Q-universal varieties of bounded lattices

A quasivariety K of algebraic systems of finite type is said to be Q-universal if, for any quasivariety M of finite type, L(M) is a homomorphic image of a sublattice of L(K), where L(M) and L(K) are the lattices of quasivarieties contained in M and K, respectively.? It is known that, for every variety K of (0, 1)-lattices, if K contains a finite nondistributive simple (0, 1)-lattice, then K is Q-universal, see [3]. The opposite implication is obviously true within varieties of modular (0, 1)-lattices. This paper shows that in general the opposite implication is not true. A family (A _i : i < 2ω) of locally finite varieties of (0, 1)-lattices is exhibited each of which contains no simple non-distributive (0, 1)-lattice and each of which is Q-universal. Received July 19, 2001; accepted in final form July 11, 2002.     

Almost ff-universality Implies Q-universality

A concrete category \mathbb Q\mathbb {Q} is finite-to-finite (algebraically) almost universal if the category of graphs and graph homomorphisms can be embedded into \mathbb Q\mathbb {Q} in such a way that finite \mathbb Q\mathbb {Q}-objects are assigned to finite graphs and non-constant \mathbb Q\mathbb {Q}-morphisms between any \mathbb Q\mathbb {Q}-objects assigned to graphs are exactly those arising from graph homomorphisms. A quasivariety \mathbb Q\mathbb {Q} of algebraic systems of a finite similarity type is Q-universal if the lattice of all subquasivarieties of any quasivariety \mathbb R\mathbb {R} of algebraic systems of a finite similarity type is isomorphic to a quotient lattice of a sublattice of the subquasivariety lattice of \mathbb Q\mathbb {Q}. This paper shows that any finite-to-finite (algebraically) almost universal quasivariety \mathbb Q\mathbb {Q} of a finite type is Q-universal.     

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.     

A quasivariety lattice of torsion-free soluble groups

Let L _q (qG) be a lattice of quasivarieties contained in a quasivariety generated by a group G. It is proved that if G is a torsion-free finitely generated group in AB\mathcal{AB} _pk , where p is a prime, p ≠ 2, and k ∈ N, which is a split extension of an Abelian group by a cyclic group, then the lattice L _q (qG) is a finite chain.     

Congruence distributive quasivarieties whose finitely subdirectly irreducible members form a universal class

By a congruence distributive quasivariety we mean any quasivarietyK of algebras having the property that the lattices of those congruences of members ofK which determine quotient algebras belonging toK are distributive. This paper is an attempt to study congruence distributive quasivarieties with the additional property that their classes of relatively finitely subdirectly irreducible members are axiomatized by sets of universal sentences. We deal with the problem of characterizing such quasivarieties and the problem of their finite axiomatizability.Presented by Joel Berman.To the memory of Basia Czelakowska.     

Algebraic modular forms

In this paper, we develop an algebraic theory of modular forms, for connected, reductive groupsG overQ with the property that every arithmetic subgroup Γ ofG(Q) is finite. This theory includes our previous work [15] on semi-simple groupsG withG(R) compact, as well as the theory of algebraic Hecke characters for Serre tori [20]. The theory of algebraic modular forms leads to a workable theory of modular forms (modp), which we hope can be used to parameterize odd modular Galois representations. The theory developed here was inspired by a letter of Serre to Tate in 1987, which has appeared recently [21]. I want to thank Serre for sending me a copy of this letter, and for many helpful discussions on the topic.     

Structure of lattices of varieties and lattices of quasivarieties: similarity and difference. III

We give representations for lattices of varieties and lattices of quasivarieties in terms of inverse limits of lattices satisfying a number of additional conditions. Specifically, it is proved that, for any locally finite variety (quasivariety) of algebras V, L _v(V)[resp., L _q(V)] is isomorphic to an inverse limit of a family of finite join semidistributive at 0 (resp., finite lower bounded) lattices. A similar statement is shown to hold for lattices of pseudo-quasivarieties. Various applications are offered; in particular, we solve the problem of Lampe on comparing lattices of varieties with lattices of locally finite ones. Translated fromAlgebra i Logika, Vol. 34, No. 6, pp. 646-666, November-December, 1995.     

Amenability of the Second Dual of Hypergroup Algebras

The amenability of the Banach algebra L ¹(G), the measure algebra M(G) and their second duals of a locally compact group have been considered by a number of authors. During these investigations it has been shown that L ¹(G)^** is amenable if and only if G is finite. If LUC (G)^*, the dual of the space of left uniformly continuous functions on G, is amenable, then G is compact and M(G) is amenable. Finally, if M(G)^** is amenable, then G is finite. The aim of this paper is to generalize all of the above results to the locally compact hypergroups.     

Finite arithmetic subgroups ofGL n,III

LetG be an algebraic group inGL _n (C) defined over Q, andK an algebraic number field with the maximal orderO _k . If the groupG(O _k ) of rational points ofG inM _n (O _k ) is a finite group and if it satisfies a certain condition, which is satisfied, for example, whenK is a nilpotent extension of Q and 2 is unramified, thenG(O _k ) is generated by roots of unity inK andG(Z). Dedicated to the memory of Professor K G Ramanathan     

Finiteness of A Set of Quasivarieties of Torsion-Free Metabelian Groups of Axiomatic Rank 2

Let M \mathcal {M} be a quasivariety of all torsion-free groups in which squares of elements are commuting. It is proved that the set of quasivarieties contained in M \mathcal {M} and defined by quasi-identities in two variables is finite.     

Groups containing a strongly embedded subgroup

An involution v of a group G is said to be finite (in G) if vv ^g has finite order for any g ∈ G. A subgroup B of G is called a strongly embedded (in G) subgroup if B and G\B contain involutions, but B ∩ B ^g does not, for any g ∈ G\B. We prove the following results. Let a group G contain a finite involution and an involution whose centralizer in G is periodic. If every finite subgroup of G of even order is contained in a simple subgroup isomorphic, for some m, to L ₂(2 ^m ) or Sz(2 ^m ), then G is isomorphic to L ₂(Q) or Sz(Q) for some locally finite field Q of characteristic two. In particular, G is locally finite (Thm. 1). Let a group G contain a finite involution and a strongly embedded subgroup. If the centralizer of some involution in G is a 2-group, and every finite subgroup of even order in G is contained in a finite non-Abelian simple subgroup of G, then G is isomorphic to L ₂(Q) or Sz(Q) for some locally finite field Q of characteristic two (Thm. 2). Supported by RFBR (project No. 08-01-00322), by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-334.2008.1), and by the Russian Ministry of Education through the Analytical Departmental Target Program (ADTP) “Development of Scientific Potential of the Higher School of Learning” (project Nos. 2.1.1.419 and 2.1.1./3023). (D. V. Lytkina and V. D. Mazurov) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 190–202, March–April, 2009.     

On equivalence of number fields

LetK be a field,G a finite group.G is calledK-admissible iff there exists a finite dimensionalK-central division algebraD which is a crossed product forG. Now letK andL be two finite extensions of the rationalsQ such that for every finite groupG, G isK-admissible if and only ifG isL-admissible. ThenK andL have the same degree and the same normal closure overQ. An erratum to this article is available at .     

Some properties of the spectrum of graphs

Let G be a graph and denote by Q(G)=D(G) A(G),L(G)=D(G)-A(G) the sum and the difference between the diagonal matrix of vertex degrees and the adjacency matrix of G,respectively. In this paper,some properties of the matrix Q(G)are studied. At the same time,anecessary and sufficient condition for the equality of the spectrum of Q(G) and L(G) is given.     

Congruence properties of lattices of quasivarieties

The congruence properties close to being lower boundedness in the sense of McKenzie are treated. In particular, the affirmative answer is obtained to a known question as to whether finite lattices of quasivarieties are lower bounded in the case where quasivarieties are congruence-Noetherian and locally finite. Namely, we state that for every congruence-Noetherian or finitely generated locally finite quasivariety K, the lattice L_q(K) possesses the Day-Pudlak-Tuma property. But if a quasivariety is locally finite without the condition of being finitely generated), that lattice satisfies only the Pudlak-Tuma property. Translated fromAlgebra i Logika, Vol. 36, No. 6, pp. 605–620, Noember, 1997.     

Rational Points on Homogeneous Varieties and Equidistribution of Adelic Periods

Let U := L\G be a homogeneous variety defined over a number field K, where G is a connected semisimple K-group and L is a connected maximal semisimple K-subgroup of G with finite index in its normalizer. Assuming that G(K _v ) acts transitively on U(K _v ) for almost all places v of K, we obtain an asymptotic for the number of rational points U(K) with height bounded by T as T → ∞, and settle new cases of Manin’s conjecture for many wonderful varieties. The main ingredient of our approach is the equidistribution of semisimple adelic periods, which is established using the theory of unipotent flows.     

p-Hypercyclically embedding and Π-property of subgroups of finite groups

Let G be a finite group, E a normal subgroup of G and p a fixed prime. We say that E is p-hypercyclically embedded in G if every p-G-chief factor of E is cyclic. A subgroup H of G is said to satisfy Π-property in G if |G∕K:N_G∕K((H∩L)K∕K)| is a π((H∩L)K∕K)-number for any chief factor L∕K in G; we say that H has Π*-property in G if H∩O^π(H)(G) has Π-property in G. In this paper, we prove that E is p-hypercyclically embedded in G if and only if some classes of p-subgroups of E have Π*-property in G. Some recent results are extended.     

A characterization of a class of [Z] groups via Korovkin theory

We characterize all the central topological groupsG for which the centreZ(L ¹(G)) of the group algebra admits a finite universal Korovkin set. It is proved thatZ(L ¹(G)) has a finite universal Korovkin set iffĜ is a finite dimensional, separable metric space. This is equivalent to the fact thatG is separable, metrizable andG/K has finite torsion free rank, whereK is a compact open normal subgroup of certain direct summand ofG.     

Groups with normal restriction property

Let G be a finite group. A subgroup M of G is said to be an NR-subgroup if, whenever K\trianglelefteq M{K\trianglelefteq M}, then K ^G ∩ M = K where K ^G is the normal closure of K in G. Using the Classification of Finite Simple Groups, we prove that if every maximal subgroup of G is an NR-subgroup then G is solvable. This gives a positive answer to a conjecture posed in Berkovich (Houston J. Math. 24 (1998), 631–638).     

Finite 2-groups with a nonabelian Frattini subgroup of order 16

According to a classical result of Burnside, if G is a finite 2-group, then the Frattini subgroup Φ(G) of G cannot be a nonabelian group of order 8. Here we study the next possible case, where G is a finite 2-group and Φ(G) is nonabelian of order 16. We show that in that case Φ(G) ≅ M × C₂, where M ≅ D₈ or M ≅ Q₈ and we shall classify all such groups G (Theorem A). Received: 16 February 2005; revised: 7 March 2005     

Characterization of Operators on the Dual of Hypergroups which Commute with Translations and Convolutions

For a locally compact group G, L^1 (G) is its group algebra and L^∞(G) is the dual of L^1 (G).Lau has studied the bounded linear operators T:L^∞(G)→L^∞(G) which commute with convolutions and translations. For a subspace H of L^∞(G), we know that M(L^∞(G),H), the Banach algebra of all bounded linear operators on L^∞(G) into H which commute with convolutions, has been studied by Pyre and Lau. In this paper, we generalize these problems to L(K)^*, the dual of a hypergroup algebra L(K) in a very general setting, i.e. we do not assume that K admits a Haar measure. It should be noted that these algebras include not only the group algebra L^1(G) but also most of the semigroup algebras.Compact hypergroups have a Haar measure, however, in general it is not known that every hypergroup has a Haar measure. The lack of the Haar measure and involution presents many difficulties; however,we succeed in getting some interesting results.