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.     

Order-bounded operators in vector lattices and in spaces of measurable functions

The survey is devoted to the presentation of the state of the art of a series of directions of the theory of order-bounded operators in vector lattices and in spaces of measurable functions. The theory of disjoint operators, the generalized Hewitt-Yosida theorem, the connection with p-absolutely summing operators are considered in detail.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 26, pp. 3–63, 1988.     

Critical points between varieties generated by subspace lattices of vector spaces

We denote by the semilattice of all compact congruences of an algebra A. Given a variety V of algebras, we denote by the class of all semilattices isomorphic to for some A∈V. Given varieties V and W of algebras, the critical point of V under W is defined as . Given a finitely generated variety V of modular lattices, we obtain an integer ?, depending on V, such that for any n≥? and any field F.In a second part, using tools introduced in Gillibert (2009) [5], we prove that:

     

Compact spaces and distributive lattices

This note presents a general construction connecting compact locales and distributive lattices, that allows us to reduce results about compactness of locales to theorems about distributive lattices. Two applications are given. One noteworthy feature of our arguments is that they can be formulated both in topos theory and in a predicative theory such as CZF.     

Consistent lattices and Steinitz spaces

It is well-known that a finite lattice L is isomorphic to the lattice of flats of a matroid if and only if L is geometric. A result due to Edelman (see [1], Theorem 3.3) states that a lattice is meet-distributive if and only if it is isomorphic to the lattice of all closed sets of a convex geometry. In this note we prove that a finite lattice is the lattice of closed sets of a closure space with the Steinitz exchange property if and only if it is a consistent lattice. Received February 28, 1997; accepted in final form February 2, 1998.     

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).     

A remark on the representation of vector lattices as spaces of continuous real-valued functions

The well-known Ogasawara-Maeda-Vulikh representation theorem asserts that for each Archimedean vector lattice L there exists an extremally disconnected compact Hausdorff space , unique up to a homeomorphism, such that L can be represented isomorphically as an order dense vector sublattice of the universally complete vector lattice C () of all extended-real-valued continuous functions f on for which % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaacmqabaGaeqyYdCNaeyicI4SaeyyQdCLaeyOoaOJaaiiFaiab% gkzaMkabgIcaOiabgM8a3jabgMcaPiaacYhacqGH9aqpcqGHEisPai% aawUhacaGL9baaaaa!4E05!\[\left\{ {\omega \in \Omega :|f(\omega )| = \infty } \right\}\] is nowhere dense. Since the early days of using this representation it has been important to find conditions on L such that consists of bounded functions only.The aim of this short article is to present a simple complete characterization of such vector lattices.     

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.     

Frames for vector spaces and affine spaces

A finite frame for a finite dimensional Hilbert space is simply a spanning sequence. We show that the linear functionals given by the dual frame vectors do not depend on the inner product, and thus it is possible to extend the frame expansion (and other elements of frame theory) to any finite spanning sequence for a vector space. The corresponding coordinate functionals generalise the dual basis (the case when the vectors are linearly independent), and are characterised by the fact that the associated Gramian matrix is an orthogonal projection. Existing generalisations of the frame expansion to Banach spaces involve an analogue of the frame bounds and frame operator.The potential applications of our results are considerable. Whenever there is a natural spanning set for a vector space, computations can be done directly with it, in an efficient and stable way. We illustrate this with a diverse range of examples, including multivariate spline spaces, generalised barycentric coordinates, and vector spaces over the rationals, such as the cyclotomic fields.     

Limit laws in vector lattices

Abstract

We prove a law of large numbers and a central limit theorem with respect to the order convergence topology in vector lattices.