A characterization of minimal locally finite varieties

In this paper we describe a one-variable Mal'cev-like condition satisfied by any locally finite minimal variety. We prove that a locally finite variety is minimal if and only if it satisfies this Mal'cev-like condition and it is generated by a strictly simple algebra which is nonabelian or has a trivial subalgebra. Our arguments show that the strictly simple generator of a minimal locally finite variety is unique, it is projective and it embeds into every member of the variety. We give a new proof of the structure theorem for strictly simple abelian algebras that generate minimal varieties.

     

Über endliche galoissche schiefkörpererweiterungen ohne endliche galoisgruppen

We define a generalized Brauer-Severi variety to be the projective Variety whose closed points correspond to rank m right ideals in a central simple algebra. We show that these varieties are forms of the Grassmann variety and their function fields are generic partial splitting fields for the associated algebras. The subgroup of the Brauer group split by the function field is calculated. Following Schofield and van den Bergh, we calculate the index of a simple ailgebra extended by the function field of a generalized Brauer-Severi variety associated with any other central simple algebra.     

Generators of existence varieties of regular rings and complemented Arguesian lattices

We proved in an earlier work that any existence variety of regular algebras is generated by its simple unital Artinian members, while any existence variety of Arguesian sectionally complemented lattices is generated by its simple members of finite length. A characterization of the class of simple unital Artinian members [members of finite length, respectively] of such varieties is given in the present paper.     

Constructions Over Tournaments

We investigate tournaments that are projective in the variety that they generate, and free algebras over partial tournaments in that variety. We prove that the variety determined by three-variable equations of tournaments is not locally finite. We also construct infinitely many finite, pairwise incomparable simple tournaments.     

Almost all minimal idempotent varieties are congruence modular

We show that, up to term equivalence, the only minimal idempotent varieties that are not congruence modular are the variety of sets and the variety of semilattices. From this it follows that a minimal idempotent variety that is not congruence distributive is term equivalent to the variety of sets, the variety of semilattices, or a variety of affine modules over a simple ring. Received March 29, 1999; accepted in final form February 8, 2000.     

Irreducible orthomodular lattices which are simple

It is well known that for a chain finite orthomodular lattice, all congruences are factor congruences, so any directly irreducible chain finite orthomodular lattice is simple. In this paper it is shown that the notions of directly irreducible and simple coincide in any variety generated by a set of orthomodular lattices that has a uniform finite upper bound on the lengths of their chains. The prototypical example of such a variety is any variety generated by a set ofn dimensional orthocomplemented projective geometries.Presented by B. Jónsson.Supported by a grant from NSERC.     

Index reduction formulas for twisted flag varieties, I.

A general formula is proved for the change in the Schur index of a central simple algebra on passing from the ground field F to the function field F(X) of a twisted flag variety X, i.e., a projective variety such that there is an adjoint semisimple algebraic group G acting on X over F such that the action becomes transitive over the separable closure of F. The general formula encompasses special cases previously proved where X is a Brauer-Severi variety, or a generic partial splitting variety of a central simple algebra, or the transfer of such a variety, a quadric, or the involution variety of an algebra with orthogonal involution. For the classical simple groups G of inner type, all the corresponding varieties X are described, and the specific index reduction formula is given for each such X.The second author would like to express his thanks to J.-L. Colliot-Thélène for stimulating discussions on this subject.Supported in part by the NSF.     

Desingularization preserving stable simple normal crossings

The subject is partial resolution of singularities. Given an algebraic variety X (not necessarily equidimensional) in characteristic zero (or, more generally, a pair (X, D), where D is a divisor on X), we construct a functorial desingularization of all but stable simple normal crossings (stable-snc) singularities, by smooth blowings-up that preserve such singularities. A variety has stable simple normal crossings at a point if, locally, its irreducible components are smooth and transverse in some smooth embedding variety. We also show that our main assertion is false for more general simple normal crossings singularities.     

On minimal simple lattices

The class of simple lattices, excluding the two-element lattice, is preordered by placing L₀L₁ iff L₀ belongs to the variety generated by L₁. We prove that the finite simple lattices minimal under this pre-ordering generate the variety of all lattices. We construct infinite simple lattices minimal under the ordering, and others that contain, in the sense of the ordering, no minimal simple lattices.Presented by G. McNulty.Research supported by National Science Foundation Grant DMS 89 04014.     

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.     

Simple solution procedures for nonlinear programming problems that are derivative decomposable

This paper describes a set of relatively simple procedures that are useful for solving a number of nonlinear programming problems. These problems are characterized by objective and constraint functions that are what we call “derivative decomposable”. Starting with a relaxed problem, we show how derivative decomposability yields a simple solution procedure. Then we demonstrate how slightly modified procedures can solve a variety of more complex problems displaying derivative decomposability. The solution procedures are easily understood in terms of their graphical representations. Furthermore, their simplicity and flexibility promise significant computational advantages for a variety of applications.     

A Short Basis for the Variety of Digroups

A digroup is an algebra defined on a set with two associative binary operations. Digroups play a prominent role in an important open problem from the theory of Leibniz algebras. We give a simple basis of independent axioms for the variety of digroups that emphasizes the two core semigroup structures undergirding each algebra in this variety.     

Wegner estimates and localization for continuum Anderson models with some singular distributions

We give a simple geometric proof of Wegner's estimate which leads to a variety of new results on localization for multi-dimensional random operators.     

Resolution except for minimal singularities I

The philosophy of this article is that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities. The idea is used in two related problems: (1) We give a proof of resolution of singularities of a variety or a divisor, except for simple normal crossings (i.e., which avoids blowing up simple normal crossings, and ends up with a variety or a divisor having only simple normal crossings singularities). (2) For more general normal crossings (in a local analytic or formal sense), such a result does not hold. We find the smallest class of singularities (in low dimension or low codimension) with which we necessarily end up if we avoid blowing up normal crossings singularities. Several of the questions studied were raised by Kollár.     

一致极小对,一致有界性及有穷基定理

本文使用最小板块的思想，揭示了泛代数领域中任一分配簇的任一代数都具有两个性质；（１）投射意义下一致极小对是存在的。（２）存在相对于整个族而言的极小对投射的一致上界。用以上的结果，给出了著名的有穷基定理的一个简单的、构造性的证明。     

Lefschetz classes of simple factors of the Jacobian variety of a Fermat curve of prime degree over finite fields

We give a necessary and sufficient condition in terms of a matrix for which all Tate classes are Lefschetz for simple abelian varieties over an algebraic closure of a finite field. As an application, we prove under an assumption that all Tate classes are Lefschetz for simple factors of the Jacobian variety of a Fermat curve of prime degree.     

Bioriented flags and resolutions of Schubert varieties

We use incidence relations running in two directions in order to construct a Kempf–Laksov type resolution for any Schubert variety of the complete flag manifold but also an embedded resolution for any Schubert variety in the Grassmannian. These constructions are alternatives to the celebrated Bott–Samelson resolutions. The second process led to the introduction of W-flag varieties, algebro-geometric objects that interpolate between the standard flag manifolds and products of Grassmannians, but which are singular in general. The surprising simple desingularization of a particular such type of variety produces an embedded resolution of the Schubert variety within the Grassmannian.     

Two finitely generated varieties having no infinite simple members

Using a method of R. McKenzie, we construct a finitely generated semisimple variety of infinite type, and a finitely generated nonsemisimple variety of finite type, both having arbitrarily large finite but no infinite simple members. This amplifies M. Valeriote's negative solution to Problem 11 from Hobby and McKenzie, The Structure of Finite Algebras.

     

Completely Regular Semigroup Varieties Whose Free Objects Have Weakly Permutable Fully Invariant Congruences

We prove that a variety of completely regular semigroups has the property from the title if and only if it consists of either completely simple semigroups or semilattices of groups.