Search

First of all, we have to note that in this article, we introduced the new concepts of relations between Jonsson theories in the class of cosemanticness for some considered Jonsson spectrum. All consideration of this new approach was done under sufficiently important class of Jonsson theories, which we called as normal Jonsson theories class. The main result, that we obtained, describes the model-theoretical properties of syntactical and semantical similarities inside the fixed cosemanticness class. For all new concepts in the article, we provided classical samples. The main result of this paper is considering normal Jonsson theories class by similarity to some fixed class of polygons (S-acts).

The issues of utilizing the central type to analyze the theoretical and model properties of the idea of heredity were examined in this research, taking into account both theories and the Jonsson spectrum. Finding solutions to issues related to the enriching language for the fixed Jonsson theory is associated with the problems of heredity of Jonsson theory. Another feature of Jonsson theories was described in the presented article. That is, the conclusion concerning J-non-multidimensional theories was presented in this study. The connection between J-P-stable theories and J-non-multidimensional theories was also characterired. In addition, the main result in the article was considered for the class of semantic pairs.

The work is related to the study of the model-theoretic properties of Jonsson theories, which, generally speaking, are not complete. In the article, on the Boolean of Jonsson subsets of the semantic model of some fixed Jonsson theory, the concept of the Jonsson closure operator Jcl was introduced, defining the J-pregeometry on these subsets, and some results were obtained describing this closure operator.

In this article, within the framework of the study of Jonsson theories, the model-theoretic properties of cosemanticness classes belonging to the factor set of the Jonsson spectrum of an existentially closed models’ subclass of some Jonsson theory in a fixed language were studied. Various results have been obtained. In particular, the properties of the cosemanticness of models and classes of models are considered; some results concerning the Jonsson equivalence in generalization for classes of existentially closed models are obtained; a criterion for the cosemanticness of J-classes in connection with their Kaiser hulls has been found.

The article is committed to the study of model-theoretic properties of stable hereditary Jonsson theories, wherein we consider Jonsson theories that retain jonssonnes for any permissible enrichment. The paper proves a generalization of stability that relates stability and classical stability for Jonsson spectrum. This paper introduces new concepts such as “existentially finite cover property” and “semantic pair”. The basic properties of e.f.c.p and semantic pairs in the class of stable perfect Jonsson spectrum are studied.

The study of syntactic and semantic properties of a first-order language, generally speaking, for incomplete theories, is one of the urgent problems of mathematical logic. In this article we study Jonsson theories, which are satisfied by most classical examples from algebra and which, generally speaking, are not complete. A new and relevant method for studying Jonson theories is to study these theories using the concepts of syntactic and semantic similarities. The most invariant concept is the concept of syntactic similarity of theories, because it preserves all the properties of the theories under consideration. The main result of this article is the fact that any perfect Jonson theory which are complete for existential sentences, is syntactically similar to some polygon theory (S-polygon, where S is a monoid). This result extends to the corresponding classes of Jonsson theories from the Jonsson spectrum of an arbitrary model of an arbitrary signature.

The article is devoted to the study of semantic Jonsson quasivarieties of universal unars and undirected graphs. The first section of the article consists of basic necessary concepts from Jonsson model theory. The following two sections are results of using new notions of semantic Jonsson quasivariety of Robinson unars JCU and semantic Jonsson quasivariety of Robinson undirected graphs JCG, its elementary theory and semantic model. In order to prove two main results of the paper, Robinson spectra RSp(JCU) and RSp(JCG) and their partition onto equivalence classes [∆]U and [∆]G by cosemanticness relation were considered. The main results are presented in the form of theorems 11 and 13 and imply following useful corollaries: countably categorical Robinson theories of unars are totally categorical; countably categorical Robinson theories of undirected graphs are totally categorical. The obtained results can be useful for continuation of the various Jonsson algebras’ research, particularly semantic Jonsson quasivariety of S-acts over cyclic monoid.

In this article, strongly minimal geometries of fragment hybrids are considered. In this article, a new concept was introduced as a family of Jonsson definable subsets of the semantic model of the Jonsson theory T, denoted by JDef(CT). The classes of the Robinson spectrum and the geometry of hybrids of central types of a fixed RSp(A) are considered. Using the construction of a central type for theories from the Robinson spectrum, we formulate and prove results for hybrids of Jonsson theories. A criterion for the uncountable categoricity of a hereditary hybrid of Jonsson theories is proved in the language of central types. The results obtained can be useful for continuing research on various Jonsson theories, in particular, for hybrids of Jonsson theories.

The paper defines a new class of algebras, the theory of which is a special case of Jonsson theories. This class applies to both varieties and Jonsson theories. The main results of this article are the following two results. In this article, an answer is obtained to the question of the equivalence of existential closure and algebraic closure of the model of the cosemantic class of a fixed spectrum of a Robinson hereditary variety. A criterion for strong minimality is obtained in the framework of the study of central types of central classes and fragments of a fixed spectrum.

Given article is devoted to the study of semantic Jonsson quasivariety of universal unars of signature containing only unary functional symbol. The first section of the article consists of basic necessary concepts. There were defined new notions of semantic Jonsson quasivariety of Robinson unars JCU, its elementary theory and semantic model. In order to prove the main result of the article, there were considered Robinson spectrum RSp(JCU) and its partition onto equivalence classes [∆] by cosemanticness relation. The characteristic features of such equivalence classes [∆]∈RSp(JCU) were analysed. The main result is the following theorem of the existence of: characteristic for every class [∆] the meaning of which is Robinson theories of unars; class [∆] for any arbitrary characteristic; criteria of equivalence of two classes [∆]1, [∆]2. The obtained results can be useful for continuation of the various Jonsson algebras’ research, particularly semantic Jonsson quasivariety of S-acts over cyclic monoid.

This article is devoted to the study of Jonsson quasivarieties in a signature enriched with new predicate and constant symbols. New concepts of semantic Jonsson quasivariety and fragment-conservativeness of the center of the Jonsson theory are introduced. The cosemanticness classes of the Jonsson spectrum constructed for a semantic Jonsson quasvariety are considered. In this case, the Kaiser hull of the semantic Jonsson quasivariety is assumed to be existentially prime. By constructing a central type for classes of theories from the Jonsson spectrum, the following results are formulated and proved. In the first main result, the necessary and sufficient condition is given for the center of the cosemanticness class of an existentially prime semantic Jonsson quasivariety to be λ-stable. The second result is the criterion for the center of the class of theories to be ω-categorical in the enriched language. The obtained theorems can be useful in continuing studies of various Jonsson algebras, in particular, Jonsson quasivarieties.

This article is devoted to the study of the forcing companions of the Jonsson AP-theories in the enriched signature. It is proved that the forcing companion of the theory does not change when expanding the theories under consideration, which have some properties, by adding new predicate and constant symbols to the language. The model-theoretic results obtained in this paper in general form are supported by examples from differential algebra. An approach in combining a Jonsson and non-Jonsson theories is demonstrated. In this paper, for the first time in the history of Model Theory. This will allow us to further develop the methods of research of Jonsson theories and expand the apparatus for studying incomplete theories.

The paper is connected with the study of Jonsson spectrum notion of the fixed class of models of Sacts signature, assuming a group as a monoid of S-acts. The Jonsson spectrum notion is effective when describing theoretical-model properties of algebras classes whose theories admit joint embedding and amalgam properties. It is usually sufficient to consider universal-existential sentences true on models of this class. Up to the present paper, the Jonsson spectrum has tended to deal only with Jonsson theories. The authors of this study define the positive Jonsson spectrum notion, the elements of which can be, non-Jonsson theories. This happens because in the definition of the existentially positive Mustafin theories considered in a given paper involve not only isomorphic embeddings, but also immersions. In this connection, immersions are considered in the definition of amalgam and joint embedding properties. The resulting theories do not necessarily have to be Jonsson. We can observe that the above approach to the Jonsson spectrum study proves to be justified because even in the case of a non-Jonsson theory there exists regular method for finding such Jonsson theory that satisfies previously known notions and results, but that will also be directly related to the existentially positive Mustafin theory in question.

The paper aims to study the model-theoretic properties of differentially closed fields of zero and positive characteristics in framework of study of Jonsson theories. The main attention is paid to the amalgam and joint embedding properties of DCF theory as specific features of Jonsson theories, namely, the implication of JEP from AP. The necessity is identified and justified by importance of information about the mentioned properties for certain theories to obtain their detailed model-theoretic description. At the same time, the current apparatus for studying incomplete theories (Jonsson theories are generally incomplete) is not sufficiently developed. The following results have been obtained: The subclasses of Jonsson theories are determined from the point of view of joint embedding and amalgam properties. Within the exploration of one of these classes, namely the AP-theories, that the theories of differential and differentially closed fields of characteristic 0, differentially perfect and differentially closed fields of fixed positive characteristic are shown to be Jonsson and perfect. Along with this, the theory of differential fields of positive characteristic is considered as an example of an AP-theory that is not Jonsson, but has the model companion, which is perfect Jonsson theory, and the sufficient condition for the theory of differential fields is formulated in the context of being Jonsson.

In this paper, new objects of research are identified, both from the standpoint of model theory and from the standpoint of universal algebra. Particularly, the Jonsson spectra of the Jonsson varieties and the Jonsson quasivarieties are considered. Basic concepts of 3 types of convexity are given: locally convex theory, ϕ(x)-convex theory, J-ϕ(x)-convex theory. Also, the inner and outer worlds of the model of the class of theories are considered. The main result is connected with the question of W. Forrest, which is related to the existential closedness of an algebraically closed variety. This article gives a sufficient condition for a positive answer to this question.

This article discusses the properties of atomic and prime models obtained with the some closure operator given on definable subsets of the semantic model some fixed Jonsson theory. The main result is to obtain the equivalence of the thus defined atomic and prime models, and this coincidence follows the assumption that there is some model with nice-defined properties.

In this paper, the model-theoretical properties of the algebra of central types of mutually model-consistent fragments are considered. Also, the connections between the center and the Jonsson theory in the permissible signature enrichment are shown, and within the framework of such enrichment, instead of some complete theory under consideration, we can obtain some complete 1-type, and we will call this type the central type, while the theories under consideration will be hereditary. Our work is divided into 3 sections: 1) the outer and inner worlds of the existentially closed model of the Jonsson theory (and the feature between these worlds is considered for two existentially closed models of this theory); 2) the λ- omparison of two existentially closed models (the Schroeder-Bernstein problem is adapted to the study of Jonsson theories in the form of a JSB-problem); 3) an algebra of central types (we carry over the results of Section 2 for the algebra (Fr(C), ×), where C is the semantic model of the theory T). Also in this article, the following new concepts have been introduced: the outer and inner worlds of one existentially closed model of the same theory (as well as the world of this model), a totally model-consistent Jonsson theory. The main result of our work shows that the properties of the algebra of Jonsson theories for the product of theories are used as an application to the central types of fixed enrichment. And it is easy to see from the definitions of the product of theories and hybrids that these concepts coincide if the product of two Jonsson theories gives a Jonsson theory.

In this paper, we consider the model-theoretical properties of the essential base of the central types of convex theory. Also shows the connection between the center and Jonsson theory in permissible enrichment signatures. Moreover, the theories under consideration are hereditary. This article is divided into 2 sections: 1) an essential types and an essential base of central types (in this case, the concepts of an essential type and an essential base are defined using the Rudin-Keisler order on the set of central types of some hereditary Jonsson theory in the permissible enrichment); 2) the atomicity and the primeness of ϕ(x)-sets. In this paper, new concepts are introduced: the ϕ(x)-Jonsson set, the AP A-set, the AP A-existentially closed model, the ϕ(x)-convex theory, the ϕ(x)-transcendental theory, the AP A-transcendental theory. One of the ideas of this article refers to the fact that in the work of Mustafin T.G. it was noticed that any universal model of a quasi-transcendental theory with a strong base is saturated, but we generalized this result taking into account that: the concept of quasi-transcendence will be replaced by the ϕ(x)-transcendence, where ϕ(x) defines some Jonsson set; and the notion of a strong base is replaced by the notion of an essential base, but in a permissible enrichment of the hereditary Jonsson theory. The main result of our work shows that the number of fragments obtained under a closure of an algebraic or definable type does not exceed the number of homogeneous models of a some Jonsson theory, which is obtained as a result of a permissible enrichment of the hereditary Jonsson theory.

This article discusses the problems of that part of Model Theory that studies the properties of countable models of inductive theories with additional properties, or, in other words, Jonsson theories. The characteristic features are analyzed on the basis of a review of works devoted to research in the field of the study of Jonsson theories and enough examples are given to conclude that the vast area of Jonsson theories is relevant to almost all branches of algebra. This article also discusses some combinations of Jonsson theories, presents the concepts of Jonsson theory, elementary theory, core Jonsson theories, as well as their combinations that admit a core model in the class of existentially closed models of this theory. The concepts of convexity, perfectness of theory semantic model, existentially closed model, algebraic primeness of model of the considered theory, as well as the criterion of perfection and the concept of rheostat are considered in this article. On the basis of the research carried out, the authors formulated and proved a theorem about the (∇1, ∇2) − cl coreness of the model for some perfect, convex, complete for existential sentences, existentially prime Jonsson theory T.

In this article discusses the model-theoretical properties of fragments of theoretical sets and the rheostat method. These two concepts: theoretical set and rheostat are new. The study of this topic in the framework of the study of Jonsson theories, the Jonsson spectrum, classes of existentially closed models of such fragments is a new promising class of problems and their solution is closely related to many problems that once defined the classical problems of model theory. The purpose of this article is to determine the rheostat of the transition from complete theory to Jonsson theory, which will be consistent with the corresponding concepts for any α and any α-Jonsson theory. For this we define a theoretical set. On the basis of research by the author formulated a model-theoretical definition of the concept of a rheostat in the transition from complete theories to ϕ(x)-theoretically convex Jonsson sets. Also was formulated an application of h-syntactic similarity to α-Jonsson theories.

When studying Jonsson theories, which are a wide subclass of inductive theories, it becomes necessary to study the so - called Jonsson sets. Similar problems are considered both in model theory and in universal algebra. This topic is related to the study of model - theoretical properties of positive fragments. These fragments are a definable closure of special subsets of the semantic model of a fixed Jonsson theory. In this article are considered model - theoretical properties of a new class of theories, namely ∆ - PJ theories of countable first - order language. These are theories that are obtained from ∆ - PJ theories by replacing in the definition of ∆- PJ theories of morphisms (∆ - continuities) with morphisms (∆ - immersions). A number of results were obtained, ∆ - PJ fragments, ∆ - PJ sets, hybrids of ∆ - PJ theories. All questions considered in this article are relevant in the study of Jonsson theories and their model classes.

In this paper, new concepts are defined in the framework of the study of Jonsson spectrum. We consider a spectrum with respect to the concept of cosemanticness, which is a generalization of elementary equivalence in the class of inductive, generally speaking, incomplete theories. Also, with the help of Jonsson spectrum, the actual directions of the study of Jonsson theories and their model classes are determined, namely, the study of classical questions of model theory, such as the completeness, model completeness, model companion of within the framework of the above conditions, which define a fairly wide subclass of inductive theories, and which Jonsson theories. Therefore, in studying the model - theoretical properties of Jonsson spectrum, we need to clarify the definition of those concepts that naturally arise when we move from the concept of elementary equivalence to the concept of cosemanticness, moreover, both theories and models. Some model - theoretical properties of the Jonsson spectrum are considered. When considering the Jonsson spectrum, all the tasks that are posed in this article make sense and their solution can be useful for solving related problems, because this problem is actively studied in the field of Jonsson theories.

The present paper concerns some properties of the so-called small models, generally speaking, not necessarily complete theories and their relationship with each other. In the well-known paper [1], R. Vaught have proved the fundamental theorem-criterion on the behavior of countable prime and atomic models for complete theories in countable language. The essence of this criterionis that in a complete theory any countable prime model is at the same time an atomic model of this theory. The result obtained in this paper is related to the classical problem of Vaught about countably prime models of complete theories but in more general formulation of the notion of countable atomicity. The main result of this paper is that it focuses on the syntactic properties on special subsets of a fragment of the semantic model the specific Jonsson theory. The concept of the so-called model-theoretic «rheostat» was also used to obtain results related to the refinement of the concept of atomicity in the framework of Jonsson theories.

In given work are considered model - theoretical properties of companions of (n1; n2) - Jonsson theory. Also were considered a communications between center and (n1; n2) - Jonsson theory. Herewith considered theories is perfect in the sense of the existence of appropriate model companion. In given article introduced new concepts: (n1; n2) - Jonsson theory, D - model companion. New results are shown with respect to model companions of - Jonsson theory and 1 - perfect 1 - Jonsson theory.

This article presents the result related to the model - theoretic properties of special subsets of the semantic model of some fixed Jonsson theory. The specialty of these sets is due to their definability and closure. Further, we consider fragments of these sets and create a hybrid of these fragments from them. The theory is subject to conditions of strongly convexity and existentially coreness. As a result, the class of existentially closed and algebraically prime models in its non - empty intersection contains a core model. By module of the above conditions, the hybrid of the considered fragments has a model that contains a special core subset, definable closure of which gives a certain existentially closed model, which is an algebraic prime model of the theory under consideration.

This article presents the result related to the model - theoretic properties of special subsets of the semantic model of some fixed Jonsson theory. The specialty of these sets is due to their definability and closure. Further, we consider fragments of these sets and create a hybrid of these fragments from them. The theory is subject to conditions of strongly convexity and existentially coreness. As a result, the class of existentially closed and algebraically prime models in its non - empty intersection contains a core model. By module of the above conditions, the hybrid of the considered fragments has a model that contains a special core subset, definable closure of which gives a certain existentially closed model, which is an algebraic prime model of the theory under consideration.

In this paper some properties of small models, generally speaking, not necessarily complete theories and their relationship with each other were considered. Under small models we will understand some modifications of the concepts of countable atomic and prime models. These models were defined in the study of countable models of complete theories. Studies were conducted by analogy with the classic result of R. Vaught on countable - prime models of complete theories, but by other technical means. This work is oriented on the syntactic properties of special subsets of the semantic model of some Jonsson theory. A new concept was also introduced, as a model - theoretic «rheostat», in order to obtain results related to the refinement of concept of atomicity within the framework of Jonsson theories. Thus, the main purpose of this article is to formally define this «rheostat» and to obtain on the basis of this concept the results having a relation to the refinement of the concept of atomicity in the frame of Jonsson theories.

Our attention in given article will be paid to the study of model - theoretic properties of hereditary Jonsson theories, while we consider Jonsson theories that retain jonsonness under any admissible enrichment. In given paper new concepts of ¾essential type¿, ¾essential geometric base¿ are introduced, the orbital types and strongly minimal sets within the framework of special subsets of the semantic model, on which a closure operator is given, defining the special geometry of Jonsson are considered. The results for the J-strongly minimal types of the semantic model in the case, when these sets are separated from the orbits of the central types of Jonsson hereditary theories are also obtained.

In this paper, the central types of convex fragments of the perfect Jonsson theory are considered. The main goal of this paper is to redefine the A.D. Taimanov’s question for complete theories in the sphere of Jonsson fragments of formula -definable subsets of a Jonsson fixed theory’s semantic model. Also the relationships between the center and Jonsson theory in the permissible enrichment of signature are considered. Herewith, the considered theories are hereditary. It is also assumed that an algebraic closure coincides with a definable closure. Within the framework of the above restrictions for the considered Jonsson theory, results on the existence of core models for such theories were obtained.

This work is an introduction to the study of the properties of a new concept, as a hybrid of Jonsson theories. We define the basic concepts and framework for studying the model-theoretic properties of these concepts. The main goal of this paper is to study the model-theoretic properties of companions of hybrids of Jonsson theories. The main objects of study are the Jonsson hybrids and their classes of models. In this paper, the main task is to consider the various links between such theories. In order to understand more deeply these connections and ultimately the connection with the primary theory itself, special algebraic constructions of semantic models of the considered fragments were identified and on this basis hybrids of these fragments were determined. In this paper, such algebraic constructions are called semantic hybrids.

In the framework of the classification of the Jonsson theories concept of interpretability and admissibility in the language of the semantic triple of the Jonsson theory was considered. A description of the syntactic and semantic similarity of perfect fragments of Jonsson subsets of the semantic model of the existential - prime convex Jonsson theory was obtained. Some model - theoretic properties for Jonsson’s theories are considered. Such theories, as group theory, the theory of Abelian groups, the theory of Boolean algebra, the theory of ordered sets, the theory of polygons, and many others satisfies Jonsson’s properties.

One of the interesting achievements among investigations from modern model theory is the implement of local properties of the geometry of strongly minimal sets. In E. Hrushovski have proved remarkable results under this ideas and this one had impacted an essential infuence for development of methods and ideas of research for global properties of structures.These new model - theoretical features and approvals play an important role in E. Hrushovski’s proof of the Mordell-Lang Conjecture for function fields. In this article, we are trying to redefine the basic concepts of the above mentioned ideas on the formul subsets of some extentional - closed model for some fixed Jonsson theory. With the help of new concepts in the frame of Jonssoness features, pregeometry is given on all subsets of Jonsson theory’s semantic model. Minimal structures and, correspondingly, pregeometry and geometry of minimal structures are determined. We consider the concepts of dimension, independence, and basis in the Jonsson strongly minimal structures for Jonsson theories.

The concept of independence plays a very important role in Model Theory for classification of a fixed complete theory. In this paper, we study the Jonsson theories, which, generally speaking, are not complete. For such theories, the concept of forking is introduced axiomatically in the framework of the study of the Jonsson subsets of the semantic model of this theory. Equivalence of forking by Shelah, by Laskar-Poizat and an axiomatically given forking for existential types over subsets of the semantic model of the Jonsson theory is given. Further, as and for complete theories, independence is defined through the notion of non - forking.

In the present paper we give a criterion of the cosemanticness relative to the Jonsson spectrum of the model in the class of Abelian groups with a distinguished predicate. This paper is devoted to the study of model - theoretic questions of Abelian groups in the frame of the study of Jonsson theories. Indeed, the paper shows that Abelian groups with the additional condition of the distinguished predicate satisfy conditions of Jonssonness and also the perfectness in the sense of Jonsson theory. It is well known that classical examples from algebra such as fields of fixed characteristic, groups, abelian groups, different classes of rings, Boolean algebras, polygons are examples of algebras whose theories satisfy conditions of Jonssonness. The study of the model - theoretic properties of Jonsson theories in the class of abelian groups is a very urgent problem both in the Model Theory itself and in an universal algebra. The Jonsson theories form a rather wide subclass of the class of all inductive theories. But considered Jonsson theories in general are not complete. The classical Model Theory mainly deals with complete theories and in case of the study of Jonsson theories, there is a deficit of a technical apparatus, which at the present time is developed for studying the modeltheoretic properties of complete theories. Therefore, the finding of analogues of such technique for the study of Jonsson theories has practical significance in thegiven research topic. In this paper the signature for one - place predicate was extended. The elements realizing this predicate form an existentially closed submodel of the considering Jonsson theory’s some model. In the final analysis, we obtain the main result of this article as a refinement of the well known W.Szmielew’s theorem on the elementary classification of Abelian groups in the frame of the study of Jonsson theories, thereby the generalization of the well known question of elementary pairs for complete theories was obtained. Also we obtained the Jonsson analogue for the joint embeddability of two models, or in another way the Schro¨der-Bernstein properties in the frame of the study of the Johnson pairs of Abelian groups’ theory.

This work is related to the concepts of admissibility, interpretability, syntactic similarity and semantic similarity of Jonsson’s theories.This new concept generalize the notion of syntactic and semantic similarities for Jonsson theories. In the frame of study of preservation of the definable formulas subsets of Jonsson theory’s semantic model regarding to considering morphisms the concepts of admissibility, interpretability, domination and similarity play a very important role. As an example, we consider the theory of Boolean algebras. These concepts are closely related to the concept of admissibility by one theory on the other. The interest of this paper is that the examples examined show the fact that admissibility, interpretability and similarity can be considered on algebras of different signatures.In this work the main definitions of notions and the basic directions of further researches are studied.

This article is concerned with the enrichment of the signature. In own time, when studying the stability of the theory and the concept of an elementary pair of models, Mustafin T.G. had noticed that these things are related to each other and he introduced the concept T ∗-stability [1]. In fact, some enrichment of the signature is considered. Generally speaking, the theories obtained in the extended language are incomplete, therefore, the number of such completions of these theories is sought. This number also determines stability in the sense of T ∗-stability. It was noted by E.A.Palyutin in [2] that the concept of T ∗-stability is not invariant with respect to definability of type. But we know that in the classical sense of S.Sellach the stability of the theory is invariant with respect to the definability of type. Therefore Palyutin E.A. had introduced the concept E ∗-stability, which preserved the definability of type...

This article, in its content, refers to the study of the theoretical - model properties of the Jonsson theories. A new approach to this study is proposed. As a new notion, the idea of a central - orbital type is used. When studying Jonsson theories are taken into account the following facts:Syntactic, concerning the Jonsson theories and the Jonsson subsets of the semantic model of the Jonsson theory under consideration.In addition, a special role in the syntactic sense is played by the enrichment of the signature associated with the given Jonsson sets. The semantic aspect of the issues under consideration is primarily concerned with the notion of convexity, strong convexity, and existential primeness.Although these definitions are related to theory, in fact we are dealing with different types of models that are existentially closed.

The main results of the article are for a new class of theories, namely existential prime strongly convex Jonsson theories. This class is quite broad in terms of algebra, for example it includes the class of all Abelian groups and groups. This article examines the issues relating to the following subjects. The language on considered a signature adds a new predicate symbol which reflects the presence of the Jonsson set. The concept of Jonsson sets in Jonsson theory is a generalization of the concept of the dimension of the linear space. T.G. Mustafin in due time, introduced and proved the basic properties of the syntactic and semantic similarity. In this paper, in the extended language we have similare to the results for the considered theories. In this direction, the main results of the work are the following results: The coincidence of P–stability for the prototype and its central-type center. The equivalence of syntactic similarity of existentially EP SCJ compleate theories and syntactical similarity of their centers was consedered. From this it can be seen a lot of useful facts. In particular semantic similarity. As well as a list of semantic properties, which are stored at the semantic similarity. For example, the semantic properties that invariant properties of the first order applies Morley rank of the central type.

In this article we consider the properties of central types for the existentially prime strongly convex Jonsson theories in some extension. This class of theories is a subclass of a broad class of Jonsson theories. In particular, the Jonsson theories include the class of all fields of a fixed characteristic. In the given work, problems related to the classical problems of the general Model Theory concerning the following topics were considered. First of all, we note the values of enrichment. Using the one-place predicate, the Jonsson subset is singled out and the concepts of P -stability and various kinds of similarities are considered for the Jonsson completion. The following results were obtained:Coincidence of P - stability for a prototype of the central type and its center. Equivalence of syntactic similarity of companions of fragments of Jonsson enrichment and syntactic similarity of their centers.The above notion of stability has an applied value for studying the properties of the central types in this enrichment. In the second place, it is necessary to note the significance of the concept of the central type in this enrichment.The very idea of a central type presupposes an additional description of the properties of incomplete Jonsson theories by means of central completion. The Jonsson subsets of the semantic model of the existentially prime convex Jonsson theory have good theoretic-model properties. This concerns the Morley rank and it is preserved in the syntactic and semantic similarity of the above theories.

This article introduced and discussed the concepts of minimal Jonsson sets and respectively strongly minimal Jonsson sets. On this basis, it introduces the concept of the independence of special subsets of existentially closed submodel of the semantic model. The notion of independence leads to the concept of basis and then we have an analogue of the Jonsson theorem on uncountable categorical. The concept of strongly minimal, as for sets and so for theories played a decisive role in obtaining results on the description of uncountable - categorical theories. It is well known that Jonsson Theories are a natural subclass of the broad class of theories, as a class of inductive theories. As is known, the basic examples theories of algebra are examples of inductive theories, and they tend to represent an example of incomplete theories. This modern apparatus of Model Theory developed mainly for complete theories, so nowadays technique studying incomplete theories noticeable poorer than for complete theories. Thus, all of the above says that the study of model - theoretic properties Jonsson theories is an actual problem. This article describes the basic properties of the Morley rank over Jonsson subsets of semantic model for some Jonsson theory.

Actually, we study the connections of the ∆-PJ-theories with their centers. The properties of various companions of some ∆-PJ-theories and their connection with this theory are considered. Also the similarity of the central types of ∆-PJ-theories in the enriched language is considered. In the class of perfect ∆-PJ-theories the conditions of coincidence of algebraic primeness with some sort of atomic models are found. In the class of ∆-PJ-theories, the concepts of syntactic and semantic similarities are introduced and the results on the relationship of these similarities in this class with their centers are obtained.

This article introduced and considered the Johnson sets and their fragments. And respectively was considered strongly minimal Jonsson sets. On this basis, introduced the concept of the independence of special subsets of existentially closed submodel of the semantic model. The notion of independence leads to the concept of base and further we develop technique for Jonssonien analog of theorem on uncountable categoricity.

Actually, we study the connections of the Δ-PM-theories with their centers in the enrich signature. The properties of various companions of some Δ-PM-theories and their connection with this theory are considered on the language of the central types of positive Jonsson theory.