On self-majorizing elements in Archimedean vector lattices

A finite element in an Archimedean vector lattice is called self-majorizing if its modulus is a majorant. Such elements exist in many vector lattices and naturally occur in different contexts. They are also known as semi-order units as the modulus of a self-majorizing element is an order unit in the band generated by the element. In this paper the properties of self-majorizing elements are studied systematically, and the relations between the sets of finite, totally finite and self-majorizing elements of a vector lattice are provided. In a Banach lattice an element \(\varphi \) is self-majorizing , if and only if the ideal and the band both generated by \(\varphi \) coincide.     

Archimedean lattices

Anarchimedean lattice is a complete algebraic latticeL with the property that for each compact elementc∈L, the meet of all the maximal elements in the interval [0,c] is 0.L ishyper-archimedean if it is archimedean and for eachx∈L, [x, 1] is archimedean. The structure of these lattices is analysed from the point of view of their meet-irreducible elements. If the lattices are also Brouwer, then the existence of complements for the compact elements characterizes a particular class of hyper-archimedean lattices. The lattice ofl-ideals of an archimedean lattice ordered group is archimedean, and that of a hyper-archimedean lattice ordered group is hyper-archimedean. In the hyper-archimedean case those arising as lattices ofl-ideals are fully characterized.     

On convergences of measurable functions in Archimedean vector lattices

For Archimedean vector lattices X, Y and the positive cone \mathbbL{\mathbb{L}} of all regular linear operators L : X → Y, a theory of sequential convergences of functions connected with an \mathbbL{\mathbb{L}} -valued measure is introduced and investigated.     

Order completions of Archimedean Riesz spaces andl-groups

It is well-known that every Archimedean Riesz space (vector lattice) can be embedded in a certain minimal Dedekind complete Riesz space (itsDedekind completion) and that this space is essentially unique. There are other nice properties that a Riesz space can enjoy besides Dedekind completeness; for example, the projection property, the principal projection property,-Dedekind completeness, and (relative) uniform completeness. It is shown that every Archimedean Riesz space has an essentially unique completion with respect to each of these properties. These completions can be viewed as universal objects in appropriate categories. As such, their uniqueness is obvious (universal objects are always unique), and their existence can be demonstrated very simply by working within the Dedekind completion. This approach is free of clutter since all it needs is theexistence of the Dedekind completion, and not its particular form (which can be quite complicated). By using the same methods within the universal completion, we can isolate further order completions; in a sense, every possible order completion can be obtained in this way, since the universal completion is the largest Riesz space in which the original space is order dense. As an added bonus, all of our results apply equally well to Archimedeanl-groups.Presented by L. Fuchs.     

Affine completions of distributive lattices

For any distributive lattice L we construct its extension ((L)) with the property that every isotone compatible function on L can be interpolated by a polynomial of ((L)). Further, we characterize all extensions with this property and show that our construction is in some sense the simplest possible.This research was supported by the GA SAV Grant 1230/95.     

Orthmodular lattices whose MacNeille completions are not orthomodular

The only known example of an orthomodular lattice (abbreviated: OML) whose MacNeille completion is not an OML has been noted independently by several authors, see Adams [1], and is based on a theorem of Ameniya and Araki [2]. This theorem states that for an inner product space V, if we consider the ortholattice ?(V,⊥) = {A \( \subseteq \) V: A = A ^⊥⊥} where A ^⊥ is the set of elements orthogonal to A, then ?(V,⊥) is an OML if and only if V is complete. Taking the orthomodular lattice L of finite or confinite dimensional subspaces of an incomplete inner product space V, the ortholattice ?(V,⊥) is a MacNeille completion of L which is not orthomodular. This does not answer the longstanding question Can every OML be embedded into a complete OML? as L can be embedded into the complete OML ?(V,⊥), where V is the completion of the inner product space V. Although the power of the Ameniya-Araki theorem makes the preceding example elegant to present, the ability to picture the situation is lost. In this paper, I present a simpler method to construct OMLs whose Macneille completions are not orthomodular. No use is made of the Ameniya-Araki theorem. Instead, this method is based on a construction introduced by Kalmbach [7] in which the Boolean algebras generated by the chains of a lattice are glued together to form an OML. A simple method to complete these OMLs is also given. The final section of this paper briefly covers some elementary properties of the Kalmbach construction. I have included this section because I feel that this construction may be quite useful for many purposes and virtually no literature has been written on it.     

Bond percolation critical probability bounds for three Archimedean lattices

Rigorous bounds for the bond percolation critical probability are determined for three Archimedean lattices: .7385 < p_c((3, 12²) bond) < .7449, .6430 < p_c((4, 6, 12) bond) < .7376, .6281 < p_c((4, 8²) bond) < .7201. Consequently, the bond percolation critical probability of the (3, 12²) lattice is strictly larger than those of the other ten Archimedean lattices. Thus, the (3, 12²) bond percolation critical probability is possibly the largest of any vertex‐transitive graph with bond percolation critical probability that is strictly less than one. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 20: 507–518, 2002     

Tensor products of Archimedean partially ordered vector spaces

We study the tensor product of two directed Archimedean partially ordered vector spaces X and Y by means of Riesz completions. With the aid of the Fremlin tensor product of the Riesz completions of X and Y we show that the projective cone in X ? Y is contained in an Archimedean cone. The smallest Archimedean cone containing the projective cone satisfies an appropriate universal mapping property.     

Convex vector lattices and l-algebras

For A an Archimedean Riesz space (=vector lattice) with distinguished positive weak unit e_A, we have the Yosida representation  as a Riesz space in D(X_A), the lattice of extended real valued functions on the space of e_A-maximal ideas. This note is about those A for which  is a convex subset of D(X_A); we call such A “convex”.Convex Riesz spaces arise from the general issue of embedding as a Riesz ideal, from consideration of uniform- and order-completeness, and from some problems involving comparison of maximal ideal spaces (which we won't discuss here; see [10]).The main results here are: (2.4) A is convex iff A is contained as a Riesz ideal in a uniformly complete Φ-algebra B with identity e_A. (3.1) Any A has a convex reflection (i.e., embeds into a convex B with a universal mapping property for Riesz homomorphisms; moreover, the embedding is epic and large).     

Envelopes and inequalities in vector lattices

The aim of this paper is to obtain a version of continuous functional calculus and some new envelope representation results in vector lattices as well as to indicate some applications.     

On nestedly complete topological vector lattices

A topological vector lattice E is called (σ-)nestedly complete if every downward directed net (resp., decreasing sequence) of order intervals in E whose ‘diameters’ tend to zero has a nonempty intersection. Some characterizations of the (σ-)nested completeness are given, and it is shown that if E is metrizable and nestedly complete, so is each of its quotients E/I, where I is a closed ideal in E. Conversely, if a closed ideal I in E is (sequentially) complete and E/I is (σ-)nestedly complete, so is E. However, the nested completeness is not a three-space property: an example is given where both I and E/I are nestedly complete while E is not. It is also shown that the nested completeness and the related notion of nested density come up quite naturally when extending some positive linear operators. Finally, the nested and other completeness type properties of vector lattices C(S) are investigated.     

Orthomodular lattices in ordered vector spaces

In a partly ordered space the orthogonality relation is defined by incomparability. We define integrally open and integrally semi-open ordered real vector spaces. We prove: if an ordered real vector space is integrally semi-open, then a complete lattice of double orthoclosed sets is orthomodular. An integrally open concept is closely related to an open set in the Euclidean topology in a finite dimensional ordered vector space. We prove: if V is an ordered Euclidean space, then V is integrally open and directed (and is also Archimedean) if and only if its positive cone, without vertex 0, is an open set in the Euclidean topology (and also the family of all order segments , a < b, is a base for the Euclidean topology). Received January 7, 2005; accepted in final form November 26, 2005.