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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. more...

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. more...

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. more...

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. more...

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. more...

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. more...

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. more...

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. more...

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. more...

The article concerns the description the new concept as core of Jonsson theories, also their combinations, which admit a core model in the class of existentially closed models of this theory. Along with core the property of an existentially algebraically prime theory is considered as an additional property to core Jonsson theory. This article also discusses some combinations of Johnson’s theories, where the authors tried to transfer some results from [1] to Johnson’s theories that satisfy the definition of core or EAP, or their combinations. From the definition of the core and the existentially algebraic primeness of Johnson theory, it can be noted that the core model from [1] in the framework of the study of any Johnson theory will be a unique and rigidly embedded model of the theory were considered. And thus, such a solution to the problem with respect to core models is considered for the first time. more...

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. more...

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. more...

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. more...

In this article the model - theoretic properties of special formula subsets of the semantic model of some fixed Jonsson theory are considered. The main purpose of this paper is the study of concepts of models’ primeness and atomness in the study of inductive theories which admit the property of joint embedding and amalgama property. For this purpose is determined special sets, each element of which realise some type which is which is the main type in the sense of an existential formulas. Definable closures of such sets form an existential closed model. The main result obtained in this paper describes the properties of atomic and prime sets regarding strongly convex Jonsson theories. more...

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. more...

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. more...

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. more...

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. more...

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. more...

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. more...

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. more...

This article is devoted to studying of the properties of model - theoretic concepts of a fragment of the Jonsson sets and their application to the lattices of existential formulas. The concept of the Jonsson set allocates as a special subset of the semantic models for the considered the Jonsson theory. Next, we present some model - theoretic properties of the fragments of the considered Jonsson sets that were previously considered for the Jonsson theories. These properties describe the relationship between the lattices of the existential formulas and a center of the considered fragment. more...