Relations compatible with near unanimity operations

Given a system of k-ary relations on a finite set A which are compatible with a (k + 1)-ary near unanimity operation on A, we provide a characterization of when is the system of all k-ary subuniverses of an algebra A on A.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived August 26, 2003; accepted in final form July 10, 2004.     

Cone-separation and star-shaped separability with applications

Let X$X$ be a (real) Banach space, A$A$ be a subset of X$X$ and x∉A$x\notin A$. We present cone-separation in terms of separation by a collection of linear functionals defined on X$X$ and obtain necessary and sufficient conditions for cone-separability A$A$ and x$x$. Also, we give characterizations for star-shaped separability. Finally, as an application of separability, we characterize best approximation problem by elements of star-shaped sets.     

Lindelöf type of generalization of separability in Banach spaces

We will introduce the countable separation property (CSP) of Banach spaces X, which is defined as follows: X has CSP if each family E of closed linear subspaces of X whose intersection is the zero space contains a countable subfamily E₀ with the same intersection. All separable Banach spaces have CSP and plenty of examples of non-separable CSP spaces are provided. Connections of CSP with Marku?evi?-bases, Corson property and related geometric issues are discussed.     

An isomorphic characterization of L-spaces

We show that a sequentially (τ)-complete topological vector lattice X_τ is isomorphic to some L¹(μ), if and only if the positive cone can be written as X₊ = ₊B for some convex, (τ)-bounded, and (τ)-closed set B X₊ {0}. The same result holds under weaker hypotheses, namely the Riesz decomposition property for X (not assumed to be a vector lattice) and the monotonic σ-completeness (monotonic Cauchy sequences converge). The isometric part of the main result implies the well-known representation theorem of Kakutani for (AL)-spaces. As an application we show that on a normed space Y of infinite dimension, the “ball-generated” ordering induced by the cone Y₊ = ₊ (for u >) cannot have the Riesz decomposition property. A second application deals with a pointwise ordering on a space of multivariate polynomials.     

Thin and thick sets in locally convex spaces

We introduce in this work some normed space notions such as norming, thin and thick sets in general locally convex spaces. We also study some effects of thick sets on the uniform boundedness-like principles in locally convex spaces such as “weak*-bounded sets are strong*-bounded if and only if the space is a Banach–Mackey space”. It is proved that these principles occur under some weaker conditions by means of thick sets. Further, we show that the thickness is a duality invariant, that is, all compatible topologies for some locally convex space have the same thick sets.     

Clones of implicit operations

There is a close connection between a variety and its clone. The clone of a variety is a multibased algebra, where the different universes are the sets of n-ary terms over this variety for every natural number n and where the operations describe the superposition of terms of different arities. All projections are added as nullary operations. Subvarieties correspond to homomorphic images of clones. Subclones can be described by reducts of varieties, isomorphic clones by equivalent varieties. Clone identities correspond to hyperidentities and varieties of clones to hypervarieties. Pseudovarieties are classes of finite algebras which are closed under taking of subalgebras, homomorphic images and finite direct products. Pseudovarieties are important in the theories of finite state automata, rational languages, finite semigroups and their connections. In a very natural way, there arises the question for the clone of a pseudovariety. In the present paper, we will describe this algebraic structure. Received April 6, 2004; accepted in final form March 28, 2005.     

SOME REMARKS CONCERNING THE DAUGAVET EQUATION

Abstract

We say that a normed space X has the Daugavet property (DP) if for every finite rank operator K in X the equality ∥I + T∥ = 1 + ∥T∥ holds. It is known that C[0,1] and L ₁[0,1] have DP. We prove that if X has DP then X has no unconditional basis. We also discuss anti-Daugavet property, hereditary DP-spaces and construct a strictly convex normed space having DP.     

Clones on three elements preserving a binary relation

We describe the clones on 3 elements that can be expressed as Pol ρ for ρ a binary relation. We present the poset of these clones ordered by inclusion. This article is a shortened version of the author’s thesis, to give an idea of the whole work. Presented by R. P?schel. Received March 24, 2005; accepted in final form December 20, 2005.     

On the weak topology of Banach spaces over non-archimedean fields

It is known that within metric spaces analyticity and K-analyticity are equivalent concepts. It is known also that non-separable weakly compactly generated (shortly WCG) Banach spaces over R or C provide concrete examples of weakly K-analytic spaces which are not weakly analytic. We study the case which totally differs from the above one. A general theorem is provided which shows that a Banach space E over a locally compact non-archimedean non-trivially valued field is weakly Lindelöf iff E is separable iff E is WCG iff E is weakly web-compact (in the sense of Orihuela). This provides a non-archimedean version of a remarkable Amir-Lindenstrauss theorem.     

Distance to spaces of continuous functions

We link here distances between iterated limits, oscillations, and distances to spaces of continuous functions. For a compact space K, a uniformly bounded set H of the space of real-valued continuous functions C(K), and ε?0, we say that H ε-interchanges limits with K, if the inequality

     

Compactness in asymmetric normed spaces

A systematic study of precompact and compact subsets on asymmetric normed linear spaces is developed, centering our attention in the case of linear lattices with an asymmetric norm.     

MV-algebras: a variety for magnitudes with archimedean units

Chang’s MV-algebras, on the one hand, are the algebras of the infinite-valued Łukasiewicz calculus and, on the other hand, are categorically equivalent to abelian lattice-ordered groups with a distinguished strong unit, for short, unital ℓ-groups. The latter are a modern mathematization of the time-honored euclidean magnitudes with an archimedean unit. While for magnitudes the unit is no less important than the zero element, its archimedean property is not even definable in first-order logic. This gives added interest to the equivalent representation of unital ℓ-groups via the equational class of MV-algebras. In this paper we survey several applications of this equivalence, and various properties of the variety of MV-algebras.Dedicated to the Memory of Wim BlokReceived August 26, 2003; accepted in final form October 3, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Banach Disks and Marrelledness Properties of Metrizalbe <Emphasis Type="Italic">c</Emphasis><Subscript>0</Subscript> (Ω, <Emphasis Type="Italic">X</Emphasis>)

Let be a non-empty set and X a metrizable locally convex space. We show that the metrizable locally convex space c₀ (, X) is p-barrelled (totally barrelled) if and only if X is p-barrelled (totally barrelled). Some applications for closed graph theorems are included.     

Fibration of points and congruence modularity

Malcev varieties, or more generally Malcev categories, have been characterized by a property of the fibres of their fibration of points. In the same way, it is shown here that congruence modular varieties, and more generally Gumm categories, are also characterized by a property of these fibres.Received June 4, 2003; accepted in final form June 18, 2004.     

Characterizing filters by convergence (with respect to filters) in Banach spaces

Let X be a topological space and let F be a filter on N, recall that a sequence (xn)_n∈N in X is said to be F-convergent to the point x∈X, if for each neighborhood U of x, {n∈N:xn∈U}∈F. By using F-convergence in ?₁ and in Banach spaces, we characterize the P-filters, the P-filters⁺, the weak P-filters, the Q-filters, the Q-filters⁺, the weak Q-filters, the selective filters and the selective⁺ filters.     

On subtractive varieties,I

A varietyV is subtractive if it obeys the laws s(x, x)=0, s(x, 0)=x for some binary terms and constant 0. This means thatV has 0-permutable congruences (namely [0]R ºS=[0]S ºR for any congruencesR, S of any algebra inV). We present the basic features of such varieties, mainly from the viewpoint of ideal theory. Subtractivity does not imply congruence modularity, yet the commutator theory for ideals works fine. We characterize i-Abelian algebras, (i.e. those in which the commutator is identically 0). In the appendix we consider the case of a classical ideal theory (comprising: groups, loops, rings, Heyting and Boolean algebras, even with multioperators and virtually all algebras coming from logic) and we characterize the corresponding class of subtractive varieties.Presented by A. F. Pixley.