Algebraic atomistic lattices of quasivarieties

The Gorbunov-Tumanov conjecture on the structure of lattices of quasivarieties is proved true for the case of algebraic lattices. Namely, for an algebraic atomistic lattice L, the following conditions are equivalent: (1) L is represented as L_q(K) for some algebraic quasivariety K; (2) L is represented as S_Λ (A) for some algebraic lattice A which satisfies the minimality condition and nearly satisfies the maximality conditions; (3) L is a coalgebraic lattice admitting an equaclosure operator. Supported by RFFR grants Nos. 96-01-01525 and 96-0-000976, and by DFG grant No. 436 (RUS) 113/2670. Translated from Algebra i Logika, Vol. 36, No. 4, pp. 363–386, July–August, 1997.     

Interpreting arithmetics in the ideal lattice of a free vector lattice ℱ<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

A vector space V over a real field R is a lattice under some partial order, which is referred to as a vector lattice if u + (v ∨ w) = (u + v) ∨ (u + w) and u + (v ∧ w) = (u + v) ∧ (u + w) for all u, v, w ∈ V. It is proved that a model N of positive integers with addition and multiplications is relatively elementarily interpreted in the ideal lattice ℱ _n of a free vector lattice ℱ _n on a set of n generators. This, in view of the fact that an elementary theory for N is hereditarily undecidable, implies that an elementary theory for ℱ _n is also hereditarily undecidable. __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 71–82, January–February, 2008.     

Covers in the lattice of varieties of ℓ-groups

We show that a variety ν in the lattice L of varieties of ℓ-groups has continuum many covers, and that the same is also true of an arbitrary o-approximable variety Χ with the property ν⊆Χ. It is proved that any o-approximable quasivariety Q of ℓ-groups, for which ν⊆Q, has the continuum of covers in the quasivariety lattice Λ. Supported by the RF State Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 253–269, May–June, 1998.     

Distributivity conditions for lattices of dominions in quasivarieties of Abelian groups

Let ℳ be any quasivariety of Abelian groups, L_q(ℳ) be a subquasivariety lattice of ℳ, dom _G ^ℳ be the dominion of a subgroup H of a group G in ℳ, and G/dom _G ^ℳ (H) be a finitely generated group. It is known that the set L(G, H, ℳ) = {dom _G ^N (H)| N ∈ L_q(ℳ)} forms a lattice w.r.t. set-theoretic inclusion. We look at the structure of dom _G ^ℳ (H). It is proved that the lattice L(G,H,ℳ) is semidistributive and necessary and sufficient conditions are specified for its being distributive. __________ Translated from Algebra i Logika, Vol. 45, No. 4, pp. 484–499, July–August, 2006.     

Minimality and unique ergodicity of homogeneous actions

Consider a connected Lie groupG, a lattice Γ inG, a connected subgroupH ofG, and the adjoint representation Ad ofG on its Lie algebra g. Suppose that Ad(H) splits into a semidirect product of a reductive subgroup and the unipotent radical. We prove that the minimality of the leftH-action onG/Γ then implies its unique ergodicity. Simultaneously, we suggest a reduction of the study of finite ergodic measures for an arbitrary action (G/Γ,H), where the subgroupH∈G is connected and Γ∈G is discrete, to the case of an Abelian subgroupH. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 293–301, August, 1999.     

Gelfand Factor Rings and Weak Zariski Topologies

Let R be any ring with identity. Let N(R) (resp. J(R)) denote the prime radical (resp. Jacobson radical) of R, and let Spec _r (R) (resp. Spec _l (R), Max _r (R), Prim _r (R)) denote the set of all right prime ideals (resp. all left prime ideals, all maximal right ideals, all right primitive ideals) of R. In this article, we study the relationships among various ring-theoretic properties and topological conditions on Spec _r (R) (with weak Zariski topology). The following results are obtained: (1) R/N(R) is a Gelfand ring if and only if Spec _r (R) is a normal space if and only if Spec _l (R) is a normal space; (2) R/J(R) is a Gelfand ring if and only if every right prime ideal containing J(R) is contained in a unique maximal right ideal.     

Spectral Characterization of Some Generalized Odd Graphs

Suppose G is a connected, k-regular graph such that Spec(G)=Spec(Γ) where Γ is a distance-regular graph of diameter d with parameters a ₁=a ₂=⋯=a _d−1=0 and a _d>0; i.e., a generalized odd graph, we show that G must be distance-regular with the same intersection array as that of Γ in terms of the notion of Hoffman Polynomials. Furthermore, G is isomorphic to Γ if Γ is one of the odd polygon C _2d+1, the Odd graph O _d+1, the folded (2d+1)-cube, the coset graph of binary Golay code (d=3), the Hoffman-Singleton graph (d=2), the Gewirtz graph (d=2), the Higman-Sims graph (d=2), or the second subconstituent of the Higman-Sims graph (d=2). Received: March 28, 1996 / Revised: October 20, 1997     

The elementary equivalence of free algebras in cantor varieties

We study properties of free algebras in the Cantor varieties C_m,n. A free algebra of rank r in C_m,n is denoted F_{C m,n}(r). We argue that the following hold: (1) any two C_m,n-free algebras F_{C m,n}(r) and F_{C m,n}(s) of ranks r and s, where r and s are arbitrary (finite or infinite) cardinals, r≥m, and s≥m, are elementary equivalent; (2) any two C_m,n-free algebras F_{C m,n}(r) and F_{C m,n}(s) of ranks r and s, where r and s are arbitrary (finite or infinite) cardinals, are universally equivalent, that is, share one ∀-theory; (3) an elementary theory Th(F_{C m,n}(r)) for an arbitrary C_m,n-free algebra of (finite or infinite) rank r, treated in a signature Ω, is decidable; (4) an elementary theory Th(K) for an arbitrary nonempty class of free algebras in C_m,n, treated in a signature Ω, is decidable. Translated fromAlgebra i Logika, Vol. 38, No. 2, pp. 228–248, March–April, 1999.     

Arithmetic Interpretability Types of Varieties and Some Additive Problems with Primes

We deal with varieties with one basic operation f(x₁,...,x_n) and one defining identity f(x₁,..., x_n) = f(x_π(1),...,x_π(n)), where π is a permutation whose cyclic set consists of distinct primes p₁,...,p_r, with the sum p₁+...+p_r = n. Their interpretability types, together with the greatest element 1 in a lattice ^int, are said to be arithmetic. It is proved that the arithmetic types constitute a distributive lattice _ar, which is dual to a lattice Sub _fΠ of finite subsets of the set Π of all primes. It is shown that for n ⩾ 2, the poset _ar( _n) of arithmetic types defined by permutations in _n, for n fixed, is a lattice iff n = 2, 3, 4, 6, 8, 9, 11. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 622–630, September–October, 2005.     

A criterion for regular sequences

LetR be a commutative noetherian ring and ƒ₁, …, ƒ_r ∃ R. In this article we give (cf. the Theorem in §2) a criterion for ƒ₁, …, ƒ_r to be regular sequence for a finitely generated module overR which strengthens and generalises a result in [2]. As an immediate consequence we deduce that if V(g ₁, …,g _r ) ⊆ V(ƒ₁, …, ƒ_r) in SpecR and if ƒ₁, …, ƒ_r is a regular sequence inR, theng ₁, …,g _r is also a regular sequence inR.     

Maximal quasivarieties of groups

We consider a lattice L_q(qG) of quasivarieties contained in the quasivariety qG, generated by a polycyclic-by-finite group G. It is proved that the lattice contains a finite set of coatoms (i.e., proper maximal elements) and that each of its elements distinct from qG is contained in some coatom. We construct an example of a finitely generated solvable group B of derived length 3, whose quasivariety lattice L_q(qB) is freed of coatoms. Supported by RFFR grant No. 96-01-00088, and by the RF Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 279–290, May–June, 1998.     

Version of the Grothendieck theorem for subspaces of analytic functions in lattices

A version of Grothendieck’s inequality says that any bounded linear operator acting from a Banach lattice X to a Banach lattice Y acts from X(ℓ²) to Y (ℓ²) as well. A similar statement is proved for Hardy-type subspaces in lattices of measurable functions. Namely, let X be a Banach lattice of measurable functions on the circle, and let an operator T act from the corresponding subspace of analytic functions X_A to a Banach lattice Y or, if Y is also a lattice of measurable functions on the circle, to the quotient space Y/Y_A. Under certain mild conditions on the lattices involved, it is proved that T induces an operator acting from X_A(ℓ²) to Y (ℓ²) or to Y/Y_A(ℓ2), respectively. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 5–16.     

Invariant symmetric restrictions of a self-adjoint operator. II

We give a criterion of invariance and symmetry of the restriction of an arbitrary unbounded self-adjoint operator in the space L ₂(ℝⁿ, dx) by using the introduced notion of support of an arbitrary vector and the notion of capacity of a subspace N ⊂ ℝⁿ. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 781–791, June, 1998. This work was partially supported by the Foundation for Fundamental Research of the Ministry of Science and Technology of the Ukraine (grant No. 1/238 “Operator”).     

Constructing Restricted Patterson Measures for Geometrically Infinite Kleinian Groups

In this paper, we study exhaustions, referred to as p-restrictions, of arbitrary nonelementary Kleinian groups with at most finitely many bounded parabolic elements. Special emphasis is put on the geometrically infinite case, where we obtain that the limit set of each of these Kleinian groups contains an infinite family of closed subsets, referred to as p-restricted limit sets, such that there is a Poincaré series and hence an exponent of convergence δp, canonically associated with every element in this family. Generalizing concepts which are well known in the geometrically finite case, we then introduce the notion of p-restricted Patterson measure, and show that these measures are non-atomic, δp-harmonic, δp-subconformal on special sets and δp-conformal on very special sets. Furthermore, we obtain the results that each p-restriction of our Kleinian group is of δp-divergence type and that the Hausdorff dimension of the p-restricted limit set is equal to δp.     

Bands of k-Archimedean semigroups

In this paper we define the radical ϱ _k (k∈Z ⁺) of a relation ϱ on an arbitrary semigroup. Also, we define various types of k-regularity of semigroups and various types of k-Archimedness of semigroups. Using these notions we describe the structure of semigroups in which ρ _k is a band (semilattice) congruence for some Green’s relation.     

Some properties of <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">φ</Emphasis></Subscript>-splines

A general approach to the construction of asymptotics of coordinate (not necessarily polynomial) B _φ -splines of an arbitrary order is proposed. Asymptotic representations for Lagrange type third order B _φ -splines are obtained. Bibliography: 4 titles. Translated from Problemy Matematicheskogo Analiza, No. 38, December 2008, pp. 11–21.     

Idempotent matrix lattices over distributive lattices

In this paper, the partially ordered set of idempotent matrices over distributive lattices with the partial order induced by a set of lattice matrices is studied. It is proved that this set is a lattice; the formulas for meet and join calculation are obtained. In the lattice of idempotent matrices over a finite distributive lattice, all atoms and coatoms are described. We prove that the lattice of quasi-orders over an n-element set Qord(n) is not graduated for n ≥ 3 and calculate the greatest and least lengths of maximal chains in this lattice. We also prove that the interval ([I, J]_≤, ≤) of idempotent (n × n)-matrices over {ie879-01}-lattices is isomorphic to the lattice of quasi-orders Qord(n). Using this isomorphism, we calculate the lattice height of idempotent {ie879-02}-matrices. We obtain a structural criterion of idempotent matrices over distributive lattices. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 4, pp. 121–144, 2007.     

Lattice Points, Dedekind Sums, and Ehrhart Polynomials of Lattice Polyhedra

   Abstract. Let σ be a simplex of R ^N with vertices in the integral lattice Z ^N . The number of lattice points of mσ (={mα : α ∈ σ}) is a polynomial function L(σ,m) of m ≥ 0 . In this paper we present: (i) a formula for the coefficients of the polynomial L(σ,t) in terms of the elementary symmetric functions; (ii) a hyperbolic cotangent expression for the generating functions of the sequence L(σ,m) , m ≥ 0 ; (iii) an explicit formula for the coefficients of the polynomial L(σ,t) in terms of torsion. As an application of (i), the coefficient for the lattice n -simplex of R ⁿ with the vertices (0,. . ., 0, a _j , 0,. . . ,0) (1≤ j≤ n) plus the origin is explicitly expressed in terms of Dedekind sums; and when n=2 , it reduces to the reciprocity law about Dedekind sums. The whole exposition is elementary and self-contained.