Basic and derived objects for ordered and generic structures, as well as their elementary theories
kulpesh@mail.ru
GOAL
Project Goal
To study the model-theoretic properties of ordered and generic structures and their elementary theories, and to describe and classify them on the basis of derived objects.
ABOUT THE PROJECT
Abstract
The project belongs to one of the important branches of mathematical logic — model theory, which studies the relationship between a formal language and its interpretations, or models. The project is aimed at solving the problem of classifying algebraic systems and their elementary theories both in general, with respect to basic characteristics, and for a number of main classes of algebraic systems, such as various kinds of weakly o-minimal structures, universal algebras and groups. The project envisages: obtaining a classification of algebraic systems based on their derived structures; classifying the algebras of distributions of formulas of a given theory; assessing the expressive power of generic classes and their influence on the structural properties of theories; and classifying countable models of complete theories. The research topics of this project are currently relevant to the development of the theory of algebraic systems and are being intensively studied at the world's leading mathematical centers. The national significance of the project lies in training highly qualified researchers in mathematical logic. Its international significance lies in obtaining new scientific results that will be submitted for publication in international peer-reviewed journals. The results obtained in this project will contribute to the further development of general model theory, which will undoubtedly influence the development of science in this field. The expected social and economic effect will consist in increasing the country's intellectual potential, training new researchers and the possible further application of the results to applied fields such as the general theory of relational databases and formal methods of software development.
RESULTS
Scientific Achievements
A theorem was proven that confirms Vaught's conjecture for weakly o-minimal theories of finite convexity rank: Let T be a weakly o-minimal theory of finite convexity rank in a countable language. Then exactly one of the following holds: (1) T is countably categorical; (2) T is Ehrenfeucht, i.e. it has k countable models, where 3≤k<ω; (3) T has countably many countable models; (4) T has continuum many countable models.
INTELLECTUAL PROPERTY
Legal Protection
All results are published in open sources and do not require patenting.
IMPLEMENTATION
Commercialization and Practical Application
The research was purely theoretical and fundamental. The results can be used for further research in model theory and in other branches of mathematics. They are also applicable in such areas of computer science as the general theory of relational databases, hybrid systems and formal methods for studying information systems.
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