It is well known that amalgamation property and the joint embedding property are independent of each other. In the case when both of these properties take place within studying of inductive theories, we have a lot of classical examples from algebra that satisfy these conditions. The authors of this article raise the question of studying classical issues of Model Theory such as categoricity, completeness, syntactic and semantic similarities within the framework of the above conditions, which define a rather wide subclass of inductive theories and which are called Jonsson theories. Since any theory can be transformed into a Jonsson theory in some enrichment, in our opinion it would be interesting to study some special invariant of any model of an arbitrary signature whose essence is a Jonsson spectrum of a class of models of an arbitrary language. This article considers the issues of countable and uncountable categoricity of a class of Jonsson spectrum’s cosemantic theories of an arbitrary signature’s the meaning of elementary equivalence. We consider this spectrum with respect to the concept of cosemanticness, which is a generalization of elementarily equivalence in the class of inductive, generally speaking, incomplete theories. As a consequence of an obtained facts, a refinement is made of the well-known results of M. Morley, D. Saracino, and P. Lindström in the framework of studying the above concepts. Also we have introduced the notion of hybrids of classes of theories by cosemanticness and two notions of similarities which preserve many model-theoretical properties. The main result of this article is a criterium of sintactically similarity for hybrids of classes from Jonsson spectrum (Theorem 21) and as applying of this result we have obtained the existence theorem of some syntactically similar algebra for any such hybrid which we considered (Theorem 22).

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.

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-comparison 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 (F r(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.

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 Delta - PJ theories of countable first-order language. These are theories that are obtained from Delta - PJ theories by replacing in the definition of Delta - PJ theories of morphisms (Delta-continuities) with morphisms (Delta-immersions). A number of results were obtained, Delta - PJ fragments, Delta - PJ sets, hybrids of Delta - PJ theories. All questions considered in this article are relevant in the study of Jonsson theories and their model classes.

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.

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.