scholarly journals EXISTENTIALLY CLOSED MODELS IN THE FRAMEWORK OF ARITHMETIC

2016 ◽  
Vol 81 (2) ◽  
pp. 774-788 ◽  
Author(s):  
ZOFIA ADAMOWICZ ◽  
ANDRÉS CORDÓN-FRANCO ◽  
F. FÉLIX LARA-MARTÍN
Keyword(s):  
Closed Model ◽  
Existentially Closed ◽  
Quantifier Complexity

AbstractWe prove that the standard cut is definable in each existentially closed model of IΔ0 + exp by a (parameter free) П1–formula. This definition is optimal with respect to quantifier complexity and allows us to improve some previously known results on existentially closed models of fragments of arithmetic.

scholarly journals Small models of convex fragments of definable subsets

2020 ◽  
Vol 100 (4) ◽  
pp. 160-167
Author(s):  
Aibat Yeshkeyev ◽  
◽  
N.V. Popova
Keyword(s):  
Model Theory ◽  
Elementary Theory ◽  
Semantic Model ◽  
Closed Model ◽  
Existentially Closed ◽  
Existential Sentences ◽  
Characteristic Features ◽  
Almost All ◽  
Small Models ◽  
Vast Area

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.

scholarly journals Existentially closed models of the theory of artinian local rings

1999 ◽  
Vol 64 (2) ◽  
pp. 825-845 ◽  
Author(s):  
Hans Schoutens
Keyword(s):  
Residue Field ◽  
Local Rings ◽  
Model Companion ◽  
Closed Model ◽  
Gorenstein Rings ◽  
Existentially Closed ◽  
Finite Set ◽  
Residue Fields

AbstractThe class of all Artinian local rings of length at most l is ∀2-elementary, axiomatised by a finite set of axioms τtl. We show that its existentially closed models are Gorenstein. of length exactly l and their residue fields are algebraically closed, and, conversely, every existentially closed model is of this form. The theory oτl of all Artinian local Gorenstein rings of length l with algebraically closed residue field is model complete and the theory τtl is companionable, with model-companion oτl.

scholarly journals An essential base of the central types of the convex theory

2021 ◽  
Vol 101 (1) ◽  
pp. 119-126
Author(s):  
A.R. Yeshkeyev ◽  
◽  
M.T. Omarova ◽  
Keyword(s):  
Universal Model ◽  
Closed Model ◽  
Strong Base ◽  
Existentially Closed ◽  
New Concepts ◽  
Homogeneous Models ◽  
Definable Type ◽  
Transcendental 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.

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

2021 ◽  
Vol 101 (1) ◽  
pp. 111-118
Author(s):  
A.R. Yeshkeyev ◽  
◽  
N.M. Mussina ◽  
Keyword(s):  
Complete Theory ◽  
Semantic Model ◽  
Closed Model ◽  
Central Type ◽  
Bernstein Problem ◽  
Existentially Closed ◽  
New Concepts ◽  
The World ◽  
Carry Over

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.

Existentially closed central extensions of locally finite p-groups

1986 ◽  
Vol 100 (2) ◽  
pp. 281-301 ◽  
Author(s):  
Felix Leinen ◽  
Richard E. Phillips
Keyword(s):  
Central Extensions ◽  
Locally Finite ◽  
Existentially Closed ◽  
Image Position

Throughout, p will be a fixed prime, and will denote the class of all locally finite p-groups. For a fixed Abelian p-group A, we letwhere ζ(P) denotes the centre of P. Notice that A is not a class in the usual group-theoretic sense, since it is not closed under isomorphisms.

Finitely generic models of TUH, for certain model companionable theories T

1985 ◽  
Vol 50 (3) ◽  
pp. 604-610
Author(s):  
Francoise Point
Keyword(s):  
Discriminator Variety ◽  
Generic Models ◽  
Elementary Class ◽  
Ordered Groups ◽  
Existentially Closed ◽  
Closed Element ◽  
Starting Point ◽  
Joint Embedding ◽  
Joint Embedding Property ◽  
Generic Elements

The starting point of this work was Saracino and Wood's description of the finitely generic abelian ordered groups [S-W].We generalize the result of Saracino and Wood to a class ∑UH of subdirect products of substructures of elements of a class ∑, which has some relationships with the discriminator variety V(∑t) generated by ∑. More precisely, let ∑ be an elementary class of L-algebras with theory T. Burris and Werner have shown that if ∑ has a model companion then the existentially closed models in the discriminator variety V(∑t) form an elementary class which they have axiomatized. In general it is not the case that the existentially closed elements of ∑UH form an elementary class. For instance, take for ∑ the class ∑0 of linearly ordered abelian groups (see [G-P]).We determine the finitely generic elements of ∑UH via the three following conditions on T:(1) There is an open L-formula which says in any element of ∑UH that the complement of equalizers do not overlap.(2) There is an existentially closed element of ∑UH which is an L-reduct of an element of V(∑t) and whose L-extensions respect the relationships between the complements of the equalizers.(3) For any models A, B of T, there exists a model C of TUH such that A and B embed in C.(Condition (3) is weaker then “T has the joint embedding property”. It is satisfied for example if every model of T has a one-element substructure. Condition (3) implies that ∑UH has the joint embedding property and therefore that the class of finitely generic elements of ∑UH is complete.)

New Tools for Cyber Security Using Blockchain Technology and Avatar-Based Management Technique

2019 ◽  
pp. 105-122 ◽  
Author(s):  
Vardan Mkrttchian ◽  
Leyla Ayvarovna Gamidullaeva ◽  
Yulia Vertakova ◽  
Svetlana Panasenko
Keyword(s):  
Cyber Security ◽  
Digital Economy ◽  
Management Technique ◽  
Closed Model ◽  
Security Issues ◽  
Blockchain Technology ◽  
Current State ◽  
Management Techniques ◽  
Visualization Techniques ◽  
Development Prospects

The chapter introduces the perspectives on the use of avatar-based management techniques for designing new tools to improve blockchain as technology for cyber security issues. The purpose of this chapter was to develop an avatar-based closed model with strong empirical grounding that provides a uniform platform to address issues in different areas of digital economy and creating new tools to improve blockchain technology using the intelligent visualization techniques. The authors show the essence, dignity, current state, and development prospects of avatar-based management using blockchain technology for improving implementation of economic solutions in the digital economy of Russia.

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