Orthomodular lattices in ordered vector spaces

In recent years, models of spatial relations, especially topological relations, have attracted much attention from the GIS community. In this paper, some basic topologic models for spatial entities in both vector and raster spaces are discussed. It has been suggested that, in vector space, an open set in 1-D space may not be an open set any more in 2-D and 3-D spaces. Similarly, an open set in 2-D vector space may also not be an open set any more in 3-D vector spaces. As a result, fundamental topological concepts such as boundary and interior are not valid any more when a lower dimensional spatial entity is embedded in higher dimensional space. For example, in 2-D, a line has no interior and the line itself (not its two end-points) forms a boundary. Failure to recognize this fundamental topological property will lead to topological paradox. It has also been stated that the topological models for raster entities are different in Z^{2} and R^{2}. There are different types of possible boundaries depending on the definition of adjacency or connectedness. If connectedness is not carefully defined, topological paradox may also occur. In raster space, the basic topological concept in vector space—connectedness—is implicitly inherited. This is why the topological properties of spatial entities can also be studied in raster space. Study of entities in raster (discrete) space could be a more efficient method than in vector space, as the expression of spatial entities in discrete space is more explicit than that in connected space.

§1. Introduction.The motivation of this work comes from two different directions: infinite abelian groups, and ordered algebraic structures. A challenging problem in both cases is that of classification. In the first case, it is known for example (cf. [KA]) that the classification of abelian torsion groups amounts to that of reducedp-groups by numerical invariants called theUlm invariants(given by Ulm in [U]). Ulm's theorem was later generalized by P. Hill to the class of totally projective groups. As to the second case, let us consider for instance the class of divisible ordered abelian groups. These may be viewed as ordered ℚ-vector spaces. Their theory being unstable, we cannot hope to classify them by numerical invariants. On the other hand, being o-minimal, the theory enjoys several good model theoretic properties (cf. [P-S]), so the search for some reasonable invariants is well motivated. The common denominator of the two cases, as well as of many others, is valuation theory. Indeed given an ordered vector space, one can consider it as a valued vector space, endowed with the natural valuation. Also, the socleG[p] of a reduced abelianp-groupG, endowed with the height functionhG, is a valued vector space over(the prime field of characteristicp) with values in the ordinals (cf. [F]).

Fuzzy topological ordered vector spaces I

Let X be a compact Hausdorff space, C(X) the class of all continuous functions f: X→R, considered as an ordered vector space over R with respect to its canonical order; furthermore let F be an order complete ordered vector space over R. A linear operator A: C(X)→F is called bounded, if it transforms each bounded set of C(X) into a bounded set of F. The purpose of this paper is to represent such a bounded linear operator A: C(X)→F as an integral Af=∫ dμ with respect to some content μ, defined on the algebraa(X) which is generated by the open sets U⊂X. This representation is unique, if μ is required to be regular (p. 11).

The sheaf-function correspondence identifies the group of constructible functions on a real analytic manifold M with the Grothendieck group of constructible sheaves on M. When M is a finite dimensional real vector space, Kashiwara-Schapira have recently introduced the convolution distance between sheaves of $\mathbf {k}$ -vector spaces on M. In this paper, we characterize distances on the group of constructible functions on a real finite dimensional vector space that can be controlled by the convolution distance through the sheaf-function correspondence. Our main result asserts that such distances are almost trivial: they vanish as soon as two constructible functions have the same Euler integral. We formulate consequences of our result for Topological Data Analysis: there cannot exist nontrivial additive invariants of persistence modules that are continuous for the interleaving distance.

If the vector space \({\mathcal{L}}^r(E, F)\) of all regular operators between the vector lattices E and F is ordered by the collection of its positive operators, then the Dedekind completeness of F is a sufficient condition for \({\mathcal{L}}^r(E, F)\) to be a vector lattice. \({\mathcal{L}}^r(E, F)\) and some of its subspaces might be vector lattices also in a more general situation. In the paper we deal with ordered vector spaces \({\mathcal{V}}(E, F)\) of linear operators and ask under which conditions are they vector lattices, lattice-subspaces of the ordered vector space \({\mathcal{L}}^r(E, F)\) or, in the case that \({\mathcal{L}}^r(E, F)\) is a vector lattice, sublattices or even Banach lattices when equipped with the regular norm. The answer is affirmative for many classes of operators such as compact, weakly compact, regular AM-compact, regular Dunford-Pettis operators and others if acting between appropriate Banach lattices. Then it is possible to study the finite elements in such vector lattices \({\mathcal{V}}(E, F)\), where F is not necessary Dedekind complete. In the last part of the paper there will be considered the question how the order structures of E, F and \({\mathcal{V}}(E, F)\) are mutually related. It is also shown that those rank one and finite rank operators, which are constructed by means of finite elements from E′ and F, are finite elements in \({\mathcal{V}}(E,F)\). The paper contains also some generalization of results obtained for the case \({\mathcal{V}}(E, F) = {\mathcal{L}}^r(E, F)\) in [10].

LetX be a real vector space,D a convex subset ofX and (Y, K) an order complete ordered vector space. The following sandwich theorem holds: Iff: D → Y is midconvex,g: D → Y ∪ {− ∞} is midconcave andg ⩽ f onD, then there exists a Jensen mappingh: D → Y ∪ {− ∞} such thatg ⩽ h ⩽ f onD. Using this theorem we show that a mappingf: D → Y is midconvex if and only if it has Jensen support at every point ofD. Moreover, ifX is a Baire topological vector space and (Y, K) is an ordered topological vector space satisfying some additional conditions, then a mappingf: D → Y is continuous whenever it has continuous Jensen support at every point ofD. As an application of these results we obtain the equality of some set-classes connected with additive and midconvex operators.

The functor KR(X), defined in [4], is a contravariant functor defined in the category of spaces with involutions.It is shown herein that this functor is classified by equivariant maps into the complex Grassmann manifold, which is given the involution induced by complex conjugation.For the case of free involutions it is shown that the classifying maps can be taken to lie outside the fixed point set of the Grassmann manifold.This fixed point set can be identified with the real Grassmann manifold.It is then shown that, for free involutions, KR(X) is an invariant of the homotopy type of the orbit space X modulo its involution.The multiplicative group of real line bundles, real in the sense of [4], is shown to be classified by equivariant maps into a quadric surface Q in complex projective space.Q carries a free involution and this classification is again valid for spaces with free involutions. I. Introduction.In [4] the definition of Atiyah-real vector bundles was given, and they, together with the associated A-theory, were studied.The purpose of this paper is twofold.First we will extend the known classification theorems for complex A-theory to the new theory.Second we will obtain new results for the case of Atiyah-real vector bundles over spaces with free involutions.It was anticipated that this functor could be used to obtain a characterization of Mk, the differentiable seven sphere defined by Milnor in [1].This question is studied in 111, where KR{Ml) is computed.To review the definitions and results of [4], we have that a real space is a topological space X together with a homeomorphism t: X -> X with t2 = identity.We will write ?(x) = x.An Atiyah-real vector bundle over a real space is a complex vector bundle E -> X which itself is a real space and such that (1) 7r: 7i-> Xis equivariant, (2) the induced map Ex -> Es is conjugate linear.The Grothendieck group of Atiyah-real vector bundles over the real space X is denoted by KR(X).There is a Bott periodicity theorem and derived cohomology theory.Define(1) Rp-" = R'i + iRp,

Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. Later on, they have been generalized to the context of real vector spaces by means of the inner points. Inner points can be seen as the most opposite concept to the extreme points. In this manuscript we solve the “faceless problem”, that is, the informally posed open problem of characterizing those convex sets free of proper faces. Indeed, we prove that a non-singleton convex subset of a real vector space is free of proper faces if and only if every point of the convex set is an inner point. As a consequence, we prove the following dichotomy theorem: a point of a convex set of a real vector space is either a extreme point or an inner point of some face. We also characterize the non-singleton closed convex subsets of a topological vector space free of proper faces, which turn out to be the linear manifolds. An application of this allows us to show that the only possible minimal faces of a linearly bounded closed convex subset of a topological vector space are the extreme points. As an application of the technique we develop to prove our results, we easily construct dense proper faces of convex sets and non-dense proper faces whose closure is not a face. This fact allows us to equivalently renorm any infinite dimensional Banach space so that its unit ball contains a non-closed face. Even more, we prove that every infinite dimensional Banach space containing an isomorphic copy of $$c_0$$ or $$\ell _p$$ , $$1<p<\infty $$ , can be equivalently renormed so that its unit ball contains a face whose closure is not a face.

In this chapter we consider ways of ‘multiplying’ two vectors in a vector space V to give a scalar. Such a product is called an inner product, or scalar product, and the theory is based initially on a familiar example (sometimes called the dot product) on three-dimensional vectors in ℝ3. To begin with, we consider real vector spaces, but we will consider complex vector spaces later on. The scalar product, inner product, or dot product, of two vectors in ℝ2 or ℝ3 is rather well known. We start this chapter with its definition and description of its main properties. Throughout this section, llvll will denote the length of the vector v in ℝ2 or ℝ3

We introduce diffeological real or complex vector spaces. We define the fine diffeology on any vector space. We equip the vector space ${\mathcal H}$ of square summable sequences with the fine diffeology. We show that the unit sphere ${\mathcal S}$ of ${\

We produce and study several sequences of equations, in the language of orthomodular lattices, which hold in the ortholattice of closed subspaces of any classical Hilbert space, but not in all orthomodular lattices. Most of these equations hold in any orthomodular lattice admitting a strong set of states whose values are in a real Hilbert space. For some of these equations, we give conditions under which they hold in the ortholattice of closed subspaces of a generalised Hilbert space. These conditions are relative to the dimension of the Hilbert space and to the characteristic of its division ring of scalars. In some cases, we show that these equations cannot be deduced from the already known equations, and we study their mutual independence. To conclude, we suggest a new method for obtaining such equations, using the tensorial product.

Curvature tensors of Kaehler type (or type K) are defined on a hermitian vector space and it has been proved that the real vector space ${\mathcal {L}_K}(V)$ of curvature tensors of type K on V is isomorphic with the vector space of sym metric endomorphisms of the symmetric product of ${V^ + }$, where ${V^{\text {C}}} = {V^ + } \oplus {V^ - }$ (Theorem 3.6). Then it is shown that ${\mathcal {L}_K}(V)$ admits a natural orthogonal decomposition (Theorem 5.1) and hence every $L \in {\mathcal {L}_K}(V)$ is expressed as $L = {L_1} + {L_W} + {L_2}$. These components are explicitly determined and then it is observed that ${L_W}$ is a certain formal tensor introduced by Bochner. We call ${L_W}$ the Bochner-Weyl part of L and the space of all these ${L_W}$ is called the Weyl subspace of ${\mathcal {L}_K}(V)$.

This paper considers metrics valued in abelian l-groups and their induced topologies. In addition to a metric into an l-group, one needs a filter in the positive cone to determine which balls are neighborhoods of their center. As a key special case, we discuss a topology on a lattice ordered abelian group from the metric dG and the positive filter consisting of the weak units of G; in the case of \({\mathbb R^{n}}\) , this is the Euclidean topology. We also show that there are many Nachbin convex topologies on an l-group which are not induced by any positive filter of the l-group.

This paper proposes a modified Gram-Schmidt algorithm for dimension reduction of a document vector space in order to do classification. We also evaluate the performance of the proposed algorithm by comparing with commonly known algorithms such as the centroid-based algorithm and latent semantic indexing. Further, to measure efficiency, a modified Gram-Schmidt training set is applied in the centroid-based algorithm and latent semantic indexing as well. Performance measurement was based on two different parameters, viz., classification accuracy and closeness of similarity with the corresponding training set. The results shows that modified Gram-Schmidt algorithm is indeed an effective method for dimension reduction prior classification. Moreover, it is easy to code and computationally inexpensive.KeywordsCentroidDocument ClassificationDimension ReductionGram-SchmidtLSIVector Space Model