Cancellative residuated lattices

Cancellative residuated lattices are natural generalizations of lattice-ordered groups ( -groups). Although cancellative monoids are defined by quasi-equations, the class of cancellative residuated lattices is a variety. We prove that there are only two commutative subvarieties of that cover the trivial variety, namely the varieties generated by the integers and the negative integers (with zero). We also construct examples showing that in contrast to -groups, the lattice reducts of cancellative residuated lattices need not be distributive. In fact we prove that every lattice can be embedded in the lattice reduct of a cancellative residuated lattice. Moreover, we show that there exists an order-preserving injection of the lattice of all lattice varieties into the subvariety lattice of .We define generalized MV-algebras and generalized BL-algebras and prove that the cancellative integral members of these varieties are precisely the negative cones of -groups, hence the latter form a variety, denoted by . Furthermore we prove that the map that sends a subvariety of -groups to the corresponding class of negative cones is a lattice isomorphism from the lattice of subvarieties of to the lattice of subvarieties of . Finally, we show how to translate equational bases between corresponding subvarieties, and briefly discuss these results in the context of R. McKenzies characterization of categorically equivalent varieties.     

Investigation of subalgebra lattices by means of Hasse constants

Hasse constants and their basic properties are introduced to facilitate the connection between the lattice of subalgebras of an algebra and the natural action of the automorphism group Aut( ) on . These constants are then used to describe the lattice of subloops of the smallest nonassociative simple Moufang loop.     

Random generators of given orders and the smallest simple Moufang loop

The probability that m randomly chosen elements of a finite power associative loop have prescribed orders and generate is calculated in terms of certain constants related to the action of Aut( ) on the subloop lattice of . As an illustration, all meaningful probabilities of random generation by elements of given orders are found for the smallest nonassociative simple Moufang loop.     

Comparing the regular and the restricted regular semidirect products

We compare the two recently introduced semidirect product operations *_r and *_rr within the lattice of e-varieties of locally inverse semigroups. For each e-variety which contains all rectangular bands and is properly contained in the e-variety of all completely simple semigroups, the inclusions are proved where is the e-variety of all semilattices and the variety of all abelian groups of exponent dividing q where q is any integer greater than one. Some consequences for the class of finite locally inverse semigroups are also obtained.     

Transformations of Grassmannians preserving the class of base subsets

Let be a finite-dimensional projective space and be the Grassmannian consisting of all k-dimensional subspaces of . In the paper we show that transformations of sending base subsets to base subsets are induced by collineations of to itself or to the dual projective space . This statement generalizes the main result of the authors paper [19].     

On the structure of level sets of uniform and Lipschitz quotient mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{R} $$

We study two questions posed by Johnson, Lindenstrauss, Preiss, and Schechtman, concerning the structure of level sets of uniform and Lipschitz quotient mappings from . We show that if , is a uniform quotient mapping then for every has a bounded number of components, each component of separates and the upper bound of the number of components depends only on and the moduli of co-uniform and uniform continuity of .Next we prove that all level sets of any co-Lipschitz uniformly continuous mapping from to are locally connected, and we show that for every pair of a constant and a function with , there exists a natural number , so that for every co-Lipschitz uniformly continuous map with a co-Lipschitz constant and a modulus of uniform continuity , there exists a natural number and a finite set with card so that for all has exactly components, has exactly components and each component of is homeomorphic with the real line and separates the plane into exactly 2 components. The number and form of components of for are also described - they have a finite tree structure.     

The dimension of the kernel in finite and infinite intersections of starshaped sets

Summary. We establish the following Helly-type result for infinite families of starshaped sets in Define the function f on {1, 2} by f(1) = 4, f(2) = 3. Let be a fixed positive number, and let be a uniformly bounded family of compact sets in the plane. For k = 1, 2, if every f(k) (not necessarily distinct) members of intersect in a starshaped set whose kernel contains a k-dimensional neighborhood of radius , then is a starshaped set whose kernel is at least k-dimensional. The number f(k) is best in each case. In addition, we present a few results concerning the dimension of the kernel in an intersection of starshaped sets in Some of these involve finite families of sets, while others involve infinite families and make use of the Hausdorff metric.     

A classification of martingale Hardy spaces associated with rearrangement-invariant function spaces

Let X be a rearrangement-invariant Banach function space over a complete probability space , and denote by the Hardy space consisting of all martingales such that . We prove that implies for any filtration if and only if Doobs inequality holds in X, where denotes the martingale defined by , n = 0, 1, 2, ..., and a.s.Received: 1 August 2000     

Compact factor congruences imply Boolean factor congruences

We prove that any variety in which every factor congruence is compact has Boolean factor congruences, i.e., for all A in the set of factor congruences of A is a distributive sublattice of the congruence lattice of A.     

On generating countable sets of endomorphisms

Sierpiski proved that every countable set of mappings on an infinite set X is contained in a 2-generated subsemigroup of the semigroup of all mappings on X. In this paper we prove that every countable set of endomorphisms of an algebra which has an infinite basis (independent generating set) is contained in a 2-generated subsemigroup of the semigroup of all endomorphisms of .     

Set of involutions arising from an egglike inversive plane

To every egglike inversive plane there is associated a family of involutions of the point set of such that circles of are the fixed point sets of the involutions in . Korchmaros and Olanda characterized a family of involutions on a set of size n² + 1to be for an egglike inversive plane of order n by four conditions. In this paper, we give an alternative proof where the Galois space PG(3,n) in which is embedded is built up directly by using concepts and results on finite linear spaces.     

Beckman-Quarles type theorems for mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{C}^n $$

Summary. Let We say that preserves the distance d 0 if for each implies Let A _n denote the set of all positive numbers d such that any map that preserves unit distance preserves also distance d. Let D _n denote the set of all positive numbers d with the property: if and then there exists a finite set S _xy with such that any map that preserves unit distance preserves also the distance between x and y. Obviously, We prove: (1) (2) for n 2 D _n is a dense subset of (2) implies that each mapping f from to (n 2) preserving unit distance preserves all distances, if f is continuous with respect to the product topologies on and      

A class of bounded operators on Sobolev spaces

We describe a class of nonlinear operators which are bounded on the Sobolev spaces , for and 1 < p < . As a corollary, we prove that the Hardy-Littlewood maximal operator is bounded on , for and 1 < p < ; this extends the result of J. Kinnunen [7], valid for s = 1. Received: 5 December 2000     

A note on the isotopy problem for $$ \mathbb{R}^2 $$ -planes

For a class of stable planes we define a notion of isotopy equivalence with respect to that class and prove that any two planes of a certain class of -planes comprising all affine -planes are isotopy equivalent. Furthermore we obtain that all affine -planes are isotopy equivalent in the class of affine -planes. Finally we give an example which shows that this approach cannot be easily generalized to 2-dimensional projective planes, and we outline a different way for a possible generalization.Received: 27 April 2001     

The Laplacian on $$ C(\overline \Omega) $$ with generalized Wentzell boundary conditions

In this note we prove that the Laplacian with generalized Wentzell boundary conditions on an open bounded regular domain in defined by generates an analytic semigroup of angle on for every > 0 and (for the definition of cf. (1.3)).Received: 13 July 2002     

On the Combinatorics of Rooted Binary Phylogenetic Trees

We study subtree-prune-and-regraft (SPR) operations on leaf-labelled rooted binary trees, also known as rooted binary phylogenetic trees. This study is motivated by the problem of graphically representing evolutionary histories of biological sequences subject to recombination. We investigate some basic properties of the induced SPR-metric on the space of leaf-labelled rooted binary trees with n leaves. In contrast to the case of unrooted trees, the number |U(T)| of trees in which are one SPR operation away from a given tree depends on the topology of T. In this paper, we construct recursion relations which allow one to determine the unit-neighbourhood size |U(T)| efficiently for any tree topology. In fact, using the recursion relations we are able to derive a simple closed-form formula for the unit-neighbourhood size. As a corollary, we construct sharp upper and lower bounds on the size of unit-neighbourhoods and investigate the diameter of . Lastly, we consider an enumeration problem relevant to population genetics.AMS Subject Classification: 05C05, 92D15.     

Cyclicity of zeros of families of analytic functions

Let be a family of holomorphic functions in the unit disk , which are also holomorphic in a parameter . We express cyclicity (=generalized multiplicity) of a zero of at via some algebraic characteristics of the ideal generated by the Taylor coefficients of . As an example we estimate the cyclicity of the family of generalized exponential polynomials.     

On an Algebra of Toeplitz Operators With Piecewise Continuous Symbols

Let and be a finite collection of smooth curves in D. Given k points consider the family of all bounded and continuous functions on with finite limits at and radial limits at z_k. We study the Toeplitz operator algebra corresponding to M_r and we prove that its Calkin algebra is isomorphic to the algebra of all continuous functions on some compact set. This fact implies that the commutator of two Toeplitz operators with this kind of symbols is compact. We also prove that the semi-commutator of such Toeplitz operators is not compact, in general.     

Equidistribution of Kronecker sequences along closed horocycles

It is well known that (i) for every irrational number the Kronecker sequence m (m = 1,...,M) is equidistributed modulo one in the limit , and (ii) closed horocycles of length become equidistributed in the unit tangent bundle of a hyperbolic surface of finite area, as . In the present paper both equidistribution problems are studied simultaneously: we prove that for any constant the Kronecker sequence embedded in along a long closed horocycle becomes equidistributed in for almost all , provided that . This equidistribution result holds in fact under explicit diophantine conditions on (e.g. for = 2) provided that , with additional assumptions on the Fourier coefficients of certain automorphic forms. Finally, we show that for , our equidistribution theorem implies a recent result of Rudnick and Sarnak on the uniformity of the pair correlation density of the sequence n² modulo one.