Norm expansion along a zero variety

The reproducing kernel function of a weighted Bergman space over domains in Cd is known explicitly in only a small number of instances. Here, we introduce a process of orthogonal norm expansion along a subvariety of (complex) codimension 1, which also leads to a series expansion of the reproducing kernel in terms of reproducing kernels defined on the subvariety. The problem of finding the reproducing kernel is thus reduced to the same kind of problem when one of the two entries is on the subvariety. A complete expansion of the reproducing kernel may be achieved in this manner. We carry this out in dimension d=2 for certain classes of weighted Bergman spaces over the bidisk (with the diagonal z₁=z₂ as subvariety) and the ball (with z₂=0 as subvariety), as well as for a weighted Bargmann-Fock space over C² (with the diagonal z₁=z₂ as subvariety).     

Degenerate (r−2)-dimensional subvarieties through the generic hyperplane section of a reduced irreducible complex curve in Pr

We prove the following result: the generic degenerate (r−2)-dimensional subvariety through the generic hyperplane section of a complex reduced irreducible curve inP ^r is smooth at each point of the section     

Congruences on ockham algebras with pseudocomplementation

The variety pOconsists of those algebras (L;?,?,f,^*,0,1) where (L;?,?,f,0,1) is an Ockham algebra, (L;?,?,f,^*,0,1) is a p-algebra, and the unary operations fand ^*. commute. For an algebra in pK _ωwe show that the compact congruences form a dual Stone lattice and use this to determine necessary and sufficient conditions for a principal congruence to be complemented. We also describe the lattice of subvarieties of pK _1,1identifying therein the biggest subvariety in which every principal congruence is complemented, and the biggest subvariety in which the intersection of two principal congruences is principal.     

Monadic MV-algebras I: a study of subvarieties

In this paper, we study and classify some important subvarieties of the variety of monadic MV-algebras. We introduce the notion of width of a monadic MV-algebra and we prove that the equational class of monadic MV-algebras of finite width k is generated by the monadic MV-algebra [0, 1] ^k . We describe completely the lattice of subvarieties of the subvariety ${\mathcal{V}([{\bf 0}, {\bf 1}]^k)}$ generated by [0, 1] ^k . We prove that the subvariety generated by a subdirectly irreducible monadic MV-algebra of finite width depends on the order and rank of ?A, the partition associated to A of the set of coatoms of the boolean subalgebra B(A) of its complemented elements, and the width of the algebra. We also give an equational basis for each proper subvariety in ${\mathcal{V}([{\bf 0}, {\bf 1}]^k)}$ . Finally, we give some results about subvarieties of infinite width.     

Singular Supports for Character Sheaves on a Group Compactification

Let G be a semisimple adjoint group over C and Ḡ be the De Concini–Procesi completion of G. In this paper, we define a Lagrangian subvariety Λ of the cotangent bundle of Ḡ such that the singular support of any character sheaf on Ḡ is contained in Λ. Received: May 2006, Accepted: November 2006     

Equational Classes of Totally Ordered Modal Lattices

A modal lattice is a bounded distributive lattice endowed with a unary operator which preserves the join-operation and the smallest element. In this paper we consider the variety CH of modal lattices that is generated by the totally ordered modal lattices and we characterize the lattice of subvarieties of CH. We also give an equational basis for each subvariety of CH.     

Global classification of the planar Lotka–Volterra differential systems according to their configurations of invariant straight lines

In this article, we study the Lotka–Volterra planar quadratic differential systems. We denote by LV systems all systems which can be brought to a Lotka–Volterra system by an affine transformation and time homotheties. All these systems possess invariant straight lines. We classify the family of LV systems according to their geometric properties encoded in the configurations of invariant straight lines which these systems possess. We obtain a total of 65 such configurations which are distinguished, roughly speaking, by the multiplicity of their invariant lines and by the multiplicities of the singularities of the systems located on these lines. We determine an algebraic subvariety of \mathbbR¹²{\mathbb{R}^{12}} which contains all these systems and we find the bifurcation diagram of the configurations of LV systems within this algebraic subvariety, in terms of polynomial invariants with respect to the group action of affine transformations and time homotheties. This geometric classification will serve as a basis for the full topological classification of LV systems.     

Degree of the divisor of solutions of a differential equation on a projective variety

Using the data schemes from [1] we give a rigorous definition of algebraic differential equations on the complex projective space Pⁿ. For an algebraic subvariety S?Pⁿ, we present an explicit formula for the degree of the divisor of solutions of a differential equation on S and give some examples of applications. We extend the technique and result to the real case.     

Semisimplicity and the discriminator in bounded BCK-algebras

By means of an adaptation of a proof given by T. Kowalski in [16], we show that the members of a relative subvariety V of bounded BCK-algebras are all semisimple if and only if V is a variety and satisfies the identity

Lower Bounds on Genera of Subvarieties of Generic Hypersurfaces

Abstract

A second-order invariant of C. Voisin gives a powerful method for bounding from below the geometric genus of a k-dimensional subvariety of a degree dhypersurface in complex projective n-space. This work uses the Voisin method to establish a general bound, which lies behind recent results of G. Pacienza and Z. Ran.     

Varieties of two-dimensional cylindric algebras II

In [2] we investigated the lattice (Df₂) of all subvarieties of the variety Df₂ of two-dimensional diagonal free cylindric algebras. In the present paper we investigate the lattice (CA₂) of all subvarieties of the variety CA₂ of two-dimensional cylindric algebras. We prove that the cardinality of (CA₂) is that of the continuum, give a criterion for a subvariety of CA₂ to be locally finite, and describe the only pre locally nite subvariety of CA2. We also characterize nitely generated subvarieties of CA2 by describing all fteen pre nitely generated subvarieties of CA₂. Finally, we give a rough picture of (CA₂), and investigate algebraic properties preserved and reected by the reduct functors .     

Characterizations of certain completely regular varieties

Abstract. We characterize orthogroups, local orthogroups and (left,right) cryptogroups within completely regular semigroups by means of absence of certain kind of subsemigroups. For each of these varieties V , we determine the complete set of minimal non-V -varieties. For each of the latter varieties, we determine the lattice of its subvarieties. We then give a generating semigroup and a basis of its identities for every variety which occurs in this way. The subvariety lattices are illustrated by three diagrams.     

An abstract version of the Morse index theorem and its application to hypersurfaces of constant mean curvature

In [5] Smale generalized the Morse index theorem (originally proved by Morse in [3]) to elliptic partial differential systems in several independent variables. Smale's result was used by Simons in [4] to obtain the index theorem for minimal submanifolds. The purpose of this paper is to give an abstract version of the Morse index theorem and use it to prove an index theorem for hypersurfaces of constant mean curvature. This was sugested by Barbosa and do Carmo in [1].     

Varieties of two-dimensional cylindric algebras.¶Part I: Diagonal-free case

We investigate the lattice of all subvarieties of the variety Df ₂ of two-dimensional diagonal-free cylindric algebras. We prove that a Df ₂-algebra is finitely representable if it is finitely approximable, characterize finite projective Df ₂-algebras, and show that there are no non-trivial injectives and absolute retracts in Df ₂. We prove that every proper subvariety of Df ₂ is locally finite, and hence Df ₂ is hereditarily finitely approximable. We describe all six critical varieties in , which leads to a characterization of finitely generated subvarieties of Df ₂. Finally, we describe all square representable and rectangularly representable subvarieties of Df ₂. Received May 25, 2000; accepted in final form November 2, 2001.     

Varieties of restriction semigroups and varieties of categories

The variety of restriction semigroups may be most simply described as that generated from inverse semigroups (S, ·, ^?1) by forgetting the inverse operation and retaining the two operations x⁺ = xx^?1 and x* = x^?1x. The subvariety B of strict restriction semigroups is that generated by the Brandt semigroups. At the top of its lattice of subvarieties are the two intervals [B₂, B₂M = B] and [B₀, B₀M]. Here, B₂ and B₀ are, respectively, generated by the five-element Brandt semigroup and that obtained by removing one of its nonidempotents. The other two varieties are their joins with the variety of all monoids. It is shown here that the interval [B₂, B] is isomorphic to the lattice of varieties of categories, as introduced by Tilson in a seminal paper on this topic. Important concepts, such as the local and global varieties associated with monoids, are readily identified under this isomorphism. Two of Tilson's major theorems have natural interpretations and application to the interval [B₂, B] and, with modification, to the interval [B₀, B₀M] that lies below it. Further exploration may lead to applications in the reverse direction.     

Generic vanishing index and the birationality of the bicanonical map of irregular varieties

We prove that any smooth complex projective variety with generic vanishing index bigger or equal than 2 has birational bicanonical map. Therefore, if X is a smooth complex projective variety φ with maximal Albanese dimension and non-birational bicanonical map, then the Albanese image of X is fibred by subvarieties of codimension at most 1 of an abelian subvariety of Alb X.     

Two Notes on the Variety Generated by Planar Modular Lattices

Let Var(M _plan) denote the variety generated by the class M _plan of planar modular lattices. In 1977, based on his structural investigations, R. Freese proved that Var(M _plan) has continuumly many subvarieties. The present paper provides a new approach to this result utilizing lattice identities. We also show that each subvariety of Var(M _plan) is generated by its planar (subdirectly irreducible) members. Dedicated to the memory of András P. Huhn This research was partially supported by the NFSR of Hungary (OTKA), grant no. T 049433, T 48809 and K 60148.     

Regular groups in varieties ofp-groups

It is shown that, if the regular groups in an irregular locally-finite varietyV ofp-groups form a subvariety, then these regular groups arep-abelian. The question of the possible structure ofV is also discussed. National Library of Australia card number and ISBN 0 86890 015 X     

Pseudocomplemented semilattices are finite-to-finite relatively universal

It is shown that the category of directed graphs is isomorphic to a subcategory of the variety S of all pseudocomplemented semilattices which contains all homomorphisms whose images do not lie in the subvariety B of all Boolean pseudocomplemented semilattices. Moreover, the functor exhibiting the isomorphism may be chosen such that each finite directed graph is assigned a finite pseudocomplemented semilattice. That is to say, it is shown that the variety S of all pseudocomplemented semilattices is finite-to-finite B-relatively universal. This illustrates the complexity of the endomorphism monoids of pseudocomplemented semilattices since it follows immediately that, for any monoid M, there exists a proper class of non-isomorphic pseudocomplemented semilattices such that, for each member S, the endomorphisms of S which do not have an image contained in the skeleton of S form a submonoid of the endomorphism monoid of S which is isomorphic to M. Received June 17, 2006; accepted in final form May 8, 2007.     

Radical theory in varieties of near-rings in which the constants form

Not all the good properties of the Kurosh-Amitsur radical theory in the variety of associative rings are preserved in the bigger variety of near-rings. In the smaller and better behaved variety of O-symmetric near-rings the theory is much more satisfactory. In this note we show that many of the results of the 0-symmetric near-ring case can be extended to a much bigger variety of near-rings which, amongst others, includes all the O-symmetric as well as the constant near-rings. The varieties we shall consider are varieties of near-rings, called Fuchs varieties, in which the constants form an ideal. The good arithmetic of such varieties makes it possible to derive more explicit conditions.

(i) for the subvariety of constant near-rings to be a semisimple class (or equivalently, to have attainable identities),

(ii) for semisimple classes to be hereditary.

We shall prove that the subvariety of 0-symmetric near-rings has attainable identities in a Fuchs variety, and extend the theory of overnilpotent radicals of 0-symmetric near-rings to the largest Fuchs variety F

The near-ring construction of [7] will play a decisive role in our investigations.