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.     

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.     

A cohomological property of <Emphasis Type="Italic">p</Emphasis>-groups

Let p be a prime, a finite p-group, any finite group with order divisible by p, and any action of on . We show that the cardinality of the set of all derivations with respect to this action is a multiple of p. This generalises theorems of Frobenius and Hall. Received: 16 June 2003     

Castelnuovo-Mumford regularity for nonhyperelliptic curves

The purpose here is to show that an irreducible, reduced, projective, nonhyperelliptic curve of degree d and genus g is n-regular for if Received: 10 July 2003     

Join congruence relations and closure relations

In this paper we show that, given a complete lattice , the following three lattices are the same: (1) the lattice of closure relations on , (2) the lattice of meet-closed subsets of , and (3) the lattice of complete join congruence relations on .     

Optimal Domains and Integral Representations of Convolution Operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(<Emphasis Type="Italic">G</Emphasis>)

Given , a compact abelian group G and a function , we identify the maximal (i.e. optimal) domain of the convolution operator (as an operator from L^p(G) to itself). This is the largest Banach function space (with order continuous norm) into which L^p(G) is embedded and to which has a continuous extension, still with values in L^p(G). Of course, the optimal domain depends on p and g. Whereas is compact, this is not always so for the extension of to its optimal domain. Several characterizations of precisely when this is the case are presented.     

Nonexistence of single blow-up solutions for a nonlinear Schrödinger equation involving critical Sobolev exponent

In this paper we shall consider the critical elliptic equation where and a(x) is a real continuous, non negative function, not identically zero. By using a local Pohozaev identity, we show that problem (0.1) does not admit a family of solutions which blows-up and concentrates as at some zero point x₀ of a(x) if the order of flatness of the function a(x) at x₀ is      

A unified approach to the topological centre problem for certain Banach algebras arising in abstract harmonic analysis

Let be a locally compact group. Consider the Banach algebra , equipped with the first Arens multiplication, as well as the algebra LUC , the dual of the space of bounded left uniformly continuous functions on , whose product extends the convolution in the measure algebra M . We present (for the most interesting case of a non-compact group) completely different - in particular, direct - proofs and even obtain sharpened versions of the results, first proved by Lau-Losert in [9] and Lau in [8], that the topological centres of the latter algebras precisely are and M , respectively. The special interest of our new approach lies in the fact that it shows a fairly general pattern of solving the topological centre problem for various kinds of Banach algebras; in particular, it avoids the use of any measure theoretical techniques. At the same time, deriving both results in perfect parallelity, our method reveals the nature of their close relation.Received: 1 January 2002     

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.     

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 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.     

On the duality between varieties and algebraic theories

It is proved that for any ultrametric space (X, d), the set L(X) of its closed balls is a lattice . It is complete, atomic, tree-like, and real graduated. For any such lattice , the set A(L) of its atoms can be naturally equipped with an ultrametric . These assignments are inverse of one another: where the first equality means an isometry while the second one is a lattice isomorphism. A similar correspondence established for morphisms, shows that there is an isomorphism of categories. The category ULTRAMETR of ultrametric spaces and non-expanding maps is isomorphic to the category LAT* of complete, atomic, tree-like, real graduated lattices and isotonic, semi-continuous, non-extensive maps. We describe properties of the isomorphism functor and its relations to the categorical operations and action of other functors. Basic properties of a space (such as completeness, spherical completeness, total boundedness, compactness, etc.) are translated into algebraic properties of the corresponding lattice L(X).     

Set Systems with few Disjoint Pairs

Let X = {1, . . . , n}, and let be a family of subsets of X. Given the size of , at least how many pairs of elements of must be disjoint? In this paper we give a lower bound for the number of disjoint pairs in . The bound we obtain is essentially best possible. In particular, we give a new proof of a result of Frankl and of Ahlswede, that if satisfies then contains at least as many disjoint pairs as X^(r).The situation is rather different if we restrict our attention to : then we are asking for the minimum number of edges spanned by a subset of the Kneser graph of given size. We make a conjecture on this lower bound, and disprove a related conjecture of Poljak and Tuza on the largest bipartite subgraph of the Kneser graph.* Research partially supported by NSF grant DMS-9971788     

Equations arising from Jordan *-derivation pairs

Summary. We study certain functional equations derived from the definition of a Jordan *-derivation pair. More precisely, if A is a complex *-algebra and M is a bimodule over A, having the structure of a complex vector space compatible with the structure of A, such that implies m = 0 and implies m = 0 and if are unknown additive mappings satisfying then E and F can be represented by double centralizers. The obtained result implies that one of the equations in the definition of a Jordan *-derivation pair is redundant. Furthermore, a remark on the extension of this result to unknown additive mappings such that is given in a special case.     

Some remarks on isologic extensions of groups

In this paper we study the concept of -isologisms among the -marginal extensions of groups, with respect to a given variety of groups . We also give some equivalent conditions under which two extensions are -isologic. Received: 9 January 2002     

On Shadows Of Intersecting Families

The shadow minimization problem for t-intersecting systems of finite sets is considered. Let be a family of k-subsets of . The -shadow of is the set of all (k-)-subsets contained in the members of . Let be a t-intersecting family (any two members have at least t elements in common) with . Given k,t,m the problem is to minimize (over all choices of ). In this paper we solve this problem when m is big enough.     

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.     

On quadratic differences that depend on the product of arguments - revisited

Summary. Let be a field of real or complex numbers and denote the set of nonzero elements of . Let be an abelian group. In this paper, we solve the functional equation f ₁ (x + y) + f ₂ (x - y) = f ₃ (x) + f ₄ (y) + g(xy) by modifying the domain of the unknown functions f ₃, f ₄, and g from to and using a method different from [3]. Using this result, we determine all functions f defined on and taking values on such that the difference f(x + y) + f (x - y) - 2 f(x) - 2 f(y) depends only on the product xy for all x and y in