Existentially closed Leibniz algebras and an embedding theorem

In this paper we introduce the notion of existentially closed Leibniz algebras. Then we use HNN-extensions of Leibniz algebras in order to prove an embedding theorem.

An essential base of the central types of the convex 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.

An algebra of the central types of the mutually model-consistent fragments

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.

Small models of convex fragments of definable subsets

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.

Method of the rheostat for studying properties of fragments of theoretical sets

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.

Differential Galois cohomology and parameterized Picard–Vessiot extensions

Assuming that the differential field [Formula: see text] is differentially large, in the sense of [León Sánchez and Tressl, Differentially large fields, preprint (2020); arXiv:2005.00888 ], and “bounded” as a field, we prove that for any linear differential algebraic group [Formula: see text] over [Formula: see text], the differential Galois (or constrained) cohomology set [Formula: see text] is finite. This applies, among other things, to closed ordered differential fields in the sense of [Singer, The model theory of ordered differential fields, J. Symb. Logic 43(1) (1978) 82–91], and to closed[Formula: see text]-adic differential fields in the sense of [Tressl, The uniform companion for large differential fields of characteristic [Formula: see text], Trans. Amer. Math. Soc. 357(10) (2005) 3933–3951]. As an application, we prove a general existence result for parameterized Picard–Vessiot (PPV) extensions within certain families of fields; if [Formula: see text] is a field with two commuting derivations, and [Formula: see text] is a parameterized linear differential equation over [Formula: see text], and [Formula: see text] is “differentially large” and [Formula: see text] is bounded, and [Formula: see text] is existentially closed in [Formula: see text], then there is a PPV extension [Formula: see text] of [Formula: see text] for the equation such that [Formula: see text] is existentially closed in [Formula: see text]. For instance, it follows that if the [Formula: see text]-constants of a formally real differential field [Formula: see text] is a closed ordered[Formula: see text]-field, then for any homogeneous linear [Formula: see text]-equation over [Formula: see text] there exists a PPV extension that is formally real. Similar observations apply to [Formula: see text]-adic fields.

TWO PROPERTIES OF EXISTENTIALLY CLOSED COMPANIONS OF STRONGLY MINIMAL STRUCTURES

The proposed article studies some properties of existentially closed companions of strongly minimal structures. A criterion for the existential closedness of an arbitrary strongly minimal structure is found in the article and it is proved that the existentially closed companion of any strongly minimal structure is itself strongly minimal. It also follows from the resulting description that all existentially closed companions of a given strongly minimal structure form an axiomatizable class whose elementary theory is complete and model-complete and, therefore, coincides with its inductive and forcing companions. This is the reason for the importance of the work done and the high international significance of the theorems obtained in it. Another equally important consequence of this research is the discovery of an important subclass of strongly minimal theories. It should be noted that a complete description of this class of theories is an independent and extremely important task. It is known that natural numbers with the following relation are an example of a strongly minimal structure in which the existential type of zero is not minimal. Then the method used in the proof of the last theorem shows that the existentially closed companion of this structure are integers with the following relation.