, the following conditions are equivalent:(1) the upper cones 5(an) form an innite sequence properly
increasing (decreasing) by inclusion;(2) the lower cones 4(an) form an innite sequence properly
decreasing (increasing) by inclusion;(3) for any n ∈ ω, the vertex an is reachable from (reaching to)
an+1, and an+1 is not reachable from (reaching to) an.Corollary 4.4.3. For any sequence an, n ∈ ω, of vertices in an
acyclic digraph Γ, the following conditions are equivalent:(1) the upper cones 5(an) form an innite sequence properly
increasing (decreasing) by inclusion;(2) the lower cones 4(an) form an innite sequence properly
decreasing (increasing) by inclusion;(3) for any n ∈ ω, the vertex an is reachable from (reaching to)
an+1.
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Clearly, the relations x 6∈ 5(y) and x 6∈ 4(y) are type denable.Thus, the following statement holds.
Proposition 4.4.4. If Γ is an acyclic digraph with an innitechain then the theory Th(Γ) has the type strict order property.
Since any powerful digraph is acyclic and any vertex has imagesand preimages, then we obtain the following:
Corollary 4.4.5. For any powerful digraph Γ, the theory Th(Γ)has the type strict order property.
2. From the type to the formula strict order property.Let Γ = 〈X, Q〉 be a graph with a transitive automorphism group,Col : X → ω ∪ ∞ be an 1-inessential coloring of Γ, ∆ be a setof formulas in the language Q ∪ Coln | n ∈ ω. A relationalexpansion M of the structure of colored graph 〈Γ, Col〉 is calledlocally (∆, 1,Col)-denable if, in expanded structure, the coloringCol is again 1-inessential and , for any new predicate symbol R(m),the formula R(x, y) ∧ Coln(x) is equivalent to a Boolean combina-tion of formulas of ∆. If ∆ consists of ∃-formulas, then a locally(∆, 1, Col)-denable expansion is called locally (∃, 1,Col)-denable.
Let Col be an 1-inessential Q-ordered coloring of a powerfuldigraph Γpg = 〈X, Q〉, p∞(x) be a type of innite elements incolor. A locally (∃, 1, Col)-denable expansion M of the structure〈Γpg, Col〉 is called p∞-powerful if the following conditions hold:
(1) the semi-isolation relation SIp∞ on realizations of p∞ is non-symmetric and it is witnessed by the formula Q(x, y);
(2) p∞ is a powerful type of the theory T Th(M);(3) each nonprincipal type q(y) ∈ S(T ) is realized in Mp∞ via
some principal formula Rq(a, y) (i.e., Rq(a, y) ` q(y)), where Rq isa new language symbol, |= p∞(a);
(4) if |= Rq(a, b), |= p∞(a), |= p∞(bi), where bi ∈ b, then thereexists an (a, bi)-Q-route.
Notice, that the properties (1)(4) are satised in Morleyzationsof structures of formula neighbourhoods of nonprincipal powerfultypes, from which structures of powerful digraphs can be taken (seeproof of Proposition 1.4.2).
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Recall (see proof of Proposition 1.3.4) that if lengths of short-est routes are bounded in a powerful digraph Γpg then the theoryTh(Γpg) has the strict order property and, in particular, is unstable.
Let M be a locally (∃, 1, Col)-denable p∞-powerful expansionof a structure of powerful digraph Γpg = 〈X, Q〉 with unboundedlengths of shortest routes. Then there exists a type q(y1, y2, y3) ∈S3(Th(M)) for which the rst coordinate of any realization is nitein color, second and third coordinates are innite in colors, and theserealizations are not linked by routes in digraph. Consider a formula
ϕ(x, y1) ∃y2 ∃y3 Rq(x, y1, y2, y3).
Since the relation Rq is locally (∃, 1, Col)-denable, the realizationsa and b1 (where |= ϕ(a, b1), |= p∞(a)) are mutually non-reachablein Γpg, and the lengths of shortest routes are unbounded, thenthe formula ϕ(x, y1) denes a binary relation that is not den-able by a formula in the language of colored digraph. Moreover,since, for a graph structure with a transitive automorphism group,∃-formulas dened only bounded lengths of shortest routes, the con-ditions |= p∞(a1) and |= Q(a1, a2) imply
|= ∀y (ϕ(a1, y) → ϕ(a2, y)) ∧ ∃y (¬ϕ(a1, y) ∧ ϕ(a2, y)). (4.11)
Since elements a1 and a2 realizes the same type p∞, the relation(4.11) implies the strict order property.
Thus the following theorem holds, in accordance of which thepreservation of 1-inessentiality for coloring of powerful digraph isinconsistent with the stability of locally (∃, 1,Col)-denable struc-ture of expanded theory, having a nonprincipal powerful type p∞.
Theorem 4.4.6. For any locally (∃, 1,Col)-denable p∞-power-ful expansion M of a structure of powerful digraph, the theoryTh(M) has the strict order property.
A mechanism of obtaining of the strict order property via anexpansion of structure of powerful digraph, represented in the proofof Theorem 4.4.6, illustrates a transformation of the type strict orderproperty, for a theory of powerful digraph, to the formula strict orderproperty, generated by specicity of the expansion.
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§ 4.5. Description of features of generic constructionof stable Ehrenfeucht theories
In the following four Sections, we prove a strengthening of thebasic result of Chapter 3 (Theorem 3.4.1) stating the realizability inthe class of stable theories of all admissible parameters described inCharacterizing Theorem 1.1.13. This strengthening, in particular,represent a positive solution of the Lachlan problem on existence ofstable Ehrenfeucht theory.
Recall that, in Chapter 3, the construction of Ehrenfeucht theo-ries with three countable models is formed by the following compo-nents:
| the nonprincipal powerful type p∞(x) with the structure con-sisting of elements of color Col∞ and obtained by inessential orderedcoloring Col of all elements in colors belonging to ω ∪ ∞;
| the binary relation Q dening the powerful digraph with un-bounded lengths of shortest Q-routes on the structure of p∞(x) andon any its neighbourhood and, as corollary, dening partial orderson these structures via transitive closures TC(Q);
| the countable set of binary relations Rq and R′q, R′
q = R−1q ,
guaranteing the corealization amalgamation (the coinci-dence of prime models over realizations of p∞(x) if these realiza-tions are connected by Rq) and dening the equivalence relationsuch that the equivalence classes form connected components w.r.t.R∗
⋃q
Rq∪ R′q; moreover, the connected components, w.r.t. R∗,
are partially ordered by the relation TC(Q) (two components, C1
and C2, are connected by TC(Q) if some their representatives areconnected by TC(Q)), and elements of each connected componentare pairwise incomparable w.r.t. TC(Q) \ idX ;
| multiplace predicates RA guaranteing realization-equivalenceof p∞(x) with all nonprincipal types.
It is not dicult to see that aforesaid attributes, possibly withthe local pairwise intersection property w.r.t. Q or with degeneratedrelations R∗ and RA, are represented in any complete theory withthree countable models.
The strategy of construction of required stable theories will berather dierent from the strategy of construction of Ehrenfeuchttheories in Chapter 3, in view of the following circumstances.
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As shown in previous Section, using an 1-inessential coloringwith consequent expansion of structure of powerful digraph by co-ordinated predicates RA, realizing all nonprincipal types, we leadthe structure out the class of stable structures. Therefore, before theconstruction of RA-expansions, we expand the structure of powerfuldigraph with an almost inessential ordered coloring by pairwise dis-joint symmetric binary relations Pi,n, i, n ∈ ω, each of which is dis-joint with TC(Q) and connects only elements of color, that equals tothe second index, with elements of more colors, so that the following(P, Q)-intersection property: for any elements a1, . . . , ak of innitecolor, any elements b1, . . . , bl of nite colors m1, . . . , ml, respectively,and any color σ > maxm1, . . . ,ml, there exists an element c ofcolor σ, for which
|=k∧
j=1
Qij (c, aj) ∧l∧
j=1
Prj ,mj (c, bj)
holds with some i1, . . . , ik, r1, . . . , rl ∈ ω \ 0.Predicates Pi,n, being constructed in a stable generic structure
on a base of linear prerank function, allow, preserving the graphstructure and the stability of theory, to realize, in a prime modelMp∞ over a realization of p∞, all nonprincipal types via RA-expan-sions, for which
∃y \ yj RA(x, y) ≡ Qn(x, yj) ∧∧
m<n
¬Qm(x, yj)
or∃y \ yj RA(x, y) ≡ Pi,n(x, yj).
The control of balance between numbers of elements and num-bers of links, satised via a prerank function and guaranteing thestability of theory, will be fullled by a fusion of generic model ofpowerful graph and of generic model in the language P (2)
i,n | n ∈ ω.Various nature of closures of nite sets in R∗- and Q-structures
(route-closures for forests and self-sucient closures w.r.t. prerankfunctions for structures of generic powerful digraphs) has dicultyfor checking the satisfaction of the nite closure property in thefusions of these structures.
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To obtain a common nature of closures for nite sets in fusions ofR∗- and Q-structures (of self-sucient closures w.r.t. prerank func-tions, reducing to a common prerank function), we dene connectedcomponents w.r.t. R∗, being variants of Hrushovski construction fora stable non-superstable countably categorical structure (see [81]).We introduce symmetric binary relations Rj in these connected com-ponents, playing the role of relations Rq. The possibility of simul-taneous control of number of Rj-links, Pi,n-links, and of number oflinks w.r.t. Q-routes via a common prerank function allows to boundthe cardinality of self-sucient closure for any nite set by a com-mon function, dependent only on cardinality of that set. Thus, thetype-denability of self-sucient closures for a fusion of R∗-, Pi,n-,and Q-structures will be guaranteed, and the nite closure property,conditioned by a stepped special system of closures, will be satised.A presence of the nite closure property, in one's turn, will allowto state that the generic models are saturated and required theoriesare stable.
§ 4.6. Stable graph extensions of colored powerful di-graphs
Denote by Γ the stable generic powerful digraph Γcpg = 〈X, Q〉with the almost inessential Q-ordered coloring Col, described in Sec-tion 4.3.
Consider the class Kf0 , dened in Section 4.3 for the construc-
tion of colored graph Γ, and expand structures in Kf0 by symmetric
binary predicates Pi,n, i, n ∈ ω, connecting only vertices a of colorn with vertices b of colors more than n and such that there are no(a, b)-Q-routes.
Like operations Tc and T ∗c on the class of cc-graphs, we deneoperations Tc and T ∗c on the class of expanded structures. Further-more, obtaining correspondent structures without furcations, anypair of vertices of Pi,n is interpreted by one edge, connecting ele-ments in J0 ∪ J1.
Consider now a monotonically decreasing sequence of positivereal numbers αk, dened for the prerank function y1(·) in Section4.3 as weights of pairs of vertices, connected by shortest Q-routes oflength k. Put elements αQ
m α2m to weights of pair of vertices, con-nected by shortest Q-routes of length m, and elements αP
m α2m+1
to weights of Pi,n-edges, where m is the value c(i, n) of Cantor enu-
180
merating function (see, for instance, [26], p. 137). After aforesaidrenotations, we dene the prerank function y(·), for structures Afrom Kf
0 , expanded by Pi,n, with the rule:
y(A) = 2 · |Af |+ |Anf | −∞∑
k=1
αQk · eQ
k (A)−∞∑
k=0
αPk · eP
k (A),
where eQ1 (A) is the number of Q-arcs in A; eQ
k (A), for k ≥ 2, is thenumber of pairs (a, a′) ∈ A2, connected only by external shortest(a, a′)-Q-routes of length k and such that there are no (a, a′)-Q-routes of length k with A-external A-compulsory furcations; eP
k (A)is the number of Pi,n-edges in A, where k = c(i, n).
A p-approximation of the prerank function y(·) is the functionyp(·) assigning a real to every structureA from Kf
0 , expanded by Pi,n,with the rule:
yp(A) = 2 · |Af |+ |Anf | −p∑
k=1
αQk · eQ
k (A)−p∑
k=0
αPk · eP
k (A).
Denote by Kf,P1 the class of all structures A from Kf
0 , expandedby predicated Pi,n and satisfying conditions y1(A′) ≥ b1
n for anystructureA′ v cfc(A), where n = |T (A′)| and v the relation \to be apart of structure". For the structure A in Kf,P
1 , we write A ∈ Kf,Pp+1,
where p > 0, i A ∈ Kf,Pp and y∆
p (A′) ≥ bpn for any structure A′ v
cfc(A), where y∆p (A′) is a minimal value yp(Tc(A0))+∆1+ . . .+∆m,
∆i = yp(Tc((Ai+1)T ∗c (Ai))) − yp(T ∗c (Ai)), A0 ⊆cc A1 ⊆cc . . . ⊆cc
Am = A′, n is a natural number, for which y∆p (A′) = n− s ·α1 with
some s ∈ ω, kp ≤ n < kp+1.Put Kf,P
0 ∞⋂
p=1Kf,P
p and denote by Kf,P the class of all graphs
in the language ΣP Coln | n ∈ ω ∪ Q ∪ Pi,n | i, n ∈ ω,for which each nite substructure (i.e., a subgraph with an informa-tion on lengths of shortest Q-routes and on compulsory furcations)belongs to the class Kf,P
0 .Let A be a nite substructure of a graph (respectively, of a nite
substructure) M (of a graph) belonging to Kf,P . We say that A isa selfsucient substructure of the graph (respectively, of the nite
181
structure) M and write A 6 M if for any nite structures A′ v A,B′ v M, A′ = A0 ⊆cc A1 ⊆cc . . . ⊆cc Am = B′, the inequality∆1 + . . . + ∆m < 0, where ∆i = y(Tc((Ai+1)T ∗c (Ai))) − y(T ∗c (Ai)),implies B′ ⊆ A. If A 6 M and M is a nite structure, then A iscalled a strong substructure of M.
Denote by Tf,P0 the class of types correspondent to all structures
in Kf,P0 and equip that class by the relation 6′, where
Φ(A) 6′ Ψ(B) ⇔ A 6 B.Repeating arguments for the class (Tf
0 ;6′cc), we state that
(Tf,P0 ; 6′) is a self-sucient generic class, for which, after adding to
types of necessary formulas, describing self-sucient closures, theuniform t-amalgamation property is satised. This fact implies thata (Tf,P
0 ; 6′)-generic model is ω-saturated and realizes all types Φ(X)correspondent to types Φ(A) in Tf,P
0 , and also the (Tf,P0 ; 6′)-generic
theory T f,P is stable. Since αk < 12 and lim
k→∞αk = 0, the theory
T f,P has the (P,Q)-intersection property.Since any two-element set a, b with (a, b) ∈ ⋃
i,n∈ωPi,n are self-
sucient, the type tp(aˆb) is dened by colors of endpoints of theedge [a, b], i.e., all formulas Pi,n(a, x), where Col(a) > n, are prin-cipal. The same property is preserved for formulas Q(a, x), whereCol(a) = ∞. Thus, the following two theorems hold:
Theorem 4.6.1. A (Tf,P0 ; 6′)-generic model M is saturated.
In addition, each nite set A ⊆ M can be extended to its self-sucient closure A ⊆ M , and the type tpX(A) is deducible from theset [Φ(A)]AX , where Φ(A) is a type in Tf,P
0 , for which M |= Φ(A).Theorem 4.6.2. The theory T f,P is stable, small, possesses the
(P, Q)-intersection property and has countably many 1-types, andeach 1-type is dened by color of any its realization. Every formulaPi,n(a, x), where Col(a) > n, and also every formula Q(b, x), forCol(b) = ∞, are principal.
Lemma 4.6.3. If A and B are self-sucient sets in a modelN |= T f,P and A 6 B, then the type tp(B/A) is isolated i B isa complete
⋃k,n∈ω
(Qk ∪ Pk,n)-graph over A, i.e., any two dierent
elements, a ∈ B and b ∈ B \A, are connected by an Qk-arc or somePi,n-edge.
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P r o o f. Let Y be a set of variables bijective with B \ A. IfB is a complete
∞⋃k=1
(Qk ∪ Pk,n)-graph over A, then, by Theorem
4.6.1, the type tp((B \ A)/A) is isolated by a principal formula oftype Φ(A, Y ), where Φ(B) is a type in Tf,P
0 such that N |= Φ(B).This formula exists in view of type denability of self-suciency,by the condition for connections elements via Pi,n-edges, and bythe niteness of records on existence of shortest Q-routes betweenelements. If B is not complete
∞⋃k=1
(Qk ∪ Pk,n)-graph over A, then
Theorem 4.6.1 implies that the type tp((B \A)/A) is isolated by thetype Φ(A, Y ), but is isolated by no nite part of this set. ¤
Lemma 4.6.3 impliesCorollary 4.6.4. Let A be a self-sucient set in a model N |=
T f,P . A prime model MA over A is a complete⋃
k,n∈ω
(Qk ∪ Pk,n)-
graph over A. The set of isomorphism types of prime models overnite sets coincides with the set of isomorphism types of the mod-els MA, where A are self-sucient sets and MA are complete⋃k,n∈ω
(Qk ∪ Pk,n)-graphs over A.
§ 4.7. Stable theories with nonprincipal powerful typesIn this Section, we describe a modication of generic construc-
tion in Section 3.2, allowing to create stable theories with nonprin-cipal types as expansions of the theory T f,P of powerful digraph〈X, Q〉 with ordered coloring Col and binary predicates Pi,n, guar-anteing that the (P, Q)-intersection property holds.
Fix a theory T f,P 8 and dene an expansion of T f,P with thesame ordered coloring, so that the type p∞(x) of innite in colorelements becomes powerful.
Below, in order to turn p∞ into a powerful type preservingthe uniqueness of nonprincipal complete 1-type over ∅, for everytype q(y2, . . . , yk) not contained in any (p∞, y1)-principal typer(y1, y2, . . . , yk), we introduce a new k-ary predicate Rq so that the
8Recall, that T f,P is uniquely dened by a special sequence of irrationalnumbers αk, 0 < αk+1 ¿ αk < 1
2, k ∈ ω \ 0.
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set Rq(y1, . . . , yk) ∪ p∞(y1) is consistent and
Rq(y1, . . . , yk) ∪ p∞(y1) ` q(y2, . . . , yk).
With this goal in mind, we renumber the set q of all typesq(y1, . . . , yk) for tuples aq with mutually distinct coordinates forwhich the sets of elements of aq are self-sucient and models Maq
are not isomorphic to a prime model or to Mp∞ : q = qm(y1, . . . ,ykm) | m ∈ ω. In this event, by Theorem 4.6.1, qm(y1, . . . , ykm) aredened by isomorphism types Am Ψm(Ym) (where Ψm(Am) ∈Tf,P
0 ) of their realizations am. In what follows, therefore, the typesqm(y1, . . . , ykm) will be identied with the isomorphism types Am.
For any type qm(y) ∈ q and correspondent type Am, we x a setΦAm
(y) of formulas ϕn(y), n ∈ ω, isolating qm(y) and describingthe following:
(a) nite colors of elements in am, and also negations of colorsless than n, for elements in am having the innite color;
(b) an existence and lengths of shortest Q-routes connecting el-ements of am;
(c) an absence of Q-routes of lengths less than n, connectingelements of am if elements are not linked by Q-routes;
(d) Pi,m-edges connecting elements in am, and also an absence ofPj,n′-edges, with c(j, n′) < n, connecting elements in am if elementsare not connected by Pi,m-edges;
(e) the self-suciency of the set of elements am.Now, consider an isomorphism type A of an arbitrary tuple a =
(a1, . . . , ak) of self-sucient set A = a1, . . . , ak of cardinality k,which is not in Mp∞ . Denote by maxA the maximal nite colorof elements in A if such elements exist, and put maxA 0 if allelements of A are innite in color. Choose for each element aj ofnite color nj a predicate Pij ,nj such that
∑
j∈j′|Col(aj′ )<ωαP
c(ij ,nj)< 1− 2 · αQ
1 .9
9Such choice of predicates is admissible in view of the (P, Q)-intersectionproperty and the condition lim
i→∞αP
c(i,nj) = 0. We choose these elements topreserve the self-suciency under adding to A an element connected with el-ements aj of nite colors by Pij ,nj -edges and with elements aj of innite colorby Q-routes.
184
We dene (k + 1)-ary relations RA as follows:
(1) `(∃y (RA(x, y) ∧ ϕm(y)) ↔ ∧
i≤maxA¬Coli(x)
)∧
∧(∃y (RA(x, y) ∧ ϕn(y)) ↔ ∧
i≤n¬Coli(x)
), m ≤ maxA ≤ n;
(2) for any n > maxA, the formula RA(x, y1, . . . , yk) ∧ Coln(x)is equivalent to a conjunction of ϕn(y) ∧ ¬ϕn+1(y) and the formuladescribing the following properties:
(a) if 〈ai1 , . . . , air〉 (where k1 < . . . < kr) is a tuple of all ele-ments ai of innite color in a, and 〈aj1 , . . . , ajs〉 (where j1 < . . . < js)is a tuple of all elements aj of nite colors in a, then there exist ele-ments z0, . . . , zr−1 such that zr−1 = ykr , Q(zm−1, zm)∧Q(zm−1, yim),m = 1, . . . , r − 1, x = z0, and Pijl
,njl(x, yjl
), l = 1, . . . , s;(b) a structure consisting of elements x, y1, . . . , yk, z1, . . . , zr−1
contains no edges with numbers c(·, ·) ≤ n and Q-arcs other thanthe edges and Q-arcs specied in (a) and in the description of Afor the elements y; nor does it contain external shortest Q-routesof length at most n but for the external shortest Q-routes connect-ing elements of y and the elements described in A; moreover, theaforesaid structure forms a self-sucient set.10
By denition, predicates RA rene the graph structure whilenot increasing sets of binary relations dened by projections such as∃y1, . . . , yi−1, yi+1, . . . , yk RA(x, y).
We claim that with the above-mentioned renement of the graphstructure (via relations RA) in hand, the required expansion maybe realized by newly constructing the generic model from the -nite structures expanded by nite records saying of positive linksbetween elements via intermediate elements through projections ofthe relations RA.
We start our construction by describing the class K∗1 of nite
structures endowed with nite records holding of interrelations ofthe elements satisfying conditions (1) and (2). Since the desiredgeneric model expands a (Tf,P
0 ; 6′)generic model, we assume thatevery nite structure A, which enters K∗
1 and is restricted to thegraph language Q ∪ Pi,n | i, n ∈ ω with coloring Col, form
10This means, in particular, that types of sets A have unique extensionsto types including elements x, z1, . . . , zr−1, where x satises p∞.
185
a structure with the record WA and belonging to Kf,P0 . Moreover,
introduction of the relations RAmrequires that the record WA is
added positive information on interrelations of the elements w.r.t.projections ∃yl1 , . . . , ylt RAm
(x, y) in accordance with item (2).Before we end to dene structures in the class K∗
1, we observe thefollowing. As shown in Theorem 4.6.1, a type in Tf,P
0 of every self-sucient structure A determines type of the set A in generic model.In dening every relation RA, the belonging of every tuple a ˆ ato that relation is specied either by a principal formula describingrelations between the elements in prime model or by a sequence offormulas (see item (2)) which locally describe the absence of linksbetween some elements of a via edges or Q-routes, while keeping thelinks between the element a and the elements of a xed in length.
The last description, as noted above for binary relations, de-pends directly on the interrelations between colors of approxima-tions an (in prime model) of an element a (these approximationsare called sources) and lengths of shortest Q-routes and numbersof P -edges between appropriate elements of approximations an (inprime model) of a tuple a (these approximations are called succes-sors). Namely, if the color number of a source an does not exceed(unbounded as n → ∞) lengths of shortest Q-routes (numbers ofP -edges) between elements of successors an, then the relation RAholds, provided that the elements z1, . . . , zk described in item (2)are in hand. But if the color number of an is greater than is some(unbounded as n → ∞) length of shortest Q-routes (number ofP -edges) between elements of an, then RA, under the same con-ditions, will fail. Below, the relation, which the color number ofa source an associates with a collection of pairwise non-bounded(as n → ∞) lengths of shortest Q-routes and numbers of P -edgesbetween elements of a successor an", is for brevity referred to asCLN-correlation.
Since, in generic theory, any type of S(∅) is expandable to a typein S(∅) of some self-sucient set, described by colors of elements,lengths of shortest Q-routes, numbers of P -edges, and the conditionof self-suciency of set, the CLN correlation may be characterizedby formulas ρ(x, yi, yj), which express the lengths of shortest (x, yi)-and (x, yj)-Q-routes and numbers of P -edges between x and yi, yj ,as well as correlations between colors of elements an and lengths ofshortest Q-routes (numbers of P -edges) connecting an
i and anj .
186
Thus, the formulas ρ(x, yi, yj) are ternary indicators for positiveor negative entries of the formulas ∃yl1 , . . . , ylt RAm
(x, y) in typesof the theory for the generic model in question. We add these in-dicators to the descriptions of types Am in dening the relationsRAm
themselves (the formulas ρ(yi, yj , yk) or add their negationsconjunctively to ϕn(y)) and to the general descriptions of the typesof tuples. The adding of ρ, in this instance, extends the very lan-guage determined by the type Am with the formulas ρδ(yi, yj , yk),δ ∈ 0, 1, added. This extended language remains countable dueto there being nitely many versions of adding positive formulas ρto every type.
Ultimately we specify that the class K∗1 consists of all nite struc-
tures of the language
Σ1 Coln | n ∈ ω ∪ Q(2) ∪ P (2)i,n | i, n ∈ ω ∪ RAm
| m ∈ ω∪
∪ρ(3) | ρ(x, yi, yj) characterizes some CLN-correlation,which are obtained from structures belonging to Kf,P
0 by adding therelations ρ, and the relations RAm
consistent with item (2) (withthe types Am extended by ρ), and all possible admissible formulasρ(ai, aj , ak).
Finite structures A with their records WA forming the class K∗1
are called c1-structures. Denote by K1 the class of all models of thelanguage Σ1, whose every nite subset forms a c1-structure in K∗
1.The concept of a c1-embedding f : A →c1 B for c1-structures A
and B, under which an appropriate record WA (Wf(A) = WB ¹ f(A))is preserved, is a natural generalization of the concept of the cc-embedding. Thereby we also dene the concept of a c1-embeddingf : A →c1 N of a c1-structure A into a model N in K1.
Two c1-structures, A and B, are said to be c1-isomorphic if thereexists a c1-embedding f : A →c1 B with f(A) = B.
A self-suciency relation 61 in K∗1 inherits the relation 6 in
Kf,P0 , i.e., for any c1-structures A and B in K∗
1 the following holds:
(A 61 B) ⇔ (A ¹ ΣP 6 B ¹ ΣP ) .
Theorem 4.7.1. There exists a saturated (K∗1; 61)-generic
model M of a stable theory satisfying the following conditions:(a) if A and B are c1-isomorphic self-sucient c1-structures
in M, then tpM(A) = tpM(B);
187
(b) the restriction of M to the language ΣP is a Kf,P0 -generic
model;(c) the theory Th(M) has countably many 1-types, and each 1-
type is dened by color of any its realization; the type p∞(x) ofelements innite in color is the unique nonprincipal 1-type, and itsown weight is innite;
(d) every formula RA(a, y), where |= p∞(a), is principal, andthe c1-isomorphism type of any realization of the formula RA(a, y)coincides with the c1-isomorphism type A;
(e) every formula RA(x, a), where A is the c1-isomorphism typeof tuple a, is principal, and each realization of RA(x, a) is a real-ization of the type p∞.
P r o o f of existence of saturated (K∗1; 61)-generic model M
in K1, satisfying (a) and (b), almost word for word repeats the proofof Theorem 4.6.1. Furthermore, the presence of new predicates RAm
and ρ is not involved in count of values of the prerank function, sincethese predicates rene correspondent graph structures of Kf,P
0 .The proof of stability of generic theory Th(M) and of item (c)
is similar the proof of Theorem 4.6.2.Repeating the proof of items (d) and (e) in Theorem 3.2.3, we
prove correspondent items. ¤Denote by T1 the theory Th(M) of (K∗
1; 61)-generic model M.Since each type of T1 over the empty set is a subtype of type,
that dened by record of some self-sucient c1-structure, and, forany type q of c1-structure not in a prime model, there ex-ists a principal formula ∃yl1 , . . . , ylt RA(a, y) (where |= p∞(a)), forwhich ∃yl1 , . . . , ylt RA(x, y)(a, y) ` q, then all types of T1 are real-ized in the model Mp∞ . Thus, the type p∞(x) is powerful.
Theorem 4.7.1(e) implies that, for any nonprincipal type q(y)of T1, the type p∞ is realized in the model Mq. Therefore, everynonprincipal type is powerful. Moreover, the introduction of pred-icates RA allows, for every tuple a, having some c1-isomorphismtype Am, to nd a realization a of p∞(x) such that |= RAm
(a, a),and so, in view of Proposition 1.1.3, the model Ma coincides with amodel Ma. Thus, all prime model over tuples, realizing nonprinci-pal types, are isomorphic to Mp∞ , and the following theorem holds.
Theorem 4.7.2. There exists a small stable theory T1 expand-ing the theory T f,P and satisfying the condition |RK(T1)| = 2.
188
§ 4.8. Stable theories with three countable modelsGeneral principles for expanding T1, which lead to the construc-
tion of a small stable theory T with the condition |RK(T )| = 2 andthe property (CEP), coincides with the same principals, that de-scribed in Section 3.3, but have reservations re ected in Section 4.5.For construction of required stable theory T , we shall use corealiza-tion amalgams, that, in dierence of construction in Chapter 3, willbe obtained automatically, in view of self-suciency of two-elementgraphs, containing new edges, and by the fact that weights of newedges are unboundedly small. Furthermore, the theory T will be var-ied on dependence of alternating weights αQ
k for lengths of shortestQ-routes, weights αP
k for Pi,n-edges, and weights αRk of Rj-edges.
Denote by Kf,P,R−1 the class of all nite structures A (including
empty structure) of language
ΣP,R Col(1)n | n ∈ ω ∪ Q(2) ∪ P (2)
i,n | i, n ∈ ω ∪ R(2)j | j ∈ ω
satisfying the following conditions:(a) the structure A ¹ ΣP belongs to Kf,P
0 ;(b) relations Rj are symmetric, irre exive, pairwise disjoint and
connect only the same vertices in color; moreover, connected⋃
j∈ωRj-
components are strictly Q-ordered, i.e., elements of the same⋃
j∈ωRj-
component are not connected by Q-routes, and if vertices a1 and a2are linked by
⋃j∈ω
Rj-route, vertices b1 and b2 are linked by⋃
j∈ωRj-
route, and there exists an (a1, b1)-Q-route, then there are no (b2, a2)-Q-routes.
Now, take a monotonically decreasing sequence of positive realnumbers αk, dene for the prerank function y1(·) in Section 4.3 asweights of pairs of vertices, connected by shortest Q-routes of lengthk. Put elements αQ
m α3m to weights of pair of vertices, connectedby shortest Q-routes of length m, elements αP
m α3m+1 to weightsof Pi,n-edges, where m = c(i, n), and elements αR
m α3m+2 toweights of Rm-edges. After aforesaid renotations, we dene theprerank function y(·) for structures A in Kf,P,R
−1 , by the rule:
y(A) = 2·|Af |+|Anf |−∞∑
k=1
αQk ·eQ
k (A)−∞∑
k=0
αPk ·eP
k (A)−∞∑
k=0
αRk ·eR
k (A),
189
where eQ1 (A) is the number of Q-arcs in A; eQ
k (A), for k ≥ 2, is thenumber of pairs (a, a′) ∈ A2, connected only by external shortest(a, a′)-Q-routes of length k and such that there are no (a, a′)-Q-routes of length k with A-external A-compulsory furcations; eP
k (A)is the number of Pi,n-edges in A, where k = c(i, n); eR
k (A) is thenumber of Rk-edges in A.
A p-approximation of the prerank function y(·) is the functionyp(·) assigning a real to every structure A ∈ Kf,P,R
−1 by the rule:
yp(A) = 2·|Af |+|Anf |−p∑
k=1
αQk ·eQ
k (A)−∞∑
k=0
αPk ·eP
k (A)−p∑
k=0
αRk ·eR
k (A).
Analogous to operations Tc and T ∗c on the class of cc-graphs, wedene operations Tc and T ∗c on the class Kf,P,R
−1 , where each pair ofvertices belonging to Pi,n of Rm is interpretable in graphs withoutfurcations by one edge, connecting elements in J0 ∪ J1.
Denote by Kf,P,R1 the class of all structuresA in Kf,P,R
−1 such thaty1(A′) ≥ b1
n for any structure A′ v cfc(A), where n = |T (A′)|. For astructure A in Kf,P,R
1 , we write A ∈ Kf,P,Rp+1 , for p ≥ 1, i A ∈ Kf,P,R
p
and y∆p (A′) ≥ bp
n for any part A′ of structure cfc(A) ∈ Kf,P,R1 ,
where y∆p (A′) is a minimal value yp(Tc(A0)) + ∆1 + . . . + ∆m, ∆i =
yp(Tc((Ai+1)T ∗c (Ai)))− yp(T ∗c (Ai)), A0 ⊆cc A1 ⊆cc . . . ⊆cc Am = A′,n is a natural number, for which y∆
p (A′) = n − s · α1 with somes ∈ ω, kp ≤ n < kp+1.
Put Kf,P,R0
∞⋂p=1
Kf,P,Rp , and denote by Kf,P,R the class of
all graphs in the language ΣP,R, for which each nite substructure(i.e., a subgraph together with an information on lengths of shortestQ-routes and on compulsory furcations) belongs to Kf,P,R
0 .Let A be a nite substructure of graph (respectively, of nite
substructure) M (of a graph) belonging Kf,P,R. We say that A is aself-sucient substructure of M and write A 6 M if for any nitestructures A′ v A, B′ v M, A′ = A0 ⊆cc A1 ⊆cc . . . ⊆cc Am = B′,the inequality ∆1 + . . .+∆m < 0, where ∆i = y(Tc((Ai+1)T ∗c (Ai)))−y(T ∗c (Ai)), implies B′ ⊆ A. If A 6 M, and M is a nite structure,then A is called a strong substructure of M.
Denote by Tf,P,R0 the class of types correspondent to all struc-
tures in Kf,P,R0 , and equip it by the relation 6′, where
Φ(A) 6′ Ψ(B) ⇔ A 6 B).
190
Repeating the arguments for the class (Tf0 ; 6′
cc), we state that(Tf,P,R
0 ; 6′) is a self-sucient generic class such that, adding to typesof formulas describing self-sucient closures, we have the uniform t-amalgamation property. It implies that a (Tf,P,R
0 ; 6′)-generic modelis ω-saturated and realizes all types Φ(X), correspondent to typesΦ(A) in Tf,P,R
0 , and the (Tf,P,R0 ; 6′)-generic theory T f,P,R is stable.
Since two-elements sets a, b with (a, b) ∈ ⋃j∈ω
Rj are self-sucient,
the type tp(aˆb) is dened by color of edge [a, b] and by color ofany its endpoint, i.e., all formulas Rj(a, x) are principal. Using theproof of Theorem 4.3.18, we obtain that each type in S1(∅) has theinnite own weight. Thus, the following theorem holds.
Theorem 4.8.1. The theory T f,P,R is stable, small, and hascountably many 1-types, each of which is dened by color of everyits realization and has the innite own weight. The type p∞(x),dened by innite color, is the unique nonprincipal 1-type. Everyformula Pi,n(a, x), where Col(a) > n, Q(b, x), where Col(b) = ∞,and Rj(c, x) is principal.
Since limk→∞
αRk = 0, any self-sucient set A in a generic model,
containing two realizations, a1 and a2, of the type p∞(x), such thata1 and a2 are not connected by Q-routes or by edges, can be trans-formed to a self-sucient set B in the same model, having the samestructure as A with the unique dierence, namely, with an Rj-edge,connecting elements b1 and b2 that correspond to a1 and a2. Sinceformulas Rj(b1, x) and Rj(x, b2) are principal, the transformationfrom A to B can be interpreted as the corealization amalgam ofmodels Ma1 and Ma2 over the type tp(A). Thus, the structure ofgraph language Rj | j ∈ ω plays in a generic model the same rolethat acyclic undigraphs in generic models of Ehrenfeucht theories,described in Chapter 3.
Similarly Corollary 4.6.4, we haveProposition 4.8.2. Let A be a self-sucient set in a model of
the theory T f,P,R. The model MA is a complete⋃
k,n∈ω
(Qk ∪ Pk,n ∪Rk)-graph over A, i.e., any two dierent elements, a ∈ MA andb ∈ MA \ A, are connected by a Qk-arc or an edge. The set ofisomorphism types of prime models over nite sets coincides with theset of isomorphism types of models MA, where A are self-sucientsets and MA are complete
⋃k,n∈ω
(Qk ∪ Pk,n ∪Rk)-graphs over A.
191
Dene the class K∗2 of nite structures in language
Σ2 ΣP,R ∪ RAm| m ∈ ω∪
∪ρ(3) | ρ(x, yi, yj) characterizes some CLN-correlarion,equipped by records on mutual connections of elements, satisfyingthe conditions (1) and (2) in previous Section, where structuresof Kf,P,R
0 are taken instead of structures in Kf,P0 . The self-suciency
relation 62 for the class K∗2 naturally inherits the self-suciency
relation 6 for the class Kf,P,R0 .
Denote by K2 the class of all models in the language Σ2, forwhich each nite subset forms a structure in K∗
2.The concept of c2-embedding f : A →c2 B for structures A and
B in K∗2 preserving the correspondent record WA (Wf(A) = WB ¹
f(A)) naturally generalizes aforesaid concepts of c-embeddings.Thus, the notion of c2-embedding f : A →c2 N of c2-structure Ainto a model N in K2 is also dened.
Two structures, A and B, are called c2-isomorphic if there existsa c2-embedding f : A →c2 B with f(A) = B.
Theorem 4.8.3. There exists a saturated (K∗2; 62)-generic
model M of a stable theory, satisfying the following conditions:(a) if A and B are c2-isomorphic self-sucient structures in M,
then tpM(A) = tpM(B);(b) the restriction of M to the language ΣP,R is a Kf,P,R
0 -genericmodel;
(c) the theory Th(M) has countably many 1-types, each of whichis dened by color of any its realization and has the innite ownweight; the type p∞(x), dened by innite color, is the unique non-principal 1-type;
(d) every formula Pi,n(a, x), where Col(a) > n, Q(b, x), whereCol(b) = ∞, and Rj(c, x) is principal;
(e) every formula RA(a, y), where |= p∞(a), is principal, andthe c2-isomorphism type of any realization of formula RA(a, y) co-incides with the c2-isomorphism type A;
(f) every formula RA(x, a), where A is the c2-isomorphism typeof a, is principal, and every realization of the formula RA(x, a) isa realization of p∞.
192
P r o o f consists in obvious combination of the proofs ofTheorems 4.7.1 and 4.8.1. ¤
Denote the (K∗2; 62)-generic theory by T2. Clearly, T2 is an
expansion of T f,P,R.Repeating the proof of Theorem 4.7.2 and using Theorem 4.8.3,
we state the following:
Theorem 4.8.4. The theory T2 satises the condition|RK(T2)| = 2.
Theorem 4.8.5. There is a limit model of the theory T2 overthe type p∞ which is unique up to isomorphism.
P r o o f. The existence of a limit model follows from Proposi-tion 1.1.8 and Corollary 1.1.9, in view of the semi-isolation relationSIp∞ being nonsymmetric w.r.t. Q(x, y). In order to prove that ourlimit model is unique, it suces to show that any limit model Mover p∞ is saturated.
Let Man , |= p∞(an), n ∈ ω, be any elementary chain over p∞whose union coincides with the limit model M. Notice, that bygeneric construction, this chain can be chosen so that each principalformula ϕn(an+1, x), for which |= ϕn(an+1, an), n ∈ ω, is equivalentto a formula Rj(an+1, x) or a formula Qk(an+1, x)∧¬Qk−1(an+1, x),k ≥ 1. If |= Rj(an+1, an), then, in models Man and Man+1 , con-nected components w.r.t.
⋃i∈ω
Rj , containing elements an and an+1
simultaneously, form a complete graph, and if there are no linksfrom an to sequential elements by relations Qk, then the model Mis prime over some element in the
⋃i∈ω
Rj-component of connectiv-ity, containing an, i.e., M is not a limit model. Thus, the sequence(an)n∈ω has innitely many transitions from an to an+1 satisfying|= Qkn(an+1, an)∧¬Qkn−1(an+1, an). Now notice, that for elementsan and an+1, satisfying |= Qkn(an+1, an) ∧ ¬Qkn−1(an+1, an), the⋃j∈ω
Rj-components of connectivity in the models Man and Man+1 ,
containing elements an and an+1 respectively, form a complete graphw.r.t.
⋃k∈ω
(Qk ∪ Rk). Therefore, the sequence (an)n∈ω contains an
innite subsequence, in which all elements are pairwise connectedby Q-routes. Condensing this subsequence by elements of short-est Q-routes, containing neighbouring elements of subsequence, we
193
obtain an elementary chain of prime models (Mbn)n∈ω, for whichM =
⋃nωMbn and |= Q(bn+1, bn), n ∈ ω.
Now, take an arbitrary 1-type q(x, c) ∈ S(c), where c is a tuplein M , and argue to show that q(x, c) is realized in M. Indeed,the tuple c belongs to some model Mbn , and, by denition of RA,for some c2-isomorphism type A, containing realizations of the typeq(x, y), and for some n′ ≥ n, there exists an element d ∈ Mbn′ suchthat some projection ∃zi1 , . . . , zim RA(d, z) s realized by c. Since theformula RA(d, z) is principal, there exists a tuple d ∈ M , realizingthat formula and extending c. By the choice of the c2-isomorphismtype A, some coordinate of d realizes the type q(x, c). Since thechosen type q is arbitrary, M is saturated. ¤
By Corollary 1.1.15 and Theorems 4.8.4 and 4.8.5, we obtain thefollowing:
Theorem 4.8.6. There exists a stable Ehrenfeucht theory T sat-isfying the condition I(T, ω) = 3.
§ 4.9. Realizations of basic characteristics of stableEhrenfeucht theories
We show that, similar Theorem 3.4.1, all possible basic charac-teristics of Ehrenfeucht theories are realized out in the class of stabletheories.
Theorem 4.9.1. For any nite preordered set 〈X;≤〉 with theleast element x0 and the greatest class x1 in the ordered factor set〈X;≤〉/∼ w.r.t. ∼ (where x ∼ y ⇔ x ≤ y and y ≤ x), and for anyfunction f : X/∼→ ω satisfying the conditions f(x0) = 0, f(x1) > 0for |X| > 1, and f(y) > 0 for |y| > 1, there exist a stable theory Tand an isomorphism g : 〈X;≤〉 →RK(T ) such that IL(g(y)) = f(y)for any y ∈ X/∼.
P r o o f. Let 〈X;≤〉 and f be as above. Without loss of gener-ality, we may assume that |X| > 1. We x a numbering ν : |X| → Xsuch that ν(m) < ν(n) and ν(m) 6∼ ν(n) imply m < n, and in corre-spondence with every∼-class is an interval in |X|. Consider a theoryT−1 of unary predicates S1, . . . , S|X|−1 forming a partition on |X|−1innite classes with inessential coloring Col : M → ω ∪ ∞ suchthat ` ∃≥ω (Pi(x) ∧ Coln(x)), i = 1, . . . , |X| − 1, n ∈ ω.
194
Clearly, using a construction above, we can expand the the-ory T−1 by symmetric binary predicates Pi,n,i′ , i, n ∈ ω (where thethird index i′ means, that elements of color exceeding n are con-nected with elements of color n, belonging to Si′), to a small sta-ble generic theory T0 with weights αP
c(c(i,n),i′) of edges, for whichany nite set of nite elements in color is connected by binarypredicated with some innite elements in color, belonging relationsS1, . . . , S|X|−1. Therefore, adding predicates RA with structures ofpowerful digraphs, we reduce the construction of required domina-tion preorder ≤RK to the statement of such preorder for 1-types ofinnite elements in color, dened by formulas S1(x), . . . , S|X|−1(x).
We claim that there exists a stable expansion T of T0 with an iso-morphism g : 〈X,≤〉 → RK(T ) such that:
(i) g(ν(i)) = Mpi , where Mpi is an isomorphism type of theprime model Mpi over a realization of the type pi(x) in S1(∅),for pi(x) being isolated by the set Si(x) ∧ ¬Coln(x) | n ∈ ω offormulas, i = 1, . . . , |X| − 1, and p1(x), . . . , p|X|−1(x) are all non-principal 1-types over ∅ in the variable x;
(ii) IL(g(y)) = f(y) for any y ∈ X/∼.We construct T =
⋃i<|X|
Ti by induction so as to respect the
numbering ν. Assume T0, . . . , Tk−1 are already constructed andν(k), ν(k + 1), . . . , ν(k + l) form a ∼-class.
If f(ν(k)) = 0, then l = 0, and we dene Tk by expanding thelanguage of Tk−1 by new binary predicate symbols Rki (where theclass ν(k) covers ν(i), i 6= 0) so as to satisfy the following conditions:
(1) each predicate Rki is injectively connected with some weightαk′ bounding via a linear prerank function y(·) (of form pointedout in Section 4.8) the number of Rki-links in dependence on thecardinality of given nite set;
(2) Rki(a, y) is a principal formula and Rki(a, y) ` pi(y) for anya |= pk;
(3) for any a, b |= pi, there are innitely many elements c |= pk
and innitely many d not realizing types p1(x), . . . , p|X|−1 such that
|= Rki(c, a) ∧Rki(c, b) ∧Rki(d, a) ∧Rki(d, b);
moreover, c |= pk and |= Rki(c, a) imply that a does not semi-isolate c.
195
Clearly, (1)(3) can be realized so that the model Mpkof Tk has
a unique realization of the type pk and hence IL(g(ν(k))) = 0 =f(ν(k)). In addition, Mpk
, by induction, will realize all types pi
dominated by pk, that is, satisfying the relation ν(i) ≤ ν(k). A pres-ence of prerank function allows on a base of arguments in previousSections to state that the theory Tk is small and stable.
Suppose f(ν(k)) = r > 0. We dene T 0k by expanding the lan-
guage of Tk−1 by new binary predicate symbols Rki (where ν(k)covers ν(i), i 6= 0) satisfying (1)(3), and by binary predicate sym-bols R′
ij (where ν(i), ν(j) ∈ ν(k)) with the following conditions:(4) each predicate Rki and R′
ij is injectively connected with someweight αk′ bounding via a linear prerank function y(·) (of formpointed out in Section 4.8) the number of Rki- and R′
ij-links independence on the cardinality of given nite set;
(5) R′ij(a, y) is a principal formula and R′
ij(a, y) ` pj(y) for anya |= pi;
(6) for any a, b |= pj , there are innitely many elements c |= pi
and innitely many d not realizing types p1(x), . . . , p|X|−1, with
|= R′ij(c, a) ∧R′
ij(c, b) ∧R′ij(d, a) ∧R′
ij(d, b);
moreover, c |= pi and |= R′ij(c, a) imply that a does not semi-
isolate c;(7) for any elements a and b not realizing formulas Sk+l+1(x), . . . ,
S|X|−1(x), there exist innitely many elements c |= pj such that|= R′
ij(a, c) ∧R′ij(b, c);
(8) the relation R′k =
⋃ν(i),ν(j)∈ν(k)
R′ij forms a digraph isomorphic
to some digraph Γcpg, and, moreover, lengths of shortest Q-routesare injectively and monotonically connected with weights αk′ .
As above, predicates Rki guarantee that types pi are dominatedby pk for ν(i) < ν(k) and ν(i) 6∼ ν(k), while relations R′
ij ensurethat pi and pj are domination equivalent and that models Mpi andMpj are non-isomorphic, for i 6= j. A presence of prerank functionwith weights αk′ also allows to state the smallness and the stabilityof theory, constructing in this step.
Now, similarly to the conditions (1) and (2) specied in Section4.7 with Qreplaced by R′
k, we extend the language by predicate
196
symbols RA so that types pi, i = k, . . . , k + l, dominate all typesq(x) ∈ S(Tk) with Sj(xi) 6∈ q(x) for j > k+l, the types q in questionthat are not dominated by pi, i < k, dominate all types pk, . . . , pk+l,and the models Mq are isomorphic to Mpk
.For the condition IL(g(ν(k))) = r to be satised, on the struc-
ture of realizations of pk, we dene a graph structure with binaryrelations R′′
1 , . . . , R′′r such that:
(9) each predicate R′′i is injectively connected with some weight
αk′ bounding via a linear prerank function y(·) (of form pointedout in Section 4.8) the number of R′′
i -links in dependence on thecardinality of given nite set;
(10) R′′i (a, y) is a principal formula and R′′
i (a, y) ` pk(y) for anya |= pk, i ≤ r;
(11) for any a, b |= pk, there are innitely many elements c |= pk
and innitely many d not realizing types p1(x), . . . , p|X|−1 for which
|= R′′i (c, a) ∧R′′
i (c, b) ∧R′′i (d, a) ∧R′′
i (d, b);
moreover, |= R′′i (c, a) and c |= pj imply that a does not semi-
isolate c;(12) relations R′′
i form digraphs isomorphic to some digraphsΓcpg, where lengths of shortest routes are injectively and monoton-ically connected with weights αk.
If Ma and Mb are prime models over realizations a and b of pk,respectively, such that |= R′′
i (a, b) and Mb ≺Ma, then we call Ma
an R′′i -extension of the model Mb. An elementary chain (Ms)s∈ω
over pk is called an R′′i -chain if Ms+1 is an R′′
i -extension of Ms forany s.
If Ma and Mb are prime models over realizations a and b of pi,respectively, with ν(i) ∼ ν(k), such that |= R′
ii(a, b) and Mb ≺Ma,then the model Ma is called an R′
ii-extension of Mb.Similarly to the conditions (a) and (b) given in Section 4.8, we
extend the language by symbols Rj so as to satisfy the following:(13) each predicate Rj is injectively connected with some weight
αk′ bounding via a linear prerank function y(·) (of form pointedout in Section 4.8) the number of Rj-links in dependence on thecardinality of given nite set;
(14) for any limit model M over a type pk+i, 0 ≤ i ≤ l, thereis a relation R′′
j such that M is the union of an R′′j -chain (Ms)s∈ω
over the type pk;
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(15) limit models M1 and M2 over pk are equivalent i there isa predicate R′′
i such that M1 and M2 are unions of R′′i -chains but
are not unions of R′′j -chains, for j > i.
Notice, that (14) and (15) are realized via predicates Rj \saying"the following:
(a) every R′kk-extensionMa ofMb contains an R′′
1-extension andvice versa;
(b) for any i, ν(i) ∼ ν(k), i 6= k, and for any nite elemen-tary chain Ma1 , . . . ,Mas , a1, . . . , as |= pi, there exist realizationsb1, . . . , bs−1 of pk such that the sequenceMa1 ,Mb1 , . . . ,Mbs−1 ,Mas
is also an elementary chain;(c) if Mb0 and Mb1 are R′′
s -extensions of Man , q is the type of atuple 〈a0, a
′0 . . . , an, a′n, a′′n, b0, b1〉 of elements which realize type pk
and are such that Mai+1 is an R′′ti-extension of a model Ma′i , equal
to Mai , |= R′′ti(ai+1, a
′i), Mb0 and Mb1 are R′′
tn-extensions of themodel Man , equal to Ma′n and to Ma′′n , |= R′′
tn(b0, a′n)∧R′′
ti(b1, a′′n),
and elements b0 and b1 are not connected by (b0, b1)-Q- or (b1, b0)-Q-routes, then Mb0 contains a corealization amalgam Mb0 ∗q Mb1 ;
(d) if Ma1 , . . . ,Mas is a nite elementary chain such that themodel Maj+1 is an R′′
ij-extension of Maj , j = 1, . . . , s − 1, and
maxi1, . . . , is−2 < is−1, then Mas contains an R′′is−1
-extensionof Ma1 and vice versa.
With the expansions exercised above we obtain a small stabletheory Tk = Tk+1 = . . . = Tk+l such that types pk, . . . , pk+l are dom-ination equivalent, Mpk
, . . . ,Mpk+lare pairwise non-isomorphic,
and the number of limit models over pk, . . . , pk+l is equal to f(ν(k)).Proceeding further with this process, at step |X| − 1, we obtain
a small stable theory T = T|X|−1 and an isomorphismg : 〈X;≤〉 → RK(T ) such that g(ν(0)) is an isomorphism type ofthe prime model for T , g(ν(m)) is an isomorphism type of Mpm ,1 ≤ m ≤ |X| − 1, and IL(g(y)) = f(y) for any y ∈ X/∼. ¤
C h a p t e r 5HYPERGRAPHS OF PRIME MODELSAND DISTRIBUTIONS OF COUNTABLEMODELS OF SMALL THEORIES
In the nal Chapter, we consider a family of hypergraphs ofprime models for an arbitrary small theory and represent a mecha-nism of structure descriptions of models of theories by these families.Thus, in particular, we verify the key role of graph-theoretic con-structions in creation of aforesaid examples of Ehrenfeucht theories.
We shall realize constructions on a base of models Meq, that,as shown, for instance, in the work by P. Tanovic [194], allow inmany cases to reduce general enough structure situations to classesof known objects.
Recall the notion of model Meq [22]. Let M be a model. Weadd to the language Σ(M) of M all possible unary predicate sym-bols PE , correspondent to ∅-denable equivalence relations E(x, y),and all possible (l(x) + 1)-ary predicate symbols πE . The obtainedlanguage Σ(M)eq is the language of Meq. The universe M eq of Meq
consists of all possible E-classes. Moreover, the elements of M areconsidered as =-classes forming the home sort of elements, and in-terpretations of symbols in Σ(M) coincide with their interpretationsin M. All other elements in M eq form imaginary sorts. Each pred-icate PE consists of all possible E-classes, and each predicate πE isa graph of function mapping tuples of elements of the home sort inE-classes containing these tuples.
The theory of Meq doesn't depend on choice of model of giventheory and is denoted by T eq.
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§ 5.1. Hypergraphs of prime modelsRecall that a hypergraph is a pair (X, Y ) of sets, where Y is
a subset of the set P(X) being the set of all subsets of X.Let M be a model of small theory T . Denote by H(M) the
set of all subsets in the universe M of M such that these subsetsare universes of prime, over tuples a ∈ M , elementary submodelsMa of M, namely, H(M) = Ma | Ma is a prime, over a tuplea ∈ M , elementary submodel of M. The pair (M, H(M)) is calleda hypergraph of all prime models of M.
Clearly, a tuple b belong to a prime model Ma ∈ H(M) i thereexists an elementary submodel Mb ∈ H(M) of Ma. Thus, theinclusion relation on the set H(M) denes a relation that consistsof all pairs (a, b) of tuples in M for which types tp(b/a) are principal.
E x a m p l e 5.1.1. (1) For an innite model M of emptylanguage, the (M, H(M)) includes all countable subsets of M .
(2) For a model M of dense linear order without endpoints, thehypergraph (M, H(M)) consists of all possible universes of count-able dense linearly ordered subsets without endpoints.
(3) A model M of the Ehrenfeucht theory with three countablemodels, constructed on dense linearly ordered set without endpointsand with constants ck, k ∈ ω, such that ck < ck+1, generates a hy-pergraph (M, H(M)) consisting of all possible universes of count-able dense linearly ordered subsets without endpoints, each of whichincludes a dense linear order for every interval (−∞; c0), (ck; ck+1),k ∈ ω.
(4) For a model M of linear space over a eld, the hypergraph(M, H(M)) is constructed by all possible subspaces of nite orcountable dimensions. ¤
Since the operation (·)eq preserves the smallness of theory andnaturally extends prime models over tuples to prime models overelements, below, for convenience, we shall often consider modelsMeq instead of M, and we include in hypergraphs of prime modelsonly universes of prime models over elements in Meq. Consideredset of universes of prime models is denoted by H(Meq).
Notice, that the following notions and arguments work for mod-els, in which every prime model over a tuple, being an elementary
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submodel, is prime over an element. For instance, any prime modelover ∅ and any model of a theory with nonprincipal 1-types, whereall nonprincipal types are powerful, have this property.
We claim, that hypergraphs (M eq,H(Meq)) of countable mod-els M allow to dene that given theory T is countably categorical.
Indeed, the primeness of Meq over an element is equivalent theproperty that the set M eq belongs to H(Meq). Since every count-able model of countably categorical theory is prime, and every smalltheory, being not countably categorical, has a countable saturatedmodel that is not prime over any tuple, we have the following:
Proposition 5.1.1. A theory T is countably categorical i, forany countable model M |= T , the set M eq belongs to H(Meq).
If given theory is not countably categorical then, as shown inaforesaid examples, structures (M eq,H(Meq)) are poorly informa-tive in general situation from classication point of view, since equiv-alence classes, relative isomorphism types of hypergraphs, are toowide w.r.t. types of mutual denability by formulas of given struc-tures, and these structures weakly take into account specicities ofgiven structures.
At the same time, for some situations, properties of hypergraphs(M eq,H(Meq)) may be useful. For instance, a presence of minimalprime models is re ected in hypergraph by minimal sets, and struc-tures with nonsymmetric semi-isolation relations imply an existencein hypergraphs of innite ⊆-chains without endpoints. For similarsituations, the problem of investigation for interrelation of classes ofstructures and of classes of correspondent hypergraphs is expectedto be perspective.
Notice the following useful property of hypergraphs(M eq,H(Meq)) for countable models M that is implied from a rep-resentation (see the proof of Proposition 1.1.7) of any countablemodel of small theory by a union of elementary chain of prime mod-els, and marks all hypergraphs of countable models of small theoriesamong all possible hypergraphs on countable universes.
Proposition 5.1.2. Any countable set M eq is a countable unionof a ⊆-chain of sets in H(Meq).
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§ 5.2. HPKB-Hypergraphs and theorem on structureof type
To obtain an additional classication information on models ofgiven theory T , we add to the hypergraph (M eq,H(Meq)) a kernel-function
kerM : H(Meq) → P(M eq),
mapping by the following rule:
kerM(Ma) b ∈ Ma | Mb = Ma.A set kerM(Ma) is called a kernel a model Ma (w.r.t. M).
Proposition 5.2.1. The kernel of any elementary submodel Ma
of Meq consists of all elements b ∈ Ma connected with a by formulasϕb(x, y) for which |= ϕb(a, b) and formulas ϕb(a, y) and ϕb(x, b) areprincipal.
P r o o f. If aforesaid formula ϕb(x, y) doesn't exist, then tp(a/b)is a nonprincipal type and the equality Mb = Ma is impossible.If we have such a formula ϕb(x, y), then take an arbitrary tuplec of elements in Ma. By denition of prime model, there existsa principal formula ψ(a, y, z) for which |= ψ(a, b, c). Then the for-mula ϕb(x, b)∧ψ(x, b, z) is principal and isolates the type tp(aˆc/b).Thus, Ma realizes only principal types over b and Ma = Mb. ¤
In view of Proposition 5.2.1, for any countable prime model Mover ∅, we have kerM(M eq) = M eq. Then this equality is true forany countable model of countably categorical theory T and, more-over, by Proposition 5.1.1, the following Corollary holds.
Corollary 5.2.2. A theory T is countably categorical i, forany countable model M |= T , the kernel of any elementary submodelMa of Meq equals to Ma.
Now, for a model Meq, we consider all possible formulas ϕ(x, y)such that ` ϕ(x, y) → ¬(x ≈ y) and there exist elements a ∈ M eq,for which ϕ(a, y) are principal formulas. For any such a formulaϕ(x, y), we dene a binary relation Rϕ (a, b) | Meq |= ϕ(a, b).If (a, b) ∈ Rϕ, the pair (a, b) is called a ϕ-arc. If ϕ(a, y) is a prin-cipal formula then ϕ-arc (a, b) is said to be principal. If, in addi-tion, ϕ(x, b) is principal, then the set [a, b] (a, b), (b, a) is called
202
a principal ϕ-edge. ϕ-Arcs and ϕ-edges are called, respectively, arcsand edges if we say about xed or some formula ϕ(x, y). If (a, b) isa principal arc and (b, a) is not a principal arc then (a, b) is calledirreversible.
A model −→Meq with an universe M eq and all possible binary re-lations Rϕ is called a principal graph of Meq.
Proposition 5.2.3. Projections of all possible binary relationsRϕ dene all sets of realizations of 1-types of T eq that realized in Meq.
P r o o f. Let p(x) be an arbitrary 1-type of T eq that is realizedin Meq. Since T is small, there exists a principal formula ϕ0(y) of T .Hence, there exists a formula ψ(x, y) and a realization a of p(x) inMeq such that ψ(a, y) is a principal formula and ψ(a, y) ` ϕ0(y).Moreover, for ψ(x, y), we can choose any formula χ(x, y) of formψ(x, y)∧ϕ(x), where ϕ(x) is an arbitrary formula of p(x). Therefore,every realization of set X of all aforesaid formulas ∃y Rψ(x, y) isa realization of p(x). ¤
Thus, besides the principal binary structure, the model −→Meq
contains the structure of all 1-types that realized in Meq.Recall [25], that an undigraph without loops is called complete
if any two dierent vertices are connected by edge.Proposition 5.2.1 implies that the kernel of any prime model Ma
with all possible principal edges forms a complete undigraph, andany vertex b ∈ Ma, not belonging to kerM(Ma), is a common end-point of all possible principal arcs (c, b), where c ∈ kerM(Ma), andnone of (b, c) is a principal arc. In particular, for any countablemodel M of countably categorical theory, the kernel kerM(M eq)forms the complete graph on the set M eq, and this property repre-sents one more characterization of countable categoricity.
For a hypergraph (M eq,H(Meq)) with a kernel-function kerM,kernels kerM(Ma) of prime models Ma over realizations a of a xed1-type p(x) form maximal connected graphs on the set M eq. Thesegraphs correspond to connected components C of undigraphs withcolored edges that composed by all possible principal edges. Restric-tions of C to the set of realizations of the type p(x) form graphswith colored edges, and having saturated model M, these graphsare called kernel-undigraphs over the type p. Furthermore, by de-nition, automorphism groups of kernel-undigraphs are transitive andtheories of these undigraphs are small.
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In view of Compactness Theorem, a kernel-undigraph on a set ofrealizations of type p(x) in a model Meq, correspondent to a satu-rated model M, is unique i the structure of the kernel-undigraph,restricted to p(Meq), has nitely many 2-types and p(x) is princi-pal. Otherwise, there are innitely many such kernel-undigraphs.Since the automorphism group of structure of a type, obtained froma saturated model, is transitive, all kernel-undigraphs over p(x) arepairwise isomorphic.
Among known kernel-undigraphs, we note the following, playingan essential role in constructions of Ehrenfeucht theories:
(a) complete α-element undigraphs, where kernel-undigraphs areobtained replacing each element, in a dense linear order withoutendpoints with a countable chain of constant, by an equivalenceclass containing α pairwise incomparable elements (see Section 1.4);
(b) acyclic undigraphs with countably many colors of edges andinnitely many edges of each color that are incident to every vertex(see Section 3.3);
(c) Hrushovski | Herwig undigraphs [81] (small stable undi-graphs with innite weight and coloring of edges by countably manycolors) that used in a solution of the Lachlan problem (see Section4.8);
(d) models of cubic theories [184].Kernel-undigraphs in the examples (a), (b), and (d) are com-
plete. Saturated Hrushovski | Herwig undigraphs form kernel-undigraphs with innite diameters relative unions of all entered bi-nary relations.
Now, we consider links between kernel- undigraphs realized byirreversible principal arcs. It is pertinent to retrace these links usingthe following object that include aforesaid components.
For a given modelM, a hypergraph of prime models with a kernel-function and a principal binary structure or a HPKB-hypergraph isa quadruple
H(M) (M eq,H(Meq), kerM, Σ
(−→Meq))
,
where Σ(−→Meq
)is the set of all binary relations in −→Meq.
We are going to show, how the structures of HPKB-hypergraphsH(M) entirely, and irreversible principal arcs, in particular, re ectstructural properties of given theory T .
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At rst, we notice that if a model M is isomorphically embeddedin a model N then the HPKB-hypergraph H(M) is isomorphicallyembedded in the HPKB-hypergraph H(N ). Therefore, in partic-ular, if M is saturated (homogeneous, universal) then H(M) hasa special structure that also can be considered as saturated (homoge-neous, universal) HPKB-hypergraph among all HPKB-hypergraphsof models of given theory. Semi-isolation relations on sets of re-alizations of types (and their properties) are also transformed tosemi-isolation relations (and correspondent properties) on sets of re-alizations of 1-types.
We claim, that having irreversible arcs, connecting realizations ofcomplete 1-type p(x), the semi-isolation relation SIp is nonsymmet-ric (in given structure as well as in correspondent HPKB-hypergraph).
Indeed, let (a, b) be a ϕ(x, y)-principal arc and the pair (b, a) isnot a principal arc, where a and b are realizations of the type p.Suppose on contrary, that SIp is symmetric. Then the consideredformula ϕ can be chosen so that ϕ(x, b) ` p(x). Since the the-ory is small, some pair (b, a′) is a principal arc, where |= ϕ(a′, b).Take a formula ψ(x, y), for which |= ψ(a′, b) and ψ(x, b) is prin-cipal, and an automorphism f mapping a′ to a. Then f maps bto an element b′ such that |= ϕ(a, b′) ∧ ψ(a, b′). Since the formulaϕ(a, y) principal, there exist an automorphism xing a and map-ping b′ to b. Hence, |= ϕ(a, b) ∧ ψ(a, b) and (b, a) is a principal arc.Obtained contradiction means, that SIp is nonsymmetric and thisproperty is guaranteed by any irreversible principal arc (a, b) withendpoints realizing p(x). Furthermore, since, for any principal type,the semi-isolation relation on the set of its realizations is symmetric,a presence of irreversible principal arc (a, b) implies that the typep(x) is not isolated.
Aforesaid arguments show that irreversible principal arcs on theset of realizations of the type p link dierent kernel-undigraphsover p. Since semi-isolation relations are transitive, these arcs de-ne a relation of partial order ≤ on the set of kernel-undigraphsover p, and each kernel-undigraph belongs to an innite ≤-chain.
Thus, the following p-decomposition theorem holds.Theorem 5.2.4. (1) A structure of realizations of any com-
plete 1-type p(x) over ∅ in a HPKB-hypergraph H(M) is composedby pairwise disjoint kernel-undigraphs over p(x) and by a partialorder ≤ on the set of these kernel-undigraphs that dened by irre-versible principal arcs.
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(2) A kernel-undigraph on the set of realizations of type p(x) inthe model Meq, correspondent to countable saturated model M, isunique if the structure of kernel-undigraph, restricted to p(Meq), hasnitely many 2-types and the type p(x) is principal. Otherwise, thereare innitely many such kernel-undigraphs. All kernel-undigraphsover p(x) are pairwise isomorphic.
(3) The partial order ≤ is identical i there are no irreversibleprincipal arcs, connecting realizations of p(x), and i the semi-isolation relation SIp is symmetric. If the partial order ≤ is notidentical then each kernel-undigraph belongs to an innite ≤-chain.
Examples of aforesaid partial orders ≤ w.r.t. irreversible princi-pal arcs are extracted from the following known examples of struc-tures with nonsymmetric semi-isolation relations:
(1) the Ehrenfeucht examples of structures on dense linear orderswith countably many constants that ordered by increase;
(2) the Peretyat'kin examples of structures on innite branchedtrees with countably many constants that ordered by increase [137],[138];
(3) the structure of free directed pseudoplane with 1-inessentialordered coloring (see Example 1.2.3);
(4) unstable and stable generic Ehrenfeucht structures, repre-sented in Chapters 3 and 4.
§ 5.3. Graph connections between typesNow, we consider graph links between structures of dierent
types.Since, for any n-type that realized in M, there exists a corre-
spondent 1-type, realized in Meq by names of realizations of then-type, the Rudin | Keisler preorder ≤RK on the set of types re-alized in M, is interpretable as the Rudin | Keisler preorder ≤eq
RKon the set of 1-types realized in Meq, and in turn, it is transformedin the structure of HPKB-hypergraph H(M).
In particular, it means, that a realizability in M of a powerfultype (or, that is equivalent, a presence in M of greatest ≤RK-classfor the theory Th(M)) means that there exists a powerful 1-typerealizable in H(M), i.e., there exists the greatest (≤eq
RK ∩ ≥eqRK)-
class for the set of 1-types realized in(M eq, Σ
(−→Meq))
, being theHPKB-hypergraph H(M) is universal.
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Thus, the following Theorem and its immediate Corollary hold.Theorem 5.3.1. For any Rudin | Keisler preorder ≤RK on
the set of types of a small theory, there exists an universal HPKB-hypergraph H(M), for which a set with a preorder ≤eq
RK, isomorphicto the set with the preorder ≤RK, is interpretable on the set of allrealizations of 1-types.
Corollary 5.3.2. A preorder ≤RK of a theory T has a greatest(≤eq
RK ∩ ≥eqRK)-class (a nonempty set of powerful types) i there
exists a greatest (≤eqRK ∩ ≥eq
RK)-class for a set of 1-types, realized in(M eq, Σ
(−→Meq))
, where M is an universal model of T .Connections between kernel-undigraphs of 1-types p and q by the
Rudin | Keisler preorder are realized by principal edges (where thetypes p and q are domination-equivalent and models Mp and Mq
are isomorphic) or by irreversible principal arcs. If elements a andb of kernel-undigraphs Γp and Γq over p and q, respectively, areconnected only by irreversible principal arcs (a, b), then models Mp
and Mq are non-isomorphic, and the domination-equivalence of thetypes p and q means that there exists a kernel-undigraph Γ′p over psuch that some element c in Γ′p belongs to a principal arc (b, c).
Hence, on the set of principal connected components C, i.e., ofconnected components formed by principal edges, we can introducea partial order ≤ dened by irreversible principal arcs. As for chainsof kernel-undigraphs, ≤-chains of principal connected components inthe model Meq, correspondent to saturated model M, can be eithersingletons (for countably categorical theory) or innite.
Indeed, in view of Compactness Theorem, if there are two prin-cipal connected components then there are innitely many of them.On the other hand, any n-element set A, formed by elements in ndierent principal connected components, belongs to a prime modelMA over A. Therefore, Meq contains an element c (a name fora tuple of all elements in A) that is a common initial vertex of nirreversible principal arcs with endpoints in A. Hence, any elementin Meq is an endpoint of some irreversible principal arc, and, usingthese arcs, one can form an innite ≤-chain of principal connectedcomponents. Moreover, an existence of aforesaid elements c impliesthat the set of principal connected components with the partial or-der ≤ is downward directed.
Thus, similar to Theorem 5.2.4, the following decomposition the-orem on structure, connecting kernel-undigraphs over dierent types,holds.
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Theorem 5.3.3. (1) The structure of set of realization of anycomplete 1-type p(x) over ∅ in HPKB-hypergraph H(M) is formedby pairwise disjoint principal connected components and by a partialorder ≤ on the set of principal connected components, that denedby irreversible principal arcs.
(2) A principal connected component in the model Meq, corre-spondent to countable saturated model M, is unique if the struc-ture M is countably categorical. Otherwise, there are innitelymany principal connected components. All principal connected com-ponents, containing realizations of the same type, are pairwise iso-morphic.
(3) The partial order ≤ is identical i there are no irreversibleprincipal arcs, and i the theory is countably categorical. If the par-tial order ≤ is not identical then each principal connected componentbelongs to an innite ≤-chain. The set of principal connected com-ponents with the partial order ≤ is downward directed.
§ 5.4. Limit modelsA model M is called limit if M is not prime over tuples and
M =⋃
n∈ωMn for some elementary chain (Mn)n∈ω of prime models
over tuples.Using the proof of Proposition 1.1.7, we haveProposition 5.4.1. Any countable model M of small theory T
is represented as the union of an elementary chain (Mai)i∈ω of primemodels over tuples ai.
In view of Proposition 5.4.1 the following theorem holds.Theorem 5.4.2. Every countable model, being not prime over
tuples, of small theory T is limit. Universes of countable models of Tare represented in HPKB-hypergraphs H(M) by marked sets or byunions of countable chains of sequential properly included markedsets Mn, n ∈ ω. Moreover, for the second case, elements of kernelskerM(Mn), n ∈ ω, are connected with elements of kerM(Mk), k < n,by irreversible principal arcs.
As noticed in Propositions 1.1.7 and 3.5.2, if a Rudin | Keislerpreorder in RK(T ) is nite, then any countable modelM of T , beingnot prime over tuples, is limit over a type p.
In general situation, limit models can not be formed by primemodels Mn over realizations of the same type, i.e., kernels of models
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Mn, after transforming in Meq, can be only inside structures ofrealizations of types pnk(x), k ∈ ω, such that pn1k1 6= pn2k2 forn1 6= n2.
Limit models M and N are called equivalent (written M∼ N )if there exist elementary chains (Mn)n∈ω and (Nn)n∈ω of primemodels over tuples, satisfying the following conditions:
(1) M =⋃
n∈ωMn, N =
⋃n∈ω
Nn;
(2) there exist constant expansions M′n+1 = 〈Mn+1, c〉c∈M ′
n
and N ′n+1 = 〈Nn+1, c〉c∈N ′
n, n ∈ ω, M′
0 = M0, N ′0 = N0, such that
M′n+1 ' N ′
n+1, n ∈ ω.The following Proposition is obvious.Proposition 5.4.3. If models M and N are limit, then
M' N i M∼ N .Now, consider hierarchies of limit models by degree of saturation,
similar to hierarchies of prime models w.r.t. Rudin | Keisler pre-orders, allowing to estimate the number of pairwise non-isomorphiclimit models.
Denote by EEL(T ) the set LM of isomorphism types of limitmodels of theory T with the relation ¹ of elementary embeddability.
Clearly, any nonempty set EEL(T ) is preordered. Since the the-ory T is small, EEL(T ) contains an element (the isomorphism typeof countable saturated model), in which all isomorphism types oflimit models are elementary embeddable. At the same time, it ispossible that the structure EEL(T ) doesn't have minimal (¹ ∩ º)-equivalence classes. Furthermore, generally speaking, such \great-est" element is not unique. Moreover, for instance, in known Ehren-feucht theories, saturated models are elementary embeddable evenin prime models over realizations of powerful types.
Thus, the number Il(T ) of pairwise non-isomorphic limit modelsof small theory T has a lower bound |EEL(T )/(¹ ∩ º)|, i.e., thenumber of (¹ ∩ º)-equivalence classes. For some cases, the estimate
Il(T ) ≥ |EEL(T )/(¹ ∩ º)| (5.1)
is not improvable, for instance, when I(T, ω) = 3. In some examplesof Chapters 3 and 4, the inequality (5.1) is strict. The problem ofcharacterization of small theories with Il(T ) = |EEL(T )/(¹ ∩ º)|is still open.
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Aforesaid arguments show, that the hierarchy of saturation oflimit models, constructed on a base of set EEL(T ) with preordersof elementary embeddability is weakly connected with the numberIl(T ).
We introduce a renement of EEL(T )-hierarchy, for which theconnections between isomorphism types of models are realized onthe base of correspondence between sequences of principal connectedcomponents, and prime models over some elements of these compo-nents form given limit models.
We say that a limit model M is elementary embeddable in coor-dination in a limit model N if there exists an elementary embeddingf of M into N mapping some tuples an ∈ M to correspondent tu-ples f(an) ∈ N , n ∈ ω, such that M =
⋃n∈ω
Man , N =⋃
n∈ωNf(an).
Denote by CEEL(T ) the set LM of isomorphism types of limit mod-els of theory T with the relation ¹c of elementary embeddability incoordination.
Similar to EEL(T )-hierarchy, the CEEL(T )-hierarchy allows tond a lower bound for the number Il(T ):
Il(T ) ≥ |CEEL(T )/(¹c ∩ c º)|. (5.2)
The following example shows that, generally speaking, similarto mutual elementary embeddability, a presence of mutual elemen-tary embeddability in coordination of models doesn't guarantee thatmodels are isomorphic, i.e., the inequality (5.2) also can be strict.
E x a m p l e 5.4.1. Take a countable model M of connectedacyclic digraph 〈M ; Q〉, for which correspondent undigraph is a treeand any element has innitely many images and innitely manypreimages (see Example 1.2.3). Expand this structure by 1-inessen-tial coloring so that, on the set of realizations of the type p∞(x),the semi-isolation relation is nonsymmetric, namely, if |= p∞(a) and|= Q(a, b) then a semi-isolates b by Q(a, y), and b doesn't semi-isolate a. Now, we expand the structure by predicates R
(n+2)n , n ∈ ω,
satisfying the following conditions:(1) R0 = Q;(2) if n ∈ ω \0, |= p∞(an), and |= Q(ai+1, ai), i = 0, . . . , n−1,
thenRn(an, . . . , a0, y) ` Rn−1(an−1, . . . , a0, y)
210
and, for realizations of p∞(x) in Q(M, an−1), the formula Rn(x,an−1, . . . , a0, y) denes on the set Rn−1(an−1, . . . , a0,M) innitelymany equivalence classes such that each class is innite.
Take a nonprincipal type q(x) containing all formulas Rn(an,. . . , a0, x), where (an)n∈ω is a sequence of realizations of p∞ and|= Q(an+1, an), n ∈ ω. There exist mutually elementary embed-dable in coordination, non-isomorphic limit models dened by thesequence (an)n∈ω and such that the type q(x) has dierent numbersof realizations. ¤
A cause for mutually elementary embeddable in coordinationlimit models to be non-isomorphic, that is found on possibilityof dierently many realizations of types q (and, as corollary, onnon-homogeneity of limit models with marked constants an), is, inessence, unique, since an absence of such types allows step-by-stepto extend nite partial isomorphisms between mutually elementaryembeddable in coordination limit models. A possibility of such ex-tensions can be observed on examples of limit models of Ehrenfeuchttheories in works [137], [138], [198], and of generic Ehrenfeucht the-ories in Chapters 3 and 4. Moreover, in generic examples, isomor-phisms of limit models are constructed on a base of existence in thesemodels of realizations of the same types over nite sets, guaranteedby special predicates RA(x, y).
§ 5.5. λ-Model hypergraphs
In this Section, we dene a link of HPKB-hypergraph structureH(M) for a countable saturated model M of a theory T with thenumber I(T, ω) of pairwise non-isomorphic countable models of T .We also point out relations allowing to expand HPKB-hypergraphssuch that the number of pairwise structural objects, dened by ex-panded HPKB-hypergraphs equals to I(T, ω).
Two complete 1-types q1(x) and q2(x) over ∅, that realized in aHPKB-hypergraph H(M), are called p-equivalent (written q1 ∼p q2)if some principal connected component contains realizations of q1(x)and q2(x).
Denote by Ip(T, ω) the number of pairwise non-isomorphic count-able models of theory T , each of which is prime over a tuple.
Proposition 5.5.1. The number Ip(T, ω) coincides with thenumber of ∼p-equivalence classes of T .
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P r o o f. Since the model M is saturated, each prime modelover a tuple a is represented by some 1-type that is realized by thename ca of a in Meq. On the other hand, prime models, repre-sented by ∼p-equivalent types, are isomorphic, since having a route,consisting of principal edges and linking realizations of 1-types q1
and q2, any model is prime over a realization of q1 i it is primeover a realization of q2. Thus, Ip(T, ω) equals to the number ofpairwise ∼p-non-equivalent complete 1-types over ∅ that realizedin H(M). ¤
Since, in view of Theorem 5.4.2, each limit model of theory T isrepresented as a union of countable chain of structures of sequentialproperly included marked sets of HPKB-hypergraph H(M), a pres-ence of isomorphism of limit models implies that correspondent limitstructures in H(M) are isomorphic. Hence, by Proposition 5.4.1,the number I(T, ω) has a lower bound, being the sum of numberof ∼p-equivalence classes and of number Il(H(M)) of pairwise non-isomorphic limit structures in H(M):
I(T, ω) ≥ Ip(T, ω) + Il(H(M)). (5.3)
This estimate is not improvable on graph level, since graph struc-tures of M are transformed in H(M).
Now, consider possibilities of expansions of HPKB-hypergraphH(M), allowing to obtain the equality in (5.3) under a substitutionof expanded HPKB-hypergraph instead of H(M).
In view of Proposition 5.4.3, a presence of isomorphism of limitmodels M and N is witnessed by the possibility of co-ordinated ex-tensions of prime models, represented in HPKB-hypergraphs H(M)and H(N ). This coordination is obtained by introduction in HPKB-hypergraphs of correspondences between tuples of element and theirnames in Meq, as well as some structure of formula-denable binaryrelations, via which all formula-denable relations of given structureare formula-denable. Thus, we obtain the equality in (5.3) for ex-panded HPKB-hypergraphs.
Now, we show that language tools, used in Chapters 3 and 4 forconstructions of realizations of basic characteristics of Ehrenfeuchttheories, are minimal for interpretations of saturated models Meq
by HPKB-hypergraphs, generating these characteristics.Indeed, unary predicates Coln(x), n ∈ ω, guarantee a presence of
nonprincipal 1-types, that are realized in Meq. A binary predicate
212
Q(x, y) (it is admissible to have at most countably many formula-independent family of predicates Qn(x, y), n ∈ ω) forms a nonsym-metric semi-isolation relation on the set of realizations of powerfultype p∞(x), isolated by the set of formulas ¬Coln(x), n ∈ ω. Thefamily Rj | j ∈ ω of binary predicates form kernel-undigraphs onthe set of realizations of p∞(x), ordered by Q in accordance withTheorems 5.2.4 and 5.3.3, and predicates RA(x, y) guarantee thecorealization of the powerful type p∞(x) with other powerful types.
Now, let T be an arbitrary theory with exactly three pairwisenon-isomorphic models. Recall, that all nonprincipal types of T arepowerful and, considering a model Meq instead of saturated modelM, we x and denote by p∞(x) a nonprincipal 1-type, realizedin Meq. For a set of formulas ϕn(x), n ∈ ω, isolating p∞(x) , whereϕn(x) | n ∈ ω ` p∞(x), we put the formulas ϕn(x)∧¬ϕn+1(x) byColn(x), and by Qn(x, y), n ∈ ω, the countable family of formulas,each of which witnesses on non-symmetry of semi-isolation relationSIp∞ and is principal after a substitution of any realization of p∞instead of rst coordinate. Moreover, we require that, for any tworealizations a and b of p∞(x), there exists a realization c of p∞(x)and formulas Qi(x, y), Qj(x, y) such that |= Qi(c, a)∧Qj(c, b), and,thus, the local pairwise intersection property holds. For predicatesRj , j ∈ ω, we take all possible binary formulas ψ(x, y) consistentwith p∞(x) ∪ p∞(y), for which all realizations on the structure ofthe type p∞(x) are principal edges and form kernel-undigraphs. Fi-nally, we take as RA(x, y), for any nonprincipal 1-type q(y) realizedin Meq, all possible formulas χq(x, y) consistent with p∞(x) ∪ q(y),connecting realizations of p∞(x) with realizations of q(y) only byprincipal edges. Adding aforesaid predicates to the hypergraph(M eq,H(Meq) as well as structure that guarantees the uniquenessof limit model, we obtain a three-model hypergraph.
In view of Corollary 1.1.15, the following theorem holds.
Theorem 5.5.2. For any small theory T , the following condi-tions are equivalent:
(1) I(T, ω) = 3;(2) T has a three-model hypergraph.As known (see Theorem 1.1.13 and Figure 1.1(b), there are three
pairwise disjoint cases for theories with four countable models:(1) two-element Rudin | Keisler preorder and two limit models
over powerful types;
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(2) three-element Rudin | Keisler preorder with two maximalelements and the unique limit model;
(3) three-element chain, forming the Rudin | Keisler preorder,and the unique limit model.
Aforesaid situations also can be characterized in terms of hyper-graphs and can be represented by four-model hypergraphs.
Continuing analysis of cases in accordance with Theorem 1.1.13w.r.t. Rudin | Keisler preorder and distribution functions of num-ber of limit models for theories having n countable models, we con-struct n-model hypergraphs as above, allowing to characterize theo-ries T with I(T, ω) = n.
The process of construction of hypergraphs, preserving givennumber of pairwise non-isomorphic countable models, can be con-tinues for innite cardinalities. It is achieved by similar introductionin hypergraphs (M eq,H(Meq) of denable information on Rudin |Keisler preorders and on number of limit models. Thus, we havethe notion of λ-model hypergraph, allowing to characterize theoriesT with I(T, ω) = λ as the following generalization of Theorem 5.5.2.
Theorem 5.5.3. For any small theory T , the following condi-tions are equivalent:
(1) I(T, ω) = λ;(2) T has a λ-model hypergraph.
§ 5.6. Distributions of prime and limit modelsRecall that, in Chapter 3, a classication of elementary complete
theories with nite Rudin | Keisler preorders is represented, and allpossible Rudin | Keisler preorders in small theories are described.At present Subsection, we describe possibilities of distributions ofnumbers of limit (not necessary over xed types) models of smalltheories and, thus, represent structural possibilities for numbers ofcountable models of small theories. Below all considered theoriesare supposed to be small.
We dene analogues of countable categoricity and of Ehren-feuchtness for prime models over tuples and limit models.
A theory T is called p-categorical (respectively, l-categorical,p-Ehrenfeucht, and l-Ehrenfeucht) if Ip(T, ω) = 1 (respectively,Il(T ) = 1, 1 < Ip(T, ω) < ω, 1 < Il(T ) < ω).
Clearly, that T is p-categorical i T countably categorical, T isp-Ehrenfeucht i the structure RK(T ) nite and has at least twoelements, and T is p-Ehrenfeucht with 1 ≤ Il(T ) < ω i T is Ehren-feucht.
214
Notice, that just an absence of p-Ehrenfeuchtness allowed in theseries of papers to prove an absence of Ehrenfeucht theories in theclasses of superstable [109], 1-based [145], pseudo-superstable [197],supersimple [105] theories, as well as for theories without the strictorder property and interpreting innite sets of pairwise dierentconstants [193].
Theorem 3.5.3 shows that, for p-Ehrenfeucht theories, the num-ber of countable models is dened by the number of prime modelsover tuples and by the distribution function IL of numbers of limitmodels over types.
Consider the class of l-categorical theories. Clearly, all such the-ories are not p-categorical and the uniqueness of limit model impliesthat it is saturated. Therefore, a union of any elementary chain ofprime models over tuples is again a prime model over a tuple or formsa limit saturated model. Hence, by denition of saturated model,the theory is l-categorical i, for any elementary chain (Mai)i∈ω ofprime models over tuples ai, for which the union is a limit model,and for any tuple b ∈ Mai , every type q(x) ∈ S1(b) is realized insome model Maj , j ≥ i.
One more criterium of l-categoricity is based on the propertythat it is impossible to construct a limit model M as a union ofelementary chain of prime models Mqn over types qn, where qn ≤RKqn+1, n ∈ ω, for which there exists a type q with qn ≤RK q and q 6≤RKqn, n ∈ ω. An absence of M is implied by that saturated modelsare limit and realize q, while q is omitted in M. Hence, if suchlimit models don't exist then the l-categoricity is equivalent to theuniqueness up to isomorphism of non-prime countable model, inwhich all types of given theory are realized.
Thus, the following theorem, representing characterizations ofl-categoricity, holds.
Theorem 5.6.1. For any small, non-ω-categorical theory T , thefollowing conditions are equivalent:
(1) T is l-categorical;(2) any limit model of T is saturated;(3) for any elementary chain (Mi)i∈ω of prime models over tu-
ples with the union forming a limit model of T , and for any tupleb ∈ Mi, every type q(x) ∈ S1(b) is realized in some model Mj, j ≥ i;
(4) the union of any elementary chain of prime models Mqn overtypes qn, n ∈ ω, is a prime model over a type or the unique, up toisomorphism, limit model, in which all types of T are realized.
Notice, that all theories T with I(T, ω) = 3 are l-categorical.
215
Also every theory is l-categorical if isomorphism types of count-able models are dened by dimensions in ω ∪ ∞ and nite valuescorrespond to prime models over tuples. In particular, in accor-dance with [33], any ω1-categorical non-ω-categorical theory is l-categorical. Since l-categorical theories don't have non-one-element∼RK-classes, not equal to ≤RK-greatest ∼RK-classes (see Corollary1.1.10), the classes of theories with three countable models andof theories with dimensions represent two disjoint possibilities forRudin | Keisler preorders in l-categorical theories:
(1) a nite or countable Rudin | Keisler preorder with boundedlengths of ≤RK-chains of ∼RK-classes such that all ∼RK-classes, notequal to the greatest class, are one-element;
(2) a countable Rudin | Keisler preorder with unboundedlengths of≤RK-chains of∼RK-classes, where each class is one-elementand each element M ∈ RK(T ) is related by M ≤RK M′ with someelement M′ of every innite ≤RK-chain.
Notice also, that since all prime and countable saturated modelsare homogeneous, all l-categorical theories are almost homogeneous,i.e., each countable model, expanded by some nite set of constants,is homogeneous.
For further investigation of an arbitrary small theory T and thecounting the number of its countable models (in particular, for aninvestigation of l-Ehrenfeuchtness), we suppose that the preorderedset RK(T ) is countable, and, in view of Theorem 3.6.1, is upwarddirected and has the least element. Since the possible numbers λof limit models over types are described in Theorem 3.5.3 (λ ∈ω, ω1, 2ω), we consider the possibilities for numbers of limit modelsthat are not limit over types.
By denition, any limit model M is a union of elementary chain(Mi)i∈ω of prime models Mi over types qi. Moreover, same typescan repeat innitely many times(that corresponds to the limit modelover a type) or the chain (Mi)i∈ω contains an innite subchain ofpairwise non-isomorphic models Mij that are prime over pairwisedierent types qij , and this subchain can not be condensed to anelementary chain (Nk)k∈ω (where M =
⋃k∈ω
Nk) of prime models
over tuples such that the last chain contains innitely many primemodels over a xed type.
Any ≤RK-sequence, i.e., a sequence of nonprincipal types (qn)n∈ω
with qn ≤RK qn+1, n ∈ ω, can dene some number λ((qn)n∈ω) ∈ω, ω1, 2ω of limit models M =
⋃n∈ω
Mqn , where (Mqn)n∈ω is an
216
elementary chain of prime models over realizations of types qn. Fur-thermore, the dependence between numbers λ((qn)n∈ω) for dier-ent ≤RK-sequences (qn)n∈ω is conditioned by the following equiva-lence. Two ≤RK-sequences (qn)n∈ω and (q′n)n∈ω are called equivalent((qn)n∈ω ∼ (q′n)n∈ω) if there exists their common ≤RK-subsequence(q′′n)n∈ω.
Notice, that, generally speaking, there are no direct dependen-cies between numbers λ((qn)n∈ω) and λ((q′n)n∈ω) for equivalent≤RK-sequences (qn)n∈ω and (q′n)n∈ω, since limit models, dened by ≤RK-sequence (qn)n∈ω, can be not dened by ≤RK-sequence (q′n)n∈ω, andvice versa. Only the following relations hold:
(1) λ((qn)n∈ω) ≤ λ((q′n)n∈ω), where (q′n)n∈ω is a ≤RK-subsequence of (qn)n∈ω;
(2) λ((qn)n∈ω) = λ((q′n)n∈ω), where, for (qn)n∈ω and (q′n)n∈ω,there exist numbers k and m such that Mqk+n
' Mq′m+n, since
some n.Indeed, by denition, any limit model over≤RK-sequence (qn)n∈ω
is limit over any its subsequence (q′n)n∈ω, but not vice versa. Fur-thermore, if for sequences (qn)n∈ω and (q′n)n∈ω, there exist numbersk and m such that Mqk+n
' Mq′m+n, since some n, then any limit
model over (qn)n∈ω is limit over (q′n)n∈ω and vice versa, since anyelementary chain (Man)n∈ω of prime models over tuples can be ex-tended to an elementary chain (Mbn
)n∈ω, where Man = Mbn+l,
n ∈ ω, tp(bi) ≤RK tp(bi+1), i = 0, . . . , l − 1.The rst relation follows that the continuum of ≤RK-sequences
(qn)n∈ω, each of which is not a proper subsequence of other ones anddene a limit model, that is not limit over other sequences, impliescontinuum many pairwise non-isomorphic models.
The following concept allows to formulate a sucient conditionfor continual number of limit models.
A theory T is called l-rich if there exist nonprincipal types qn,n ∈ ω, over which prime models Mqn are pairwise non-isomorphicand such that, for any sequence (nk)k∈ω, there exists an elementarychain of models Mnk
, isomorphic to Mqnk, with the union forming
a limit model.Notice, that all types qn, in the denition of l-rich theory, are
RK-equivalent, and the correspondent ∼RK-class in RK(T ) is count-able.
217
Proposition 5.6.2. If T is a rich theory then Il(T ) = 2ω.P r o o f. By denition of l-rich theory, it suces to show, that
there are continuum many pairwise non-equivalent ≤RK-sequences,formed by types qn. We partition the set of naturals, indexingtypes qn, on two parts X0 and X1, and again each of these partson countably many parts Xj,k
i , i = 0, 1, j, k ∈ ω. Now, consid-ering sequences of types qn with indexes from X0,k0
i0, X1,k1
i1, . . .,
Xm,km
im, . . . such that dierent sequences of qn correspond to dier-
ent sets Xm,km
im, we have that these sequences are pairwise disjoint
and subindexes code all possible binary sequences. Thus, the num-ber of non-equivalent ≤RK-sequences of types qn is continual. ¤
Notice that, in view of Theorem 3.5.3 and Proposition 5.6.2,there are three cases of generation of continual number of limit mod-els over ∼RK-classes:
(1) continuum of limit models over some type in given∼RK-class;(2) continuum of limit models over some ≤RK-sequence of pair-
wise dierent types in given ∼RK-class;(3) continuum of limit models over pairwise dierent ≤RK-
sequences of pairwise dierent types in given ∼RK-class.In view of Morley Theorem [127], numbers of limit models over
given ≤RK-sequences of nonprincipal types q (consisting of same orpairwise dierent types) can be varied in the set ω ∪ ω, ω1, 2ω.Thus, the number Il(T ) is represented as the sum
∑q
ILq, where ILq
is the number of limit models related to ≤RK-sequence q and notrelated to restrictions of q, that used in count of Il(T ).
Similar Corollary 1.1.10 using the proof of Proposition 1.1.8, wehave the following:
Lemma 5.6.3. If q is a ≤RK-sequence of types qn, n ∈ ω, and(Mqn)n∈ω is an elementary chain such that any co-nite subchaindoesn't consist of pairwise isomorphic models, then ILq ≥ 1.
We say that a family Q of ≤RK-sequences q of types representsa ≤RK-sequence q′ of types if any limit model over q′ is limit oversome q ∈ Q.
In view of aforesaid, we have the following Theorem, that gener-alize Theorem 3.5.3 for theories with arbitrary Rudin | Keisler pre-orders and describes all possible distributions of numbers of count-
218
able models of theory in dependence on given sequences of nonprin-cipal types.
Theorem 5.6.4. Any small theory T satises the following con-ditions:
(a) the structure RK(T ) is upward directed and has the least ele-ment M0 (the isomorphism type of prime model of T ), IL(M0) = 0;
(b) if q is a ≤RK-sequence of nonprincipal types qn, n ∈ ω,such that each type q of T is related by q ≤RK qn for some n, thenthere exists a limit model over q; in particular, Il(T ) ≥ 1 and thecountable saturated model is limit over q, if q exists;
(c) if q is a ≤RK-sequence of types qn, n ∈ ω, and (Mqn)n∈ω isan elementary chain such that any co-nite subchain doesn't consistof pairwise isomorphic models, then there exists a limit model over q;
(d) if q′ = (q′n)n∈ω is a subsequence of ≤RK-sequence q, thenany limit model over q is limit over q′;
(e) if q = (qn)n∈ω and q′ = (q′n)n∈ω are ≤RK-sequences of typessuch that for some k, m ∈ ω, since some n, any types qk+n and q′m+nare related by Mqk+n
' Mq′m+n, then any model M is limit over q
i M is limit over q′.Moreover the following decomposition formula holds:
I(T, ω) = |RK(T )|+∑
q∈Q
ILq,
where ILq ∈ ω ∪ ω, ω1, 2ω is the number of limit models related tothe ≤RK-sequence q and not related to extensions and to restrictionsof q that used for the counting of all limit models, and the family Qof ≤RK-sequences of types represents all ≤RK-sequences, over whichlimit models exist.
In view of Theorem 5.6.4, nite values∑
q∈Q
ILq characterize the
class of l-Ehrenfeucht theories.By Theorem 5.6.4, similar to theories with nite Rudin | Keisler
preorders, small theories are classied by Rudin | Keisler preordersand distribution functions of numbers of limit (not necessary overxed types) models.
Recall, that Theorem 3.5.1 represents all possible realizations oftheories with nite Rudin | Keisler preorders w.r.t. these preordersand distribution functions of numbers of limit models, where valuesof functions belong to ω ∪ ω, 2ω.
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The following Theorem generalizes Theorem 3.5.1 for arbitraryRudin | Keisler preorders and correspondent distribution functionsof numbers of limit models, where values of functions belong toω ∪ ω, 2ω.
Theorem 5.6.5. Let 〈X,≤〉 be a nite or countable upward di-rected preordered set with a least element x0, f : Y → ω ∪ ω, 2ωbe a function with the set Y of all ≤0-sequences, i.e. of sequencesin X \ x0, forming all ≤-chains, and satisfying the following con-ditions:
(a) f(y) ≥ 1 if for any x ∈ X there exists some x′ in the sequencey such that x ≤ x′;
(b) f(y) ≥ 1 if any co-nite subsequence of y doesn't contain ofpairwise equal elements;
(c) f(y) ≤ f(y′) if y′ is a subsequence of y;(d) f(y) = f(y′) if y = (yn)n∈ω and y′ = (y′n)n∈ω are sequences
such that there exist some k,m ∈ ω for which yk+n = y′m+n sincesome n.
Then there exists a small theory T and an isomorphism
g : 〈X,≤〉 → RK(T )
such that any value f(y) is equal to the number of limit models over≤RK-sequence (qn)n∈ω, correspondent the ≤0-sequence y = (yn)n∈ω,where g(yn) is the isomorphism type of the model Mqn, n ∈ ω.
P r o o f consists of a modication of generic constructionsin the proofs of Theorems 3.4.1 and 3.6.1 satisfying the followingconditions:
(1) any element x ∈ X \x0 is represented by an 1-type qx thatdened by the innite color and unary predicate Px;
(2) every prime model over a tuple is prime or isomorphic toa model Mqx ;
(3) given ≤RK-interconnection between types is formed bya countable family of binary predicates;
(4) given distribution of numbers of limit models is dened bysequences of binary predicates, identied via calculus of identities(see Section 3.5) using corealization amalgams.
Since numbers of limit models may vary depending on sequencesof pairwise dierent types q = (qn)n∈ω, considering identities inSection 3.5 to obtain given number of limit models over q, we haveto use only identities connecting words of same lengths.
220
To obtain n countable models, it suces to use the followingsystem of identities:
n− 1 ≈ m,
m ≥ n, andn0n1 . . . ns ≈ ns . . . ns︸ ︷︷ ︸
s+1 times,
maxn0, n1, . . . , ns−1 < ns, reducing all sequences in ωω to n con-stant sequences.
Considering the system
n0n1 . . . ns ≈ ns . . . ns︸ ︷︷ ︸s+1 times
,
maxn0, n1, . . . , ns−1 < ns,
n0n1 . . . ns ≈ n0(n0 + 1) . . . (n0 + s),
n0 + s ≤ ns,
n0n1 . . . ns ≈ n0(n0 + 1) . . . (n0 + t) (n0 + t) . . . (n0 + t)︸ ︷︷ ︸s−t times
,
n0 + t = ns, t > 0, s > t, of identities, we get a theory with ωcountable models over q, where all sequences in ωω are reducible toconstant or diagonal sequences. ¤
The constructions of Chapter 4 allow to create realizations forthe proof of Theorem 5.6.5 in the class of stable theories.
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236
Index
(∗), 157(a0, an)-route, 38(n, p)-type, 39(K0;6), 55(T1;61) FESS
(T0;60)(T2; 62), 76
(T1;61) F(T0;60) (T2; 62), 72(FESS)n
i=1(Ti; 6i), 77A ↓r
Z B, 131, 148Ain, 154H(Meq), 200I(T, λ), 17Il(T ), 209Il(H(M)), 212Ip(T, ω), 211Lp, 123Q-cone
lower, 175upper, 175
Qn, 43R′′i -chain, 108, 197R′′i -extension, 108, 197R′ii-extension, 108, 197Rp
ψ(M), 46Rϕ, 41S(T ), 17S(∅), 17Sn(T ), 17Sn(∅), 17SI , 112S∅, 112Sp(T ), 39Sn,p(T ), 39T (A), 152T (BA′), 154
T (M), 152T 0, 111T 0/I, 113T ∗, 41T ∗(A), 154T ∗c (A), 159T f , 167T bp, 128T cpg, 173T eq, 199T nf , 143T f,P,R, 191T0, 83T1, 92, 188T1 FT0 T2, 72T2, 94, 193T3, 102T4, 104Ta, 96Tc(A), 159Tc(BA′), 160Tp, 40X ∗Z Y , 133, 169X ↓r
Z Y , 132, 148X ↓Z Y , 169X 6 N , 131, 147Γcpg, 173Γgen, 83Σ(M)eq, 199Σ1, 187Σ2, 99, 192ΣP , 181ΣP,R, 189Υ(A), 153
237
∗i∈IΓi, 137a |= p, 245(a), 1755Q(a), 1754(a), 1754Q(a), 175K, 119K∗
0, 119K0, 119M, 119K∗, 116K∗
0, 116Kf
0 , 173K0, 117M, 117〈M, Col〉, 34〈Cl1, Cl2〉, 73(
limn→∞
an
)M
, 33≤RK-sequence, 216≤RK-sequences equivalent, 217A, 64, 143H(M), 200S, 64X, 131, 169X
c, 148Φ(A), 64T0, 58A, 164M, 60Mn, 18−→Meq, 203⊆c, 79⊆cv, 118⊆fin, 129⊆c0 , 79⊆cc, 159ϕ-arc, 202
principal, 202ϕ-edge principal, 203c-(AP), 141c-AP, 80c-amalgam free, 79, 141
c-embedding, 80canonical, 152strong, 141
c-graph, 79J-closed, 141expanded, 139tandem without furcations, 152,
154c-graphs c-isomorphic, 80
canonically, 152c-isomorphism, 80
canonical, 152c-subgraph, 79
canonical, 154self-sucient, 139strong, 139
c0-subgraph, 78c1-embedding, 91, 187c1-isomorphism, 91c1-structure, 91, 187c1-structures c1-isomorphic, 187c2-embedding, 93, 192c2-structure, 93c3-AP, 101c3-amalgam free, 100c3-embedding, 101c3-graph, 98
closed, 100c3-graphs c3-isomorphic, 101c3-isomorphism, 101c3-subgraph, 99c3-tracing, 100c4-embedding, 103c4-structure
closed, 103c4-structures, 103c4-structures c4-isomorphic, 103ca-AP, 97ca-amalgam free, 97ca-embedding, 97ca-graph, 96
closed, 96ca-graphs ca-isomorphic, 97ca-isomorphism, 97ca-subgraph, 96
238
cc(Γ1), 79f -chain, 111f : Φ(A) → Ψ(B), 57f : A →c B, 80, 141f : A →c N , 80f : A →c1 B, 91, 187f : A →c1 N , 91f : A →c2 B, 93, 192f : A →c2 N , 93f : A →c3 B, 101f : A →c3 N , 101f : A →c4 B, 103f : A →c4 N , 103f : A →ca B, 97f : A →cc B, 164f : A →cc N , 165g : T (A) →c,f0 T (B), 152n-extension of model, 111nA, 73p-approximation, 122, 138, 155,
159, 181, 190p-type, 39p(M), 40, 46p(M), 33, 40p ≤RK q, 20p ∼RK q, 20p∞, 36, 44, 46q1 ∼p q2, 211r(A/B), 130, 145r(A/X), 130, 147rN , 129, 144w(p), 134y∆(·), 162y∆
p (·), 161, 181, 190T f,P , 182K∗, 79K∗
0, 80K∗
1, 91, 185, 187K∗
2, 93, 192K∗
3, 101K∗
4, 103Kf , 161Kf
0 , 161Kf
1 , 161
Kfp , 161
Kbp, 125Kbp
0 , 125Knf , 139Knf
0 , 139Kf,P,R, 190Kf,P,R
0 , 190Kf,P,R−1 , 189
Kf,P , 181Kf,P
0 , 181K0, 55, 81K1, 91, 187K2, 93, 192K3, 101K4, 103LM, 209M1 ∼RK M2, 21PM, 21Tbp
0 , 125Tnf
0 , 139, 142Tf,P,R
0 , 190Tf,P
0 , 182Γ≤2, 137Γbp, 122Γbp
c , 137Γnf , 137Γnf
0 , 137A 6c M, 139A 6cc M, 161A 6 M, 190A v B, 160A vM, 160A ⊆f B, 154A ⊆cc B, 159B ∗A C, 79, 97, 100, 126, 141, 165B ¹ A, 153C(a,Q), 42C(a,Γ), 42H(M), 204M∼ N , 25, 209Mn, 18Meq, 199M1 FM0 M2, 72
239
Mni , 18
Mp ≤RK Mq, 20Mp ∼RK Mq, 20Mb0 ∗q Mb1 , 95P(X), 200(CEP), 27(GPIP), 47(LPIP), 47AIEC(M1,M2), 31CEEL(T ), 210Comb(M1,M2), 30EEL(T ), 209HPKB-hypergraph, 204
homogeneous, 205saturated, 205universal, 205
IECTM, 31IEC(M1,M2), 31IL(M), 26ILk(T 0/I), 113Ind(β), 122RK(T ), 21SIp, 19TC(Γ), 51acl(A), 35ccl(A), 139cc-(AP), 165, 173cc-amalgam free, 165, 169cc-embedding, 164, 165
strong, 165cc-graph, 158
closed, 158self-sucient, 164
cc-graphs cc-isomorphic, 164cc-isomorphism, 165cc-subgraph, 159
self-sucient, 161strong, 162
cc(A), 96cfc(A), 158cf-closure, 158cv-amalgam, 118cv-graph, 118cv-subgraph, 118dcl(A), 35
iclM(S), 64kerM, 202maxA, 87, 184⊆S(A), 29( -AP), 119( -AP), 116AP, 127CLN-correlation, 186CL-correlation, 90Cl, 72ESSM-system, 75ESS-system, 76MI-principal, 75NPE-class, 72PE-class, 72facc-theory, 96
Amalgamcorealization, 95, 98free, 126, 133, 137
Amalgamation, 55Approach
semantic, 56syntactic, 56
Approximation best rational, 122Arc, 37, 203
principal, 202irreversible, 203
Aref'ev R., 14Ash C. J., 11
Baisalov E. R., 10, 11Baldwin J. T., 9, 1214, 56, 58,
62, 63, 67, 68, 72, 131,132, 134, 168
Baudisch A., 13, 14, 70, 73Belegradek O. V., 9, 45Benda M., 6, 9, 12Bonato A. C. J., 14Bouscaren E., 45Buechler S., 9, 10
Calvert W., 11Chain
elementary, 22
240
over a type, 22Chang C. C., 9, 16Chapuis O., 13Chatzidakis Z., 12, 14Class
generic, 57complete, 66dominating, 66hereditary, 65minimal hereditary, 65quantier free, 59self-sucient, 60witnessing on the nite clo-
sure property, 71having nite closures, 59of types
conal in a class of mod-els, 58
conal in a model, 58Closure
J-self-sucient, 143algebraic, 35denable, 35intrinsic, 64, 131, 148, 169route, 139self-sucient, 64, 164transitive, 51
Colorarc, 116of element, 34
Coloringϕ-ordered, 36of model, 34
n-inessential, 36almost inessential, 34inessential, 34innerly almost inessential,
34innerly inessential, 34
of universe, 34Combination
of models, 30almost inessential, 31inessential, 31
of theories, 30
almost inessential, 31inessential, 31
of typesinessential, 30
Componentconnected
of graph, 42principal, 207
of connectivity of graph, 42Cone, 175Conjecture
Lachlan, 6Pillay, 10Vaught, 10Zilber, 13
ConstructionHerwig, 121Hrushovski, 13, 121Jonsson | Frasse, 13
Copiesc-isomorphic, 80
canonically, 152c3-isomorphic, 101ca-isomorphic, 97cc-isomorphic, 165
Cycle, 38
de Bonis M. J., 13Detracing, 80Diagram, 56Digraph, 37
acyclic, 38powerful, 45prepowerful, 47
Edge, 203principal, 203
Ehrenfeucht A., 5, 11Element
xed, 170mobile, 170
Embeddingelementary, in coordination, 210of set strong, 62of type in model, 57
241
strong, 57of type in type, 57
strong, 57strong, 127
Ershov Yu. L., 79, 16Estimate majorizing, 75Evans D. M., 13, 14Examples Ehrenfeucht, 18Expansion
p∞-powerful, 176locally (∆, 1, Col)-denable, 176locally (∃, 1,Col)-denable, 176of generic class, 71
Extension hereditary in furcations,154
Family of ≤RK-sequences repre-senting≤RK-sequence, 218
Formula(q, p)-principal, 20copied over set, 53decomposition, 26, 110, 219principal, 37witnessing on non-symmetry
of relation SIp, 19witnessing on semi-isolation,
18Frasse R., 13Function
preprerank, 155, 159prerank, 122, 138, 181, 189
relative, 129, 145rank, 129, 144
relative, 129, 145Ryll-Nardzewski, 10sequence, 43
Furcation, 137compulsory, 156
external, 156lower, 136upper, 136
Furcation-tandem, 151Fusion
Hrushovski, 73of generic
classes, 71models, 71, 179theories, 71
Gavryushkin A. N., 12Generation of closure operation,
73Goncharov S. S., 6, 7, 9, 11, 16Goode J. B., 14Graph, 37
complete, 172, 182, 191, 203bipartite, 136
connected, 38directed, 37Herwig, 73principal, 203tandem without furcations, 152undirected, 37without furcations, 137
Group of automorphismstransitive, 43, 45
Harary F., 16Harizanov V. S., 11, 12Harnik V., 69Harrington L., 10, 69Hart B., 5, 9, 10Hasson A., 13, 70, 73Herwig B., 6, 7, 9, 1214, 121,
122Hils M., 13, 70, 73Hodges W., 9, 16Holland K., 13, 14, 70, 73Hrushovski E., 5, 6, 9, 10, 12, 13,
70, 73, 121Hypergraph, 200
λ-model, 214n-model, 214four-model, 214of prime models, 200
with a kernel-function anda principal binary struc-ture, 204
three-model, 213
Identity, 112
242
deducible from set, 112Ikeda K., 9, 11, 13, 14Index of number, 122Isomorphism types
domination-equivalent, 21Ivanov A. A., 13
Jonsson B., 13
Kechris A. S., 13Keisler H. J., 9, 16Kernel of model, 202Kernel-function, 202Kernel-undigraph
over a type, 203Khusainov B., 9, 11Kikyo H., 14Kim B., 6, 9, 12, 13Knight J. F., 11Knight R. W., 10, 114Koiran P., 13Kolesnikov A. S., 13Kueker D. W., 14
Lachlan A. H., 6, 9, 12, 13Lascar D., 6, 9, 12, 13Laskowski M. S., 5, 9, 10, 14Lattice, 122Lemma
-amalgamation, 119-amalgamation, 116
c3-amalgamation, 101ca-amalgamation, 97amalgamation, 80, 127, 141,
165, 173Lempp S., 11Length of route, 38Limit of sequence in a model, 33Loveys J., 9, 12Low L. F., 9, 10
Makkai M., 10Mal'tsev A. I., 7, 45Marcus L., 10Martin-Pizarro A., 13, 70, 73Mayer L., 9, 10
Mazucco M., 14Millar T., 9, 11, 12Miller A., 10Miller S., 11Model
(K0; 6)-generic, 55(T0; 6)-generic, 59(T0; 6)-homogeneous, 59(T0; 6)-universal, 59K∗
0-generic, 83K∗
1-generic, 92K∗
2-generic, 94K∗
3-generic, 102K∗
4-generic, 104colored, 34dominated, 20generic, 55having nite closures, 59limit, 208limit over a type, 24not exceed q under the Rudin |
Keisler preorder, 20universal, 29
Modelsdomination-equivalent, 20limit equivalent, 25, 209Rudin | Keisler equivalent,
20Morley M., 9, 110Morleyzation, 41Mustan T. G., 6, 9, 12
Nesin A., 13Newelski L., 10Nies A., 9, 11Number iterative, 73
Omarov B., 6, 9, 11Omarova G. A., 11Ore O., 16Ovchinnikova E. V., 16
Pair of types minimal, 61Palyutin E. A., 8, 9, 16, 29, 30Pantano M. E., 13Part
243
of cc-graph, 160of graph, 160
Peateld N., 13Peretyat'kin M. G., 6, 9, 11, 45Pillay A., 6, 914, 16, 38Pinus A. G., 7, 8Poizat B. P., 9, 13, 14, 16, 45, 72Pourmahdian M., 6, 9, 11, 12, 14Preorder
Rudin | Keisler, 20semi-isolation, 19
Principal of minimization of iter-ations, 75
ProblemLachlan, 6, 12
for simple theories, 12spectrum, 5, 9Vaught, 5
Property(P, Q)-intersection, 179-amalgamation, 119-amalgamation, 116
c-amalgamation, 80, 141c3-amalgamation, 101ca-amalgamation, 97t-amalgamation, 57t-covering, 61t-uniqueness, 57cc-amalgamation, 165, 173amalgamation, 55, 127consistent extension of chains
of models prime over tu-ples, 27
nite closure, 70hereditary, 161joint embedding, 59local realizability, 57pairwise ψ-intersection, 135pairwise intersection, 45
global, 47local, 47
uniform t-amalgamation, 67Pseudoplane free directed, 38Puninskaya V. A., 10Pustovoy N. V., 8
Rajani R., 14Recoloring admissible, 81Reed R., 9, 12Rosendal C., 13Route, 38
external, 78Rudin M. E., 13Rusaleev M. A., 7Ryaskin A. N., 7, 9Ryll-Nardzewski C., 10, 17
Sacks G. E., 9, 16Sae U., 6, 10, 12, 29, 30Self-suciency relation, 55Sequence
dening, 33irregular in a model, 33regular in a model, 33
SequencesI-equivalent, 113almost similar, 113equivalent, 39strongly I-equivalent, 113
SetJ-self-sucient, 143m-inconsistent, 53closed, 61, 131, 147, 168of arcs, 37of vertices, 37partially ordered directed
downward, 51upward, 51
self-sucient, 58, 168Sets independent, 131, 132, 148,
169Shelah S., 5, 6, 9, 10, 12, 13, 16,
134Shi N., 14, 56, 58, 62, 63, 67, 68,
131, 132, 134, 168Shore R. A., 9, 12Slaman T., 11Slope, 123Solecki S., 13Sort
home, 199
244
imaginary, 199Source, 89, 186Starchenko S. S., 9, 10, 29, 30Steel J., 10Structure binary principal, 203Structures c2-isomorphic, 192Subgraph
self-sucient, 125strong, 125
Subset strong, 58Substructure, 181, 190
self-sucient, 181, 190strong, 182, 190
Subtype strong, 57Successor, 89, 186Sudoplatov S. V., 16System of closures E-stepped spe-
cial, 76with minimality condition, 75
Tanovic P., 6, 912, 199Theorem
p-decomposition, 205decomposition, 207Ryll-Nardzewski, 10
Theories similar, 47Theory
(n, p)-invariant, 47(T0;6)-generic, 59∆-based, 29λ-stable, 32l-Ehrenfeucht, 214l-categorical, 214l-rich, 217p-Ehrenfeucht, 214p-categorical, 214K∗
0-generic, 83K∗
1-generic, 92K∗
2-generic, 94K∗
3-generic, 102K∗
4-generic, 104almost ∆-based, 29almost homogeneous, 216colored, 34Ehrenfeucht, 6, 11, 17
free acyclic, 96generic, 56, 59having the strict order prop-
erty, 41having the type
independence property, 175order property, 175strict order property, 175
of acyclic pairing function, 45reduced over a type, 40small, 18transitive, 36type
stable, 175unstable, 175
Thomas S., 10, 11Toalori C., 10Tracing, 80Tsuboi A., 6, 914Tuple
semi-isolating, 18Tuples
dependent, 53independent, 53
Type(p∞, y1)-principal, 86∆-based, 29p-principal, 40dened by a set of formulas,
29dominated, 19embeddable in a model, 57
strongly, 57embeddable in a type, 57
strongly, 57free, 45having innite own weight, 53having the independence prop-
erty, 174having the order property, 174having the strict order prop-
erty, 174isolated by a set of formulas,
29isomorphism, 184
245
not exceed q under the Rudin |Keisler preorder, 20
powerful, 18self-sucient, 58unstable, 174
Typesp-equivalent, 211domination-equivalent, 20realization-equivalent, 20Rudin | Keisler equivalent,
20
Undigraph, 37acyclic, 38complete, 203
Universe of c-graph, 79
Valeriote M., 9, 10Vaught R., 10, 11Verbovskiy V. V., 13, 14Vershik A. M., 14Vertex, 37
nal, 137initial, 137intermediate, 137
Vikent'ev A. A., 6, 7, 12Villaveces A., 14Vlasov D. Yu., 7Vostrikov A. S., 8
Wagner F. O., 9, 12, 14, 16, 84Weight
of arc, 122of type, 134
Witness on property, 71Woodrow R. E., 6, 10, 11, 41
Yoneda I., 14
Zambrano P., 14Ziegler M., 13, 14, 70, 73Zil'ber B. I., 13, 14
246