Ordered Vector Spaces Research Articles

Multimorphisms on pre-Riesz spaces of continuous functions

In the theory of vector lattices, mostly three classes of lattice ordered algebras are investigated, called f-algebras, d-algebras, and almost-f-algebras. Each class is endowed with a different type of positive bilinear map as the multiplication, namely with bi-orthomorphisms, bi-Riesz homomorphisms, and orthosymmetric maps, respectively. On the space C(K), with K a compact Hausdorff space, representations as weighted composition or integral operators are known. We give generalized definitions of the above mentioned maps in the setting of partially ordered vector spaces, where we deal with n-linear maps. We generalize the representations to the case of certain subspaces of some C(K) space, with K a compact Hausdorff space. With the representations at hand, we deduce basic properties of these maps. The results also cover the case of order unit spaces, which can be seen as order dense subspaces of some C(K) space via the functional representation.

A generalization of Riesz* homomorphisms on order unit spaces

Riesz homomorphisms on vector lattices have been generalized to Riesz* homomorphisms on ordered vector spaces by van Haandel using a condition on sets of finitely many elements. Van Haandel attempted to prove that it suffices to take sets of two elements. We show that this is not true, in general. The description by two elements motivates to introduce mild Riesz* homomorphisms. We investigate their properties on order unit spaces, where the geometry of the dual cone plays a crucial role. Hereby, we mostly focus on the finite-dimensional case.

Unbounded Order Convergence in Ordered Vector Spaces

We consider an ordered vector space X. We define the net xα⊆X to be unbounded order convergent to x (denoted as xα⟶uox). This means that for every 0≤y∈X, there exists a net yβ (potentially over a different index set) such that yβ↓0, and for every β, there exists α0 such that ±xα−xu,yl⊆yβl whenever α≥α0. The emergence of a broader convergence, stemming from the recognition of more ordered vector spaces compared to lattice vector spaces, has prompted an expansion and broadening of discussions surrounding lattices to encompass additional spaces. We delve into studying the properties of this convergence and explore its relationships with other established order convergence. In every ordered vector space, we demonstrate that under certain conditions, every uo-convergent net implies uo-Cauchy, and vice versa. Let X be an order dense subspace of the directed ordered vector space Y. If J⊆Y is a uo-band in Y, then we establish that J∩X is a uo-band in X.

Mixed lattice structures and cone projections

Problems related to projections on closed convex cones are frequently encountered in optimization theory and related fields. To study these problems, various unifying ideas have been introduced, including asymmetric vector-valued norms and certain generalized lattice-like operations. We propose a new perspective on these studies by describing how the problem of cone projection can be formulated using an order-theoretic formalism developed in this paper. The underlying mathematical structure is a partially ordered vector space that generalizes the notion of a vector lattice by using two partial orderings and having certain lattice-type properties with respect to these orderings. In this note we introduce a generalization of these so-called mixed lattice spaces, and we show how such structures arise quite naturally in some of the applications mentioned above.

Companionability characterization for the expansion of an o-minimal theory by a dense subgroup

This paper provides a full characterization for when the expansion of a complete o-minimal theory, one that extends the theory of ordered divisible abelian groups, by a unary predicate that picks out a divisible, dense and codense group has a model companion. This result is motivated by criteria and questions introduced in the recent works [14] and [10] concerning the existence of model companions, as well as preservation results for some neostability properties when passing to the model companion. Examples are included both in which the predicate is an additive subgroup of a real ordered vector space, and where it is a multiplicative subgroup of the nonzero elements of an o-minimal expansion of a real closed field. The paper concludes with a brief discussion of neostability properties and examples that illustrate the lack of preservation (from the base o-minimal theory to the model companion of the expansion we define) for properties such as strong, NIP, and NTP2, though there are also examples for which some or all three of those properties are preserved.

Ideals, bands and direct sum decompositions in mixed lattice vector spaces

A mixed lattice vector space is a partially ordered vector space with two partial orderings and certain lattice-type properties. In this paper we first give some fundamental results in mixed lattice groups, and then we investigate the structure theory of mixed lattice vector spaces, which can be viewed as a generalization of the theory of Riesz spaces. More specifically, we study the properties of ideals and bands in mixed lattice spaces, and the related idea of representing a mixed lattice space as a direct sum of disjoint bands. Under certain conditions, these decompositions can also be given in terms of order projections.

Order preserving maps on quantum measurements

We study the partially ordered set of equivalence classes of quantum measurements endowed with the post-processing partial order. The post-processing order is fundamental as it enables to compare measurements by their intrinsic noise and it gives grounds to define the important concept of quantum incompatibility. Our approach is based on mapping this set into a simpler partially ordered set using an order preserving map and investigating the resulting image. The aim is to ignore unnecessary details while keeping the essential structure, thereby simplifying e.g. detection of incompatibility. One possible choice is the map based on Fisher information introduced by Huangjun Zhu, known to be an order morphism taking values in the cone of positive semidefinite matrices. We explore the properties of that construction and improve Zhu's incompatibility criterion by adding a constraint depending on the number of measurement outcomes. We generalize this type of construction to other ordered vector spaces and we show that this map is optimal among all quadratic maps.

Compatible topologies on mixed lattice vector spaces

A mixed lattice vector space is a partially ordered vector space with two partial orderings, generalizing the notion of a Riesz space. The purpose of this paper is to develop the basic topological theory of mixed lattice spaces. A vector topology is said to be compatible with the mixed lattice structure if the mixed lattice operations are continuous. We give a characterization of compatible mixed lattice topologies, similar to the well known Roberts-Namioka characterization of locally solid Riesz spaces. We then study locally convex topologies and the associated seminorms, as well as connections between mixed lattice topologies and locally solid topologies on Riesz spaces. In the locally convex case, we obtain a more complete characterization of compatible mixed lattice topologies. We also briefly discuss asymmetric norms and cone norms on mixed lattice spaces with a particular application to finite dimensional spaces.

Gelfand–Naimark theorems for ordered -algebras

Abstract The classical Gelfand–Naimark theorems provide important insight into the structure of general and of commutative $C^*$ -algebras. It is shown that these can be generalized to certain ordered $^*$ -algebras. More precisely, for $\sigma $ -bounded closed ordered $^*$ -algebras, a faithful representation as operators is constructed. Similarly, for commutative such algebras, a faithful representation as complex-valued functions is constructed if an additional necessary regularity condition is fulfilled. These results generalize the Gelfand–Naimark representation theorems to classes of $^*$ -algebras larger than $C^*$ -algebras, and which especially contain $^*$ -algebras of unbounded operators. The key to these representation theorems is a new result for Archimedean ordered vector spaces V: If V is $\sigma $ -bounded, then the order of V is induced by the extremal positive linear functionals on V.

On partially ordered vector spaces with the Riesz interpolation property

On partially ordered vector spaces with the Riesz interpolation property

Incompatibility in General Probabilistic Theories, Generalized Spectrahedra, and Tensor Norms

In this work, we investigate measurement incompatibility in general probabilistic theories (GPTs). We show several equivalent characterizations of compatible measurements. The first is in terms of the positivity of associated maps. The second relates compatibility to the inclusion of certain generalized spectrahedra. For this, we extend the theory of free spectrahedra to ordered vector spaces. The third characterization connects the compatibility of dichotomic measurements to the ratio of tensor crossnorms of Banach spaces. We use these characterizations to study the amount of incompatibility present in different GPTs, i.e. their compatibility regions. For centrally symmetric GPTs, we show that the compatibility degree is given as the ratio of the injective and the projective norm of the tensor product of associated Banach spaces. This allows us to completely characterize the compatibility regions of several GPTs, and to obtain optimal universal bounds on the compatibility degree in terms of the 1-summing constants of the associated Banach spaces. Moreover, we find new bounds on the maximal incompatibility present in more than three qubit measurements.

Riesz representation theorems for positive linear operators

We generalise the Riesz representation theorems for positive linear functionals on \(\text {C}_{\text {c}}(X)\) and \(\text {C}_{\text {0}}(X)\), where X is a locally compact Hausdorff space, to positive linear operators from these spaces into a partially ordered vector space E. The representing measures are defined on the Borel \(\sigma \)-algebra of X and take their values in the extended positive cone of E. The corresponding integrals are order integrals. We give explicit formulas for the values of the representing measures at open and at compact subsets of X. Results are included where the space E need not be a vector lattice, nor a normed space. Representing measures exist, for example, for positive linear operators into Banach lattices with order continuous norms, into the regular operators on KB-spaces, into the self-adjoint linear operators on complex Hilbert spaces, and into JBW-algebras.

DETECTION OF BIO ELEMENTS PRESENT IN HUMAN BIOLOGICAL TISSUE-TOOTH AND ITS USAGE FOR ELEMENT BIOMETRIC AUTHENTICATION

Biometric authentication system uses some technique that measures the physical and biological characteristics of human to identify individuals and thus provide security to a system against fraud or intrusion. Common biometric authentication processes are vulnerable and possibility for imitation. Teeth are an important biological entity that plays a major role in forensic research to identify an individual whom cannot be identified visually. There are different algorithms used in biometric authentication. This paper proposes a unique method to recognize the human teeth by using a combination of Discrete Fourier Transform (DFT) and Discrete Cosine Transform (DCT) to extract significant features and an improved version of Binary Particle Swarm Optimization (BPSO) for feature selection is employed to search the feature vector space in order to obtain optimal feature subset to increase the performance rate. A combination of image pre-processing techniques like background removal, gamma intensity correction and Laplacian of Gaussian (LoG) filter are used to help in correct feature extraction. Using the shift invariance property of DFT, a circular feature extraction technique and the energy compaction property of DCT, a circular sector feature extraction method is presented. Experimental results on IvisionLab/dental-image standard database are shown which exhibit promising performance of the teeth recognition system.

Order integrals

We define an integral of real-valued functions with respect to a measure that takes its values in the extended positive cone of a partially ordered vector space E. The monotone convergence theorem, Fatou’s lemma, and the dominated convergence theorem are established; the analogues of the classical \({\mathscr {L}}^1\)- and \({\mathrm L}^1\)-spaces are investigated. The results extend earlier work by Wright and specialise to those for the Lebesgue integral when E equals the real numbers. The hypothesis on E that is needed for the definition of the integral and for the monotone convergence theorem to hold (\(\sigma \)-monotone completeness) is a rather mild one. It is satisfied, for example, by the space of regular operators between a directed partially ordered vector space and a \(\sigma \)-monotone complete partially ordered vector space, and by every JBW-algebra. Fatou’s lemma and the dominated convergence theorem hold for every \(\sigma \)-Dedekind complete space. When E consists of the regular operators on a Banach lattice with an order continuous norm, or when it consists of the self-adjoint elements of a strongly closed complex linear subspace of the bounded operators on a complex Hilbert space, then the finite measures as in the current paper are precisely the strongly \(\sigma \)-additive positive operator-valued measures. When E is a partially ordered Banach space with a closed positive cone, then every positive vector measure is a measure in our sense, but not conversely. Even when a measure falls into both categories, the domain of the integral as defined in this paper can properly contain that of any reasonably defined integral with respect to the vector measure using Banach space methods.

EMBEDDING ORTHOMORPHISMS d-ALGEBRA IN BIORTHOMORPHISMS AS ORDERED IDEAL

In the historical development of Riesz spaces, we can trace the history of ordered vector spaces to the International Mathematical Congress in Bologna in 1928. Studies related with f-Algebras for the Dedekind complete ordered vector space defined in Riesz spaces were initiated by Nakano and their current definition was made by Amemiya, Birkhoff and Pierce.The revival of f-algebras, which had a tendency to slow down for a period of time, emerged as a result of Pagter's doctoral thesis [9] and the examination of Alkansas lecture notes by Luxemburg. 
 The concepts of homomorphism, isomorphism, orthomorphism and biorthomorphism in Riesz spaces are defined by Zaanen, Huijsmans, Boulabiar, Buskes and Triki. Algebraic structure of biorthomorphisms defined on Riesz space examined by [8]. f-Algebra on Orth(X,X) were studied by [8] and [6]. [6] demonstrated that biorthomorphisms space have an f-algebraic structure with the help of the product defined as (T_1 *_e T_2)(x,y)=T_1 (x,T_2 (e,y)) for e∈X^+, ∀x,y∈X ve T_1,T_2∈Orth(X,X).
 [8] showed that if orthomorphisms are semiprime Dedekind complete f-algebras, it is an ordered ideal in biorthomorphisms. [6] developed an alternative proof for this situation. If X Archimedean Riesz space, Orth(X) is an f-algebra according to compound operation with unit element.
 [10] showed that if the orthomorphism X is a semi-prime f-algebra, it is a d-algebra. In this study, we investigated embedding orthomorphism in biorthomorphisms when X is uniformly complete d-algebra.

Topological concepts in partially ordered vector spaces

In the context of partially ordered vector spaces one encounters different sorts of order convergence and order topologies. This article investigates these notions and their relations. In particular, we study and relate the order topology presented by Floyd, Vulikh and Dobbertin, the order bound topology studied by Namioka and the concept of order convergence given in the works of Abramovich, Sirotkin, Wolk and Vulikh. We prove that the considered topologies disagree for all infinite dimensional Archimedean vector lattices that contain order units. For reflexive Banach spaces equipped with ice cream cones we show that the order topology, the order bound topology and the norm topology agree and that order convergence is equivalent to norm convergence.

Operators in pre-Riesz spaces: moduli and homomorphisms

We focus on two topics that are related to moduli of elements in partially ordered vector spaces. First, we relate operators that preserve moduli to generalized notions of lattice homomorphisms, such as Riesz homomorphisms, Riesz* homomorphisms, and positive disjointness preserving operators. We also consider complete Riesz homomorphisms, which generalize order continuous lattice homomorphisms. Second, we characterize elements with a modulus by means of disjoint elements and apply this result to obtain moduli of functionals and operators in various settings. On spaces of continuous functions, we identify those differences of Riesz* homomorphisms that have a modulus. Many of our results for pre-Riesz spaces of continuous functions lead to results on order unit spaces, where the functional representation is used.

Order continuity from a topological perspective

We study three types of order convergence and related concepts of order continuous maps in partially ordered sets, partially ordered abelian groups, and partially ordered vector spaces, respectively. An order topology is introduced such that in the latter two settings under mild conditions order continuity is a topological property. We present a generalisation of the Ogasawara theorem on the structure of the set of order continuous operators.

A tetrachotomy for expansions of the real ordered additive group

Let \(\mathcal {R}\) be an expansion of the ordered real additive group. When \(\mathcal {R}\) is o-minimal, it is known that either \(\mathcal {R}\) defines an ordered field isomorphic to \((\mathbb {R},<,+,\cdot )\) on some open subinterval \(I\subseteq \mathbb {R}\), or \(\mathcal {R}\) is a reduct of an ordered vector space. We say \(\mathcal {R}\) is field-type if it satisfies the former condition. In this paper, we prove a more general result for arbitrary expansions of \((\mathbb {R},<,+)\). In particular, we show that for expansions that do not define dense \(\omega \)-orders (we call these type A expansions), an appropriate version of Zilber’s principle holds. Among other things we conclude that in a type A expansion that is not field-type, every continuous definable function \([0,1]^m \rightarrow \mathbb {R}^n\) is locally affine outside a nowhere dense set.

5* Knowledge Graph Embeddings with Projective Transformations

Performing link prediction using knowledge graph embedding models has become a popular approach for knowledge graph completion. Such models employ a transformation function that maps nodes via edges into a vector space in order to measure the likelihood of the links. While mapping the individual nodes, the structure of subgraphs is also transformed. Most of the embedding models designed in Euclidean geometry usually support a single transformation type -- often translation or rotation, which is suitable for learning on graphs with small differences in neighboring subgraphs. However, multi-relational knowledge graphs often include multiple subgraph structures in a neighborhood (e.g.~combinations of path and loop structures), which current embedding models do not capture well. To tackle this problem, we propose a novel KGE model 5*E in projective geometry, which supports multiple simultaneous transformations -- specifically inversion, reflection, translation, rotation, and homothety. The model has several favorable theoretical properties and subsumes the existing approaches. It outperforms them on most widely used link prediction benchmarks