Linear mappings on ordered vector spaces

§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

It is a standard and very useful technique to approximate geometric objects and maps by linear objects and maps. A principal (trivial) example in basic analysis is approximating an open subspace U of \(\mathbb{K}^{n}\) at a point by the linear space \(\mathbb{K}^{n}\) itself by visualizing \(\mathbb{K}^{n}\) as a tangent space to U at the given point. Another (less trivial) example is the derivative of a differentiable map f on U in a point that is the best approximation of f nearby this point by a linear map. In the first section we extend this technique to premanifolds and morphisms of premanifolds. Thus we first define the tangent space of a premanifold at a point, which will be a \(\mathbb{K}\)-vector space that we visualize as the ”best linear approximation“ of the premanifold at the given point. Then we define the derivative in a point x of a morphism f. This will be a linear map from tangent space at x to the tangent space at \(f(x)\).

Riesz homomorphisms between vector lattices are characterized by interval preserving order adjoints. Buskes, van Rooij, and van Haandel generalized the notion of a Riesz homomorphism to linear maps between general ordered vector spaces. In this paper, we discuss how far the characterization between vector lattices extends to directed ordered vector spaces and pre-Riesz 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 \( \{ z \in V:a < z < b\} \), a < b, is a base for the Euclidean topology).

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

Learning bilingual sentiment lexicon for online reviews

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].

This chapter studies, in the setting of vector spaces over a field, the basics concerning multilinear functions, tensor products, spaces of linear functions, and algebras related to tensor products. Sections 1–5 concern special properties of bilinear forms, all vector spaces being assumed to be finite-dimensional. Section 1 associates a matrix to each bilinear form in the presence of an ordered basis, and the section shows the effect on the matrix of changing the ordered basis. It then addresses the extent to which the notion of “orthogonal complement” in the theory of inner-product spaces applies to nondegenerate bilinear forms. Sections 2–3 treat symmetric and alternating bilinear forms, producing bases for which the matrix of such a form is particularly simple. Section 4 treats a related subject, Hermitian forms when the field is the complex numbers. Section 5 discusses the groups that leave some particular bilinear and Hermitian forms invariant. Section 6 introduces the tensor product of two vector spaces, working with it in a way that does not depend on a choice of basis. The tensor product has a universal mapping property—that bilinear functions on the product of the two vector spaces extend uniquely to linear functions on the tensor product. The tensor product turns out to be a vector space whose dual is the vector space of all bilinear forms. One particular application is that tensor products provide a basis-independent way of extending scalars for a vector space from a field to a larger field. The section includes a number of results about the vector space of linear mappings from one vector space to another that go hand in hand with results about tensor products. These have convenient formulations in the language of category theory as “natural isomorphisms.” Section 7 begins with the tensor product of three and then $n$ vector spaces, carefully considering the universal mapping property and the question of associativity. The section defines an algebra over a field as a vector space with a bilinear multiplication, not necessarily associative. If $E$ is a vector space, the tensor algebra $T(E)$ of $E$ is the direct sum over $n\geq0$ of the $n$-fold tensor product of $E$ with itself. This is an associative algebra with a universal mapping property relative to any linear mapping of $E$ into an associative algebra $A$ with identity: the linear map extends to an algebra homomorphism of $T(E)$ into $A$ carrying 1 into 1. Sections 8–9 define the symmetric and exterior algebras of a vector space $E$. The symmetric algebra $S(E)$ is a quotient of $T(E)$ with the following universal mapping property: any linear mapping of $E$ into a commutative associative algebra $A$ with identity extends to an algebra homomorphism of $S(E)$ into $A$ carrying 1 into 1. The symmetric algebra is commutative. Similarly the exterior algebra $\bigwedge(E)$ is a quotient of $T(E)$ with this universal mapping property: any linear mapping $l$ of $E$ into an associative algebra $A$ with identity such that $l(v)^2=0$ for all $v\in E$ extends to an algebra homomorphism of $\bigwedge(E)$ into $A$ carrying 1 into 1. The problems at the end of the chapter introduce some other algebras that are of importance in applications, and the problems relate some of these algebras to tensor, symmetric, and exterior algebras. Among the objects studied are Lie algebras, universal enveloping algebras, Clifford algebras, Weyl algebras, Jordan algebras, and the division algebra of octonions.

The almost split triangles for perfect complexes over gentle algebras

Subdifferentiability of Convex Functions with Values in an Ordered Vector Space.

A general extended functional is a functional which is allowed to take on infinite values; in other words such functionals are similar to the Lebesgue integral on the space of all integrable functions. The problem of representing a general extended functional f on a vector lattice X as an operator with finite values only has been solved in [12]. In fact, this problem has been solved for a larger class of general extended operators on an ordered vector space. The solution of this problem was given by means of an extension of the range R of the functional f to some ultrapower of R. Notice, however, that it is not always the case that a functional f can be considered as a trace of some internal functional *f:*X→R. (We remark, without going into details, that such an internal functional exists exactly in the case when the results of the nonstandard analysis can be used for investigation of the given functional.) In [12] a ‘standard’ necessary and sufficient condition was given for solving this latter problem on the existence of *f. Namely, f is a trace of an internal *f if and only if it is finitely extendable. This result makes the finite extendability problem worthy of study. The present paper is a first attempt in this direction. Simultaneously we introduce and study a more general notion of weak finite extendability that coincides with finite extendability, for instance, for vector lattices with a strong unit.

Book Review: Barrelledness in topological and ordered vector spaces

A (partially) ordered vector space A (over ℝ) is said to be Archimedean if the negative elements a∈A− are the only ones for which {αa|α∈R+} has an upper bound. A vector subspace J of ordered vector space is said to be an order ideal if $$ a,b \in J,{\text{ }}c \in A,{\text{ }}a \leqq c \leqq b \Rightarrow c \in J. $$ (1.1)

Approximate solutions of vector optimization problems