arXiv subjects
Publications and source records attributed to Sergey V. Sudoplatov.
We study possibilities of preservations for properties, their links and related connections between semantic and syntactic ones, both in general and as characterizations for subalgebras, congruence relations, Henkin construction, Tarski-Vaught test, some graphs, and graded structures. Traces for types and related objects are studied, too.
We introduce and study some general principles and hierarchical properties of expansions and restrictions of structures and their theories The general approach is applied to describe these properties for classes of $ω$-categorical theories and structures, Ehrenfeucht theories and their models, strongly minimal, $ω_1$-theories, and stable ones.
We introduce and study the class of spherically ordered groups. The notions of spherically ordered groups and their spectra of spherical orderability are introduced. Values of these spectra are found for a series of natural groups.
We study possibilities for semantic and syntactic rigidity, i.e., the rigidity with respect to automorphism group and with respect to definable closure. Variations of rigidity and their degrees are studied in general case, for special languages and for some natural operations with structures.
We study possibilities for algebraic closures, differences between definable and algebraic closures in first-order structures, and variations of these closures with respect to the bounds of cardinalities of definable sets and given sets of formulae. Characteristics for these possibilities and differences are introduced and described. These characteristics are studied for some natural classes of theories. Besides algebraic closure operators with respect to sets of formulae are introduced and studied. Semilattices and lattices for families of these operators are introduced and characteristics of these structures are described.
We study semantic and syntactic properties of spherical orders and their elementary theories, including finite and dense orders and their theories. It is shown that theories of dense $n$-spherical orders are countably categorical and decidable. The values for spectra of countable models of unary expansions of $n$-spherical theories are described. The Vaught conjecture is confirmed for countable constant expansions of dense $n$-spherical theories.
We study applications of a general approach for arities and arizabilities of theories to group and monoid theories. It is proved that a theory of a group $G$ is aritizable if and only if $G$ is finite. It is shown that this criterion does not hold for monoids/groupoids: there is an infinite monoid/groupoid having a binary theory.
We study possibilities for almost $n$-ary and $n$-aritizable theories. Their dynamics both in general case, for $ω$-categorical theories, and with respect to operations for theories are described.
We study and describe possibilities for arities of elementary theories and of their expansions. Links for arities with respect to Boolean algebras, to disjoint unions and to compositions of structures are shown. The dynamics for arities of theories is described.
We study links between first-order formulas and arbitrary properties for families of theories, classes of structures and their isomorphism types. Possibilities for ranks and degrees for formulas and theories with respect to given properties are described. Characteristics for generic sentences and generic theories with respect to properties are described and characterized.
We consider and characterize classes of finite and countably categorical structures and their theories preserved under $E$-operators and $P$-operators. We describe $e$-spectra and families of finite cardinalities for structures belonging to closures with respect to $E$-operators and $P$-operators.
We define and study semilattices and lattices for $E$-closed families of theories. Properties of these semilattices and lattices are investigated. It is shown that lattices for families of theories with least generating sets are distributive.
We define the notions of relative $e$-spectra, with respect to $E$-operators, relative closures, and relative generating sets. We study properties connected with relative $e$-spectra and relative generating sets.
We investigate combinations of structures by families of structures relative to families of unary predicates and equivalence relations. Conditions preserving $ω$-categoricity and Ehrenfeuchtness under these combinations are characterized. The notions of $e$-spectra are introduced and possibilities for $e$-spectra are described.
We investigate closure operators and describe their properties for $E$-combinations and $P$-combinations of structures and their theories. We prove, for $E$-combinations, that the existence of a minimal generating set of theories is equivalent to the existence of the least generating set, and characterize syntactically and semantically the property of the existence of the least generating set. For the class of linearly ordered language uniform theories we solve the problem of the existence of least generating set with respect to $E$-combinations and characterize that existence in terms of orders.
We apply a general approach for distributions of binary isolating and semi-isolating formulas to families of isolated types and to the class of countably categorical theories.
We apply a general approach for distributions of binary isolating and semi-isolating formulas to the class of strongly minimal theories.
We present distributions of countable models and correspondent structural characteristics of complete theories with continuum many types: for prime models over finite sets relative to Rudin-Keisler preorders, for limit models over types and over sequences of types, and for other countable models of theory.