Varieties and Vector Measures

A locally convex space L has the property ? if equicontinuous subsets of L* are weak-star sequentially compact. (L*, σ(L*, L)) is a MAZUR space if given F ∈ L** with F weak-star sequentially continuous then F∈ L. If L is complete with the property ∈, then (L*, σ (L*, L)) is a MAZUR space. The class of locally convex spaces with the property ? forms a variety ??_? and this variety is generated by the BANACH spaces it contains. Weakly compactly generated locally convex spaces and SCHWARTZ spaces belong to ??_?. MAZUR spaces are used to give a characterization of GROTHENDIECK BANACH spaces. The last section contains a characterization of the variety generated by the reflexive BANACH spaces.     

Hausdorff properties of topological algebras

Let P be a property of topological spaces. Let [P] be the class of all varieties having the property that any topological algebra in has underlying space satisfying property P. We show that if P is preserved by finite products, and if is preserved by ultraproducts, then [P] is a class of varieties that is definable by a Maltsev condition.?The property that all T ₀ topological algebras in are j-step Hausdor. (Hj) is preserved by finite products, and its negation is preserved by ultraproducts. We partially characterize the Maltsev condition associated to by showing that this topological implication holds in every (2j + 1)-permutable variety, but not in every (2j + 2)-permutable variety.?Finally, we show that the topological implication holds in every k-permutable, congruence modular variety. Received March 1, 2000; accepted in final form October 18, 2001.     

Definable principal subcongruences

For varieties of algebras, we present the property of having "definable principal subcongruences" (DPSC), generalizing the concept of having definable principal congruences. It is shown that if a locally finite variety V of finite type has DPSC, then V has a finite equational basis if and only if its class of subdirectly irreducible members is finitely axiomatizable. As an application, we prove that if A is a finite algebra of finite type whose variety V(A) is congruence distributive, then V(A) has DPSC. Thus we obtain a new proof of the finite basis theorem for such varieties. In contrast, it is shown that the group variety V(S ₃ ) does not have DPSC. Received May 9 2000; accepted in final form April 26, 2001.     

Persistency of linear programming relaxations for the stable set problem

The Nemhauser–Trotter theorem states that the standard linear programming (LP) formulation for the stable set problem has a remarkable property, also known as (weak) persistency: for every optimal LP solution that assigns integer values to some variables, there exists an optimal integer solution in which these variables retain the same values. While the standard LP is defined by only non-negativity and edge constraints, a variety of other LP formulations have been studied and one may wonder whether any of them has this property as well. We show that any other formulation that satisfies mild conditions cannot have the persistency property on all graphs, unless it is always equal to the stable set polytope.

     

Closedness Properties of Internal Relations III: Pointed Protomodular Categories

A pointed variety of universal algebras is protomodular in the sense of D. Bourn, if and only if it is classically ideal determined in the sense of A. Ursini (this result is due to D. Bourn and G. Janelidze). We prove a characterization theorem for pointed protomodular categories, which is a (pointed) categorical version of Ursini’s characterization theorem for classically ideal determined varieties, involving classically 0-regular algebras. A suitable simplification of the property of a pair of relations, which is used to define a classically 0-regular algebra, yields a new closedness property of a single binary relation – we show that a finitely complete pointed category is protomodular if and only if every binary internal relation R→A ² in it has this closedness property. Partially supported by South African National Research Foundation, and Georgian National Science Foundation (GNSF/ST06/3-004).     

Almost isometric actions,property (<Emphasis Type="Italic">T</Emphasis>), and local rigidity

Let Γ be a discrete group with property (T) of Kazhdan. We prove that any Riemannian isometric action of Γ on a compact manifold X is locally rigid. We also prove a more general foliated version of this result. The foliated result is used in our proof of local rigidity for standard actions of higher rank semisimple Lie groups and their lattices in [FM2].One definition of property (T) is that a group Γ has property (T) if every isometric Γ action on a Hilbert space has a fixed point. We prove a variety of strengthenings of this fixed point properties for groups with property (T). Some of these are used in the proofs of our local rigidity theorems. To Yakov G. Sinai on his 70th birthday     

FINITE REPLACEMENT AND FINITE HILBERT-STYLE AXIOMATIZABILITY

We define a property for varieties V, the f.r.p. (finite replacement property). If it applies to a finitely based V then V is strongly finitely based in the sense of [14], see Theorem 2. Moreover, we obtain finite axiomatizability results for certain propositional logics associated with V, in its generality comparable to well-known finite base results from equational logic. Theorem 3 states that each variety generated by a 2-element algebra has the f.r.p. Essentially this implies finite axiomatizability of a 2-valued logic in any finite language.     

Property (H) in Köthe-Bochner spaces

The property (H) in Köthe-Bochner space E(X), where E is a locally uniformly rotund Köthe function space and X is an arbitrary Banach space, is discussed. Specifically, the question of whether or not this geometrical property lifts from X to E(X) is examined. Among others it is proved that E(X) has the property (H) whenever X has the property (G). Moreover, it is shown that the property (H) does not lift from X to E(X) when the Köthe space E is over a measure space in which the measure is not purely atomic.     

Semiautomorphic inverse property loops

We define a variety of loops called semiautomorphic, inverse property loops that generalize Moufang and Steiner loops. We first show an equivalence between a previously studied variety of loops. Next we extend several known results for Moufang and Steiner loops. That is, the commutant is a subloop and if a is in the commutant, then a² is a Moufang element, a³ is a c-element and a⁶ is in the center. Finally, we give two constructions for semiautomorphic inverse property loops based on Chein’s and de Barros and Juriaans’ doubling constructions.     

Schreier varieties of algebras

Some necessary and sufficient conditions for a variety of algebras to possess the Schreier property are given. It is proved that the variety of algebras defined by the identity xx ² = 0is a Schreier variety. Translated fromAlgebra i Logika, Vol. 33, No. 3, pp. 317–340, May–June, 1994.     

On the difference property of the family of functions with the Baire property

We show that it is consistent with ZFC that the family of functions with the Baire property has the difference property. That is, every function for which f(x + h)-f(x) has the Baire property for every h∈R is of the form f=g + Awhere g has the Baire property and A is additive. This revised version was published online in June 2006 with corrections to the Cover Date.     

W v paths on the torus

We say that a cell complex has theW _v property provided any two vertices can be joined by a path that never returns to a facet once it leaves it. The boundaries of convex polytopes have been shown to have theW _v property for polytopes of dimensions at most 3. We extend the three-dimensional result to polyhedral maps on the torus by showing that all such maps have theW _v property. We also show that theW _v property for all polyhedral maps on manifolds of a given genus is equivalent to the property that for all maps on manifolds of that genus each two faces lie in a subcomplex that is a cell.     

Making doughnuts of Cohen reals

For a ? b ? ω with b\ a infinite, the set D = {x ∈ [ω]^ω : a ? x ? b} is called a doughnut. A set S ? [ω]^ω has the doughnut property ?? if it contains or is disjoint from a doughnut. It is known that not every set S ? [ω]^ω has the doughnut property, but S has the doughnut property if it has the Baire property ?? or the Ramsey property ?. In this paper it is shown that a finite support iteration of length ω₁ of Cohen forcing, starting from L , yields a model for CH + (??) + (??) + (?).     

Simple <Emphasis Type="Italic">SL</Emphasis>(<Emphasis Type="Italic">n</Emphasis>)-modules with normal closures of maximal torus orbits

Let T be the subgroup of diagonal matrices in the group SL(n). The aim of this paper is to find all finite-dimensional simple rational SL(n)-modules V with the following property: for each point v∈V the closure [`(Tv)]\overline{Tv} of its T-orbit is a normal affine variety. Moreover, for any SL(n)-module without this property a T-orbit with non-normal closure is constructed. The proof is purely combinatorial: it deals with the set of weights of simple SL(n)-modules. The saturation property is checked for each subset in the set of weights.     

On the Unitary Binary Group

One consider the variety of the unitary binary group in n variables and it is shown that this algebraic variety is rational and it has n² parameters. Then it is given a parametrical rational representation of this variety.AMS Subject Classification (1991) 20G20     

Generic points of quadrics and Chow groups

In this article we study the sufficient conditions for the k̅-defined element of the Chow group of a smooth variety to be k-rational (defined over k). For 0-cycles this question was addressed earlier. Our methods work for cycles of arbitrary dimension. We show that it is sufficient to check this property over the generic point of a quadric of sufficiently large dimension. Among the applications one should mention the uniform construction of fields with all known u-invariants.     

MCD-finite Domains and Ascent of IDF Property in Polynomial Extensions

An integral domain is said to have the IDF property when every non-zero element of it has only a finite number of non-associate irreducible divisors. A counterexample has already been found showing that the IDF property does not necessarily ascend in polynomial extensions. In this paper, we introduce a new class of integral domains, called MCD-finite domains, and show that for any domain D, D[X] is an IDF domain if and only if D is both IDF and MCD-finite. This result entails all the previously known sufficient conditions for the ascent of the IDF property. Our new characterization of polynomial domains with the IDF property enables us to use a different construction and build another counterexample which strengthen the previously known result on this matter.     

Confident Search

Abstract

The task of searching for the best element or a good element in a large set P is central to many problems in artificial intelligence and related fields. Often, heuristic information is used to reduce the scope of the search; however, in many instances, this information carries no guarantee of good performance. This article begins with an arbitrary heuristic search procedure and supplies it with a confidence statement of the following form: With specified high probability β, the output of the confidence procedure will be among the best 100α% of the elements of P. The confidence procedure will report either the outcome of the heuristic search or a better alternative with the required properties; that is, it will either certify that the heuristic answer has the desired confidence property or it will produce a better answer having the property. The approach involves combining heuristic search with a form of heuristic sampling that tends to sample the better elements of P. The sample is designed in such a way that the best element in the sample has the desired confidence property—if the answer produced by the heuristic search is better still, it inherits the confidence property. Various devices permit the sampling procedure to retain its confidence property while (1) moving the sample in the direction suggested by the heuristic, (2) adjusting the heuristic preference in response to what is learned during sampling, and (3) reorganizing the sampling whenever promising discoveries are made by chance.     

A weak form of interpolation in equational logic

The notions of a weak interpolation property and of weak amalgamation are introduced. It is proved that in varieties with the congruence extension property, the weak interpolation property is equivalent to the weak amalgamation property. In turn, weak amalgamability of a variety is equivalent to amalgamability of a class of finitely generated simple algebras in this variety. Supported by RFBR (grant Nos. 06-01-00358 and 05-01-04003-NNIOa) and by INTAS (grant No. 04-77-7080). __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 94–107, January–February, 2008.     

Reflexive domains and fixed points

In a recent paper in this journal, J. Soto-Andrade and F. J. Varela draw attention to the fact that ifR is a retract of a reflexive domain in a suitable category thenR has the fixed point property. They suggest [1], pp. 1 and 18, that conversely every structure with the fixed point property is a retract of a reflexive domain. In this note it is shown that ifR is a retract of a reflexive domain thenR ^R has the fixed point property. This leads to counterexamples to the suggestion of Soto-Andrade and Varela in the categoryPo of partially ordered sets and monotone maps.