Dominions of universal algebras and projective properties

Let A be a universal algebra and H its subalgebra. The dominion of H in A (in a class {ie304-01}) is the set of all elements a ∈ A such that every pair of homomorphisms f, g: A → ∈ {ie304-02} satisfies the following: if f and g coincide on H, then f(a) = g(a). A dominion is a closure operator on a set of subalgebras of a given algebra. The present account treats of closed subalgebras, i.e., those subalgebras H whose dominions coincide with H. We introduce projective properties of quasivarieties which are similar to the projective Beth properties dealt with in nonclassical logics, and provide a characterization of closed algebras in the language of the new properties. It is also proved that in every quasivariety of torsion-free nilpotent groups of class at most 2, a divisible Abelian subgroup H is closed in each group 〈H, a〉 generated by one element modulo H. Translated from Algebra i Logika, Vol. 47, No. 5, pp. 541–557, September–October, 2008.     

Dominions in quasivarieties of metabelian groups

The dominion of a subgroup H of a group A in a quasivariety ℳ is the set of all a ∈ A with equal images under all pairs of homomorphisms from A into every group in ℳ which coincide on H. The concept of dominion provides some closure operator on the lattice of subgroups of a given group. We study the closed subgroups with respect to this operator. We find a condition for the dominion of a divisible subgroup in quasivarieties of metabelian groups to coincide with the subgroup.     

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.     

Quasivarieties of groups closed under wreath products and wreathZ-products

LetL _q(M) be a lattice of quasivarieties contained in a quasivarietyM. The quasivariety is closed under direct wreath Z-products if together with a group G, it contains its wreath product G ≀ Z with an infinite cyclic group Z. We prove the following: (a) ifM is closed under direct wreath Z-products then every quasivariety, which is a coatom inL _q(M), is likewise closed under these; (b) ifM is closed under direct wreath products thenL _q(M) has at most one coatom. An example of a quasivariety is furnished which is closed under direct wreath Z-products and whose subquasivariety lattice contains exactly one coatom. Also, it turns out that the set of quasivarieties closed under direct wreath Z-products form a complete sublatttice of the lattice of quasivarieties of groups. Supported by RFFR grant No. 96-01-00088, and by the RF Committee of Higher Education. Translated fromAlgebra is Logika, Vol. 38, No. 3, pp. 257–268, May–June, 1999.     

Sharp and Meager Elements in Orthocomplete Homogeneous Effect Algebras

We prove that every orthocomplete homogeneous effect algebra is sharply dominating. Let us denote the greatest sharp element below x by x ^↓. For every element x of an orthocomplete homogeneous effect algebra and for every block B with x ∈ B, the interval [x ^↓,x] is a subset of B. For every meager element (that means, an element x with x ^↓ = 0), the interval [0,x] is a complete MV-effect algebra. As a consequence, the set of all meager elements of an orthocomplete homogeneous effect algebra forms a commutative BCK-algebra with the relative cancellation property. We prove that a complete lattice ordered effect algebra E is completely determined by the complete orthomodular lattice S(E) of sharp elements, the BCK-algebra M(E) of meager elements and a mapping h:S(E)→2 ^M(E) given by h(a) = [0,a] ∩ M(E).     

On 2-Closedness of the Rational Numbers in Quasivarieties of Nilpotent Groups

The dominion of a subgroup H of a group G in a class M is the set of all elements a ∈ G that have equal images under every pair of homomorphisms from G to a group of M coinciding on H. A group H is said to be n-closed in M if for every group G = gr(H, a₁,..., a _n ) of M that contains H and is generated modulo H by some n elements, the dominion of H in G (in M) is equal to H. We prove that the additive group of the rational numbers is 2-closed in every quasivariety M of torsion-free nilpotent groups of class at most 3 whenever every 2-generated group of M is relatively free.     

The twisted conjugacy problem for endomorphisms of metabelian groups

Let M be a finitely generated metabelian group explicitly presented in a variety of all metabelian groups. An algorithm is constructed which, for every endomorphism φ ∈ End(M) identical modulo an Abelian normal subgroup N containing the derived subgroup M′ and for any pair of elements u, v ∈ M, decides if an equation of the form (xφ)u = vx has a solution in M. Thus, it is shown that the title problem under the assumptions made is algorithmically decidable. Moreover, the twisted conjugacy problem in any polycyclic metabelian group M is decidable for an arbitrary endomorphism φ ∈ End(M). Supported by RFBR (project No. 07-01-00392). (V. A. Roman’kov) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 157–173, March–April, 2009.     

On the Existence of Towers in Pseudo-Tree Algebras

A tower in a Boolean algebra (BA) is a strictly increasing sequence, of regular order type, of elements of the algebra different from 1 but with sum 1. A pseudo-tree is a partially ordered set T such that the set T↓t = {s ∈ T:s < t} is linearly ordered for every t ∈ T. If that set is well-ordered, then T is a tree. For any pseudo-tree T, the BA Treealg(T) is the algebra of subsets of T generated by all of the sets T↑t = {s ∈ T:t ≤ s}. The main theorem of this note is a characterization in tree terms of when Treealg(T) has a tower of order type κ (given in advance).     

Homomorphisms between JC＊-algebras and Lie C＊-algebras

It is shown that every almost ＊-homomorphism h ： A→B of a unital JC＊-algebra A to a unital JC＊-algebra B is a ＊-homomorphism when h（rx） = rh（x） （r 〉 1） for all x∈A, and that every almost linear mapping h ： A→B is a ＊-homomorphism when h（2^nu o y） - h（2^nu） o h（y）, h（3^nu o y） - h（3^nu） o h（y） or h（q^nu o y） = h（q^nu） o h（y） for all unitaries u ∈A, all y ∈A, and n = 0, 1,.... Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings. We prove that every almost ＊-homomorphism h ： A→B of a unital Lie C＊-algebra A to a unital Lie C＊-algebra B is a ＊-homomorphism when h（rx） = rh（x） （r 〉 1） for all x ∈A.     

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.     

Convolution properties of some classes of analytic functions

Let A denote the class of functions which are analytic in |z|<1 and normalized so that f(0)=0 and f′(0)=1, and let R(α, β)⊂A be the class of functions f such thatRe[f′(z)+αzf″(z)]>β,Re α>0, β<1. We determine conditions under which (i) f ∈ R(α₁, β₁), g ∈ R(α₂, β₂) implies that the convolution f×g of f and g is convex; (ii) f ∈ R(0, β₁), g ∈ R(0, β₂) implies that f×g is starlike; (iii) f≠A such that f′(z)[f(z)/z]^μ-1 ≺ 1 + λz, μ>0, 0<λ<1, is starlike, and (iv) f≠A such that f′(z)+αzf″(z) ≺ 1 + λz, α>0, δ>0, is convex or starlike. Bibliography: 16 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 138–154.     

On Dominions of the Rationals in Nilpotent Groups

The dominion of a subgroup H of a group G in a class M is the set of all a ∈ G that have the same images under every pair of homomorphisms, coinciding on H from G to a group in M. A group H is n-closed in M if for every group G = gr(H, a₁,..., a_n) in M that includes H and is generated modulo H by some n elements, the dominion of H in G (in M) is equal to H. We prove that the additive group of the rationals is 2-closed in every quasivariety of torsion-free nilpotent groups of class at most 3.     

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.     

Spectral synthesis in the sobolev space generated by an integral metric

Let l ∈ ℕ, A ⊂ ℝⁿ . The main goal of this paper is to describe (in inner terms) the closure of the set {f ∈ W ₁ ^l : f=0 in a neighborhood of the set A} with respect to the norm of the space W ₁ ^l (ℝⁿ). Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 217, 1994, pp. 92–111.     

Matrices with zero trace

LetM _n(F) denote the algebra ofn-square matrices with elements in a fieldF. In this paper we show that ifM ∈M _n(F) has zero trace thenM=AB−BA for certainA, B ∈ M _n(F), withA nilpotent and traceB=0, apart from some exceptional cases whenn=2 or 3. We also determine whenM=MB−BM for someB ∈ M _n(F). The preparation of this paper was supported in part by the U.S. Air Force under contract AFOSR 698-65.     

Riemann�CHilbert for tame complex parahoric connections

A local Riemann–Hilbert correspondence for tame meromorphic connections on a curve compatible with a parahoric level structure will be established. Special cases include logarithmic connections on G-bundles and on parabolic G-bundles. The corresponding Betti data involves pairs (M, P) consisting of the local monodromy M ∈ G and a (weighted) parabolic subgroup P ⊂ G such that M ∈ P, as in the multiplicative Brieskorn–Grothendieck–Springer resolution (extended to the parabolic case). The natural quasi-Hamiltonian structures that arise on such spaces of enriched monodromy data will also be constructed.     

Complex algebras of subalgebras

Let V be a variety of algebras. We specify a condition (the so-called generalized entropic property), which is equivalent to the fact that for every algebra A ∈ V, the set of all subalgebras of A is a subuniverse of the complex algebra of the subalgebras of A. The relationship between the generalized entropic property and the entropic law is investigated. Also, for varieties with the generalized entropic property, we consider identities that are satisfied by complex algebras of subalgebras. Dedicated to George Gr?tzer on the occasion of his 70th birthday Supported by INTAS grant No. 03-51-4110. Supported by MŠMTČR (project MSM 0021620839) and by the Grant Agency of the Czech Republic (grant No. 201/05/0002). Translated from Algebra i Logika, Vol. 47, No. 6, pp. 655–686, November–December, 2008.     

Lattices of dominions in quasivarieties of Abelian groups

Let M be any quasivariety of Abelian groups, (H) be the dominion of a subgroup H of a group G in M, and L_q(M) be the lattice of subquasivarieties of M. It is proved that (H ) coincides with a least normal subgroup of the group G containing H, the factor group with respect to which is in M. Conditions are specified subject to which the set L(G,H,M) = { (H) | N L_q(M)} forms a lattice under set-theoretic inclusion and the map : L_q(M) L(G,H,M) such that (N) = (H) for any quasivariety N L_q(M)is an antihomomorphism of the lattice L _q (M) onto the lattice L(G, H, M).__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 238–251, March–April, 2005.