We study enriched Boolean algebras. We describe subalgebras of Boolean algebras determined by fixed elements of automorphisms. We construct a continuum of superatomic Boolean algebras with a distinguished subalgebra whose elementary theories have no prime model. We investigate semigroups of elementary types of Boolean algebras with a distinguished subalgebra and of Boolean algebras with distinguished ideals.
Let G be the group of all bounded permutations of the set of natural numbers ℕ . Suppose that a group H contains elements h, h1, . . . , hn, . . . such that |h| = ∞, h_1^p = h, h_2^p = h1, . . . , h_n^p = hn−1, . . . , where p is a prime number. We prove that H is not isomorphic to a subgroup of G.
We prove that G2(2m) and 3D4(23m) are automorphism groups of regular polytopes of rank n = 4, 5, except for the group 3D4(23m) when n = 5 and m is a power of 3. The minimal number of generating conjugate involutions whose product equals 1 is found, and the existence of new regular polytopes of rank 3 with these automorphism groups is established.
We consider the problem of existence of single-valued, computable, positive, and negative representations of the family of all computably enumerable subsets on admissible structures. Additionally, a description of the properties of the admissible structures under consideration is given.
We investigate the independence of axioms defining a quandle – an algebraic structure important in knot theory and combinatorial algebra. It is proved that all four axioms included in the standard definition of a quandle are independent. For finite quandles, one of the solvability axioms turns out to be a consequence of the others, owing to which it is possible to construct a minimal system of three axioms. An example of a right-distributive left quasigroup is constructed. As an application, for the constructed right-distributive systems, set-theoretic solutions to the Yang–Baxter equations and the corresponding associated groups are indicated.
The note concerns the comparative study of two approaches to generalized computability over the real numbers: computable analysis and Σ-definability in hereditarily finite su-perstructures. As known, there exist total computable functions that are not Σ-definable and discontinuous Σ-definable functions that are not computable. In this note, we show that even among continuous real functions, there exists a Σ-definable function that is not computable in the sense of computable analysis. The proof uses a diagonalization construction with infinite computable disjunctions of Δ0-formulas.
We prove that the quasivariety qF(𝒩c) generated by the free non-abelian c-step nilpotent group (c ≥ 4) contains a continuum of quasivarieties. All quasivarieties contained in qF(𝒩3) are described.
In this paper, we prove that for any computable numberings β < α of an arbitrary family of c.e. sets, where α is equivalent to the completion of some noncomplete numbering, there exists a chain of its computable numberings (with respect to the reducibility of numberings) bounded above by the numbering α and having β as its least element whose order type is the first nonconstructive ordinal. We also establish that this statement is true for any ∑_n^0- computable numberings (n > 1) β < α provided that α is ∅(n−1)-precomplete.
With the use of the notion of an order ideal, we prove that many types of approximation spaces are projects or retracts of certain approximation spaces with a base consisting of compacts.
We prove that every nonassociative Novikov algebra can be equipped with a nontrivial structure of a Novikov–Poisson algebra. Using this result, we show that, under certain finiteness conditions, every simple Novikov algebra is obtained by the Gelfand–Dorfman construction applied to an associative commutative differentially simple algebra. We also introduce and study the Witt doubles of Novikov–Poisson algebras. The description of the isomorphisms of simple Novikov algebras over an algebraically closed field is reduced to the description of some special automorphisms of the underlying associative commutative algebras.
We consider effective versions of Stone duality for distributive c-posets and almost semispectral spaces with base. For any oracle Z ⊆ ω, the notions of Z-computably enumerable c-posets and Z-computably enumerable topological spaces with base are introduced. The dual equivalence of the categories AS and DP restricts to a dual equivalence of their full subcategories of Z-computably enumerable objects. Effective dualities are also obtained for distributive lattices, distributive meet semilattices, and distributive join semilattices. As an application, for any nontrivial countable structure 𝒮 , there exists a separable almost semispectral space with base whose degree spectrum coincides with that of 𝒮 .
We consider the uniformity of constructions in computability theory. Examples of uniform and nonuniform constructions in Turing degree theory are considered, including the completeness criterion for computably enumerable sets, the Sacks Jump Theorem, constructions of effectively nowhere simple sets, and the problem of uniformly obtaining a computably enumerable set below an arbitrary 2-computably enumerable set. Results by Arslanov, Yamaleev, Downey, Stob, Terwijn, and others are analyzed. Open questions on Σ1-embeddability of structures of n-computably enumerable degrees are discussed.
We obtain essential completions of topological T0-spaces with the help of ideal spaces. We also establish a connection between the essential completeness of function spaces and the essential completeness, as well as with the existence of approximation in the semilattice of open sets.
Invariant properties of limit models of Ehrenfeucht theories are studied. Partial answers are obtained to the Ash–Millar question on the arithmetical decidability of models of Ehrenfeucht theories with arithmetical types.
We study the problem of elementary equivalence of countable atomless Boolean algebras with a distinguished nonprincipal and nondense ideal, constructed from different elementarily equivalent countable infinite atomic Boolean algebras, under the condition that both algebra and its quotient algebra by the given ideal are atomless. We prove the existence of a continuum family of such countable Boolean algebras with a distinguished ideal,
We introduce a definition of an model-theoretic property and present the set ModThProp of all model-theoretic properties concerning complete theories. We justify that the pro-posed approach corresponds to the practice of research in model theory.
We establish a duality between the category of algebraic lattices with complete lattice homomorphisms as morphisms and the category of join semilattices whose morphisms are join-semilattice homomorphisms possessing an additional refinement property. We draw several conclusions from this duality to benefit Lattice Theory.
We prove that there exists a partial order having no computable copies in which any effectively interpreted linearly ordered structure has a computable copy, but there are linearly ordered structures with the same degree spectrum which are obtained by a natural algorithmic transformation not directly related to effective interpretability.
We study tensor completions G⊗_𝒩_2,RR of finitely generated torsion-free 2-nilpotent groups G in the class 𝒩_2,R of all 2-nilpotent R-groups over a binomial domain R. We show that G⊗_𝒩_2,RR is isomorphic to the group ( G⊗_ℋR ) × D, where G⊗_ℋR is the classical Hall R-completion of G, D is an Abelian R-group, and the direct product is a product of abstract groups (not R-groups!). In particular, this answers an old question of Remeslennikov about the algebraic structure of free 2-nilpotent R-groups in the quasivariety 𝒩_2,R (they are precisely the tensor R-completions of free 2-nilpotent groups).
For computable families of computably enumerable sets we find sufficient conditions under which, for any n > 1, the Rogers semilattices of these families have ideals containing exactly n minimal elements, each specified by positive undecidable numberings. It is shown that the Rogers semilattice of every Σ_m^0 -computable family, m > 1, possessing a Friedberg Σ_m^0 -computable numbering, for any n > 1 possesses an ideal that also contains exactly n minimal elements specified by positive undecidable numberings.
