Small sublattices with large dominions

Two examples of finite sublattices with infinite dominions are given. It is proven that this cannot occur in a finitely generated lattice variety. Received May 10, 2001; accepted in final form October 6, 2005.     

Varieties minimal over representable varieties of lattice-ordered groups

Weinberg showed that the variety of abelian lattice-ordered groups is the minimal nontrivial variety in the lattice of varieties of lattice-ordered groups. Scrimger showed that the abelian variety of lattice-ordered groups has countably infinitely many nonrepresentable covering varieties, and it is now known that his varieties are the only nonrepresentable covers of the abelian variety.

In this paper, a variation of the method used to construct the Scrimger varieties is developed that is shown to produce every nonrepresentable cover of any representable variety. Using this variation, all nonrepresentable covers of any weakly abelian l-variety are specifically identified, as are the nonrepresentable covers of any l-metabelian representable l-variety. In both instances, such il-varieties have only countably infinitely many such covers.

Any nonrepresentable cover of a representable il-variety is shown to be a subvariety of a quasi-representable il-variety as defined by Reilly. The class of these quasi-representable l-varieties is shown to contain the well-known L_n l-varieties and to generalize many of their properties.     

Vector bundles on toric varieties

Following Sam Payne?s work, we study the existence problem of nontrivial vector bundles on toric varieties. The first result we prove is that every complete fan admits a nontrivial conewise linear multivalued function. Such functions could potentially be the Chern classes of toric vector bundles. Then we use the results of Cortiñas, Haesemeyer, Walker and Weibel to show that the (non-equivariant) Grothendieck group of the toric 3-fold studied by Payne is large, so the variety has a nontrivial vector bundle. Using the same computation, we show that every toric 3-fold X either has a nontrivial line bundle, or there is a finite surjective toric morphism from Y to X, such that Y has a large Grothendieck group.     

Modular varieties with the Fraser-Horn property

The notion of central idempotent elements in a ring can be easily generalized to the setting of any variety with the property that proper subalgebras are always nontrivial. We will prove that if such a variety is also congruence modular, then it has factorable congruences, i.e., it has the Fraser-Horn property. (This property is well known to have major implications for the structure theory of the algebras in the variety.)

     

Lattices of dominions of universal algebras

We fix a universal algebra A and its subalgebra H. The dominion of H in A (in a class M) is the set of all elements a ∈ A such that any pair of homomorphisms f, g: A → M ∈ M satisfies the following: if f and g coincide on H then f(a) = g(a). In association with every quasivariety, therefore, is a dominion of H in A. Sufficient conditions are specified under which a set of dominions form a lattice. The lattice of dominions is explored for down-semidistributivity. We point out a class of algebras (including groups, rings) such that every quasivariety in this class contains an algebra whose lattice of dominions is anti-isomorphic to a lattice of subquasivarieties of that quasivariety. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 26–45, January–February, 2007.     

E-Unitary Covers over Group Varieties

In this paper we show that for every nontrivial group variety V different from the variety of all groups, there exists an E-unitary regular semigroup whose greatest group homomorphic image belongs to V, but it has no embeddable E-unitary cover over V.     

Convergence and finite determination of formal CR mappings

It is shown that a formal mapping between two real-analytic hypersurfaces in complex space is convergent provided that neither hypersurface contains a nontrivial holomorphic variety. For higher codimensional generic submanifolds, convergence is proved e.g. under the assumption that the source is of finite type, the target does not contain a nontrivial holomorphic variety, and the mapping is finite. Finite determination (by jets of a predetermined order) of formal mappings between smooth generic submanifolds is also established.

     

Projective algebras and primitive subquasivarieties in varieties with factor congruences

We prove that in the varieties where every compact congruence is a factor congruence and every nontrivial algebra contains a minimal subalgebra, a finitely presented algebra is projective if and only if it has every minimal algebra as its homomorphic image. Using this criterion of projectivity, we describe the primitive subquasivarieties of discriminator varieties that have a finite minimal algebra embedded in every nontrivial algebra from this variety. In particular, we describe the primitive quasivarieties of discriminator varieties of monadic Heyting algebras, Heyting algebras with regular involution, Heyting algebras with a dual pseudocomplement, and double-Heyting algebras.     

Warped products with special Riemannian curvature

We study the geometry of particular classes of Riemannian manifolds obtained as warped products. We focus on the case of constant curvature which is completely classified and on the Einstein case. This study provides nontrivial instances of Einstein manifolds which are warped product of Einstein factors.Supported by a grant from Università di Parma     

Relative congruence formulas and decompositions in quasivarieties

Quasivarietal analogues of uniform congruence schemes are discussed, and their relationship with the equational definability of principal relative congruences (EDPRC) is established, along with their significance for relative congruences on subalgebras of products. Generalizing the situation in varieties, we prove that a quasivariety is relatively ideal iff it has EDPRC; it is relatively filtral iff it is relatively semisimple with EDPRC. As an application, it is shown that a finitary sentential logic, algebraized by a quasivariety K, has a classical inconsistency lemma if and only if K is relatively filtral and the subalgebras of its nontrivial members are nontrivial. A concrete instance of this result is exhibited, in which K is not a variety. Finally, for quasivarieties \({\sf{M} \subseteq \sf{K}}\), we supply some conditions under which M is the restriction to K of a variety, assuming that K has EDPRC.     

A semidefinite programming method for integer convex quadratic minimization

We consider the NP-hard problem of minimizing a convex quadratic function over the integer lattice \({\mathbf{Z}}^n\). We present a simple semidefinite programming (SDP) relaxation for obtaining a nontrivial lower bound on the optimal value of the problem. By interpreting the solution to the SDP relaxation probabilistically, we obtain a randomized algorithm for finding good suboptimal solutions, and thus an upper bound on the optimal value. The effectiveness of the method is shown for numerical problem instances of various sizes.     

Many varieties of lattices with nonsurjective epimorphisms

We use dominions to show that many varieties of lattices have nonsurjective epimorphisms. The variety D of distributive lattices is treated in detail. We show that the dominion in D of a sublattice is the closure of M under relative complementation in L. This dominion is also the largest sublattice of L in which M is epimorphically embedded. In any variety of lattices larger than D, the dominion of M in L is just M. Received May 1, 2001; accepted in final form October 4, 2005.     

A characterization of locally finite varieties that satisfy a nontrivial congruence identity

We show that a locally finite variety satisfies a nontrivial congruence identity if and only if it satisfies an idempotent Mal'tsev condition that fails in the variety of semilattices. Received January 27, 1999; accepted in final form June 11, 1999.     

Covers in the Lattice of Varieties of m-Groups

We consider some questions on covers in the lattice of varieties of m-groups. We prove the existence of a nonabelian cover of the smallest nontrivial variety of m-groups. We show that there exists an uncountable set of o-approximable varieties of m-groups each of which has continuum many o-approximable covers. In the lattice of o-approximable varieties of m-groups we find a variety that has no covers in this variety and no independent basis of identities.     

The equational theory of a nontrivial discriminator variety is co-NP-hard

Discriminator varieties play a central role in the classification of decidable varieties; and they arise naturally in the study of algebraic logics. There are also important connections with the reduction of theorem proving to equational logic. In this paper we show, for any nontrivial discriminator variety, that the problem of determining if an equation holds in the variety is co-NP-hard.Received August 24, 2002; accepted in final form August 5, 2004.     

Every Moufang loop of odd order with nontrivial commutant has nontrivial center

We get a partial result for Phillips’ problem: does there exist a Moufang loop of odd order with trivial nucleus? First we show that a Moufang loop Q of odd order with nontrivial commutant has nontrivial nucleus, then, by using this result, we prove that the existence of a nontrivial commutant implies the existence of a nontrivial center in Q. Introducing the notion of commutantly nilpotence, we get that the commutantly nilpotence is equivalent to the centrally nilpotence for the Moufang loops of odd order.     

A new model for automated examination timetabling

Automated examination timetabling has been addressed by a wide variety of methodologies and techniques over the last ten years or so. Many of the methods in this broad range of approaches have been evaluated on a collection of benchmark instances provided at the University of Toronto in 1996. Whilst the existence of these datasets has provided an invaluable resource for research into examination timetabling, the instances have significant limitations in terms of their relevance to real-world examination timetabling in modern universities. This paper presents a detailed model which draws upon experiences of implementing examination timetabling systems in universities in Europe, Australasia and America.     

On Cantor identities

In this paper, some special varieties which generalize Jónsson?CTarski algebras are considered. We prove that every nontrivial algebra from such a variety is term infinite and contains infinitely many distinct proper diagonal term operations of every arity.     

Completions of GBL-algebras: negative results

We provide a simple sufficient criterion to show that a given variety of GBL-algebras does not admit (local) completions. As corollaries, we obtain that no variety of GBL-algebras containing Chang’s chain, no nontrivial variety of ℓ-groups, nor the variety of product algebras admit completions. The first result strengthens a result of Gehrke and Priestley. Received August 10, 2006; accepted in final form March 8, 2007.