Search

The paper investigates uniformly closed subspaces, sublattices, and ideals of finite codimension in Archimedean vector lattices. It is shown that every uniformly closed subspace (or sublattice) of finite codimension may be written as an intersection of uniformly closed subspaces (respectively, sublattices) of codimension one. Every uniformly closed sublattice of codimension n contains a uniformly closed ideal of codimension at most $ 2n $ 2 n. If the vector lattice is uniformly complete then every ideal of finite codimension is uniformly closed. Results of the paper extend (and are motivated by) results of Abramovich Y.A., Lipecki Z. [On ideals and sublattices in linear lattices and F-lattices. Math Proc Cambridge Philos Soc. 1990;108(1):79–87.; On lattices and algebras of simple functions. Comment Math Univ Carolin. 1990;31(4):627–635.], as well as Kakutani's characterization of closed sublattices of $ C(K) $ C (K) spaces. [ABSTRACT FROM AUTHOR]

It was showed by Donner in [ Extension of positive operators and Korovkin theorems , Lecture Notes in Mathematics, vol. 904, Springer-Verlag, Berlin-New York, 1982] that every order complete vector lattice X may be embedded into a cone X^s, called the sup-completion of X. We show that if one represents the universal completion of X as C^\infty (K), then X^s is the set of all continuous functions from K to [-\infty,\infty ] that dominate some element of X. This provides a functional representation of X^s, as well as an easy alternative proof of its existence. [ABSTRACT FROM AUTHOR]

Let X be an Archimedean vector lattice and X + denote the positive cone of X. A unary operation ϖ on X + is called a truncation on X if x ∧ ϖ y = ϖ x ∧ y for all x , y ∈ X +. Let X u denote the universal completion of X with a distinguished weak element e > 0. It is shown that a unary operation ϖ on X + is a truncation on X if and only if there exists an element u ∈ X u and a component p of e such that p ∧ u = 0 and ϖ x = p x + u ∧ x for all x ∈ X +. Here, px is the product of p and x with respect to the unique lattice-ordered multiplication in X u having e as identity. As an example of illustration, if ϖ is a truncation on some L p μ -space then there exists a measurable set A and a function u ∈ L 0 μ vanishing on A such that ϖ x = 1 A x + u ∧ x for all x ∈ L p μ. [ABSTRACT FROM AUTHOR]

Orthogonally biadditive operators preserving disjointness are studied. It is proved that, that for a Dedekind complete vector lattice and order ideals and in , the set of all orthogonally biadditive operators commuting with projections is a band in the Dedekind complete vector lattice of all regular orthogonally biadditive operators from the Cartesian product of and to . A general form of the order projection onto this band is obtained, and an operator version of the Radon–Nikodym theorem for disjointness-preserving positive orthogonally biadditive operators is proved. [ABSTRACT FROM AUTHOR]

A monomial in finite collection of homogeneous polynomials is a point-wise product of these polynomials each raised to some power. It was established in [14] and [15] that, given a finite set of positive linear operators acting from an Archimedean vector lattice into an f-algebra with multiplicative unit, the corresponding monomial is orthogonally additive polynomial if and only if the collection of these operators is a disjointness preserving set. It is shown in this paper that a similar result is also true in the case when the target space is a uniformly complete vector lattice. Moreover, the result remains valid if we replace the linear operators with homogeneous orthogonally additive polynomials. The correct definition of monomials in homogeneous polynomials is guaranteed by the homogeneous functional calculus and the concavification procedure. [ABSTRACT FROM AUTHOR]

Denote by I d c G the lattice of all principal ℓ -ideals of an Abelian ℓ -group G. Our main result is the following. Theorem For every countable Abelian ℓ-group G, every countable completely normal distributive 0 -lattice L, and every closed 0 -lattice homomorphism φ : I d c G → L , there are a countable Abelian ℓ-group H, an ℓ-homomorphism f : G → H , and a lattice isomorphism ι : I d c H → L such that φ = ι ∘ I d c f. We record the following consequences of that result: (1) A 0-lattice homomorphism φ : K → L , between countable completely normal distributive 0-lattices, can be represented, with respect to the functor I d c , by an ℓ -homomorphism of Abelian ℓ -groups iff it is closed. (2) A distributive 0-lattice D of cardinality at most ℵ 1 is isomorphic to some I d c G iff D is completely normal and for all a , b ∈ D the set { x ∈ D | a ≤ b ∨ x } has a countable coinitial subset. This solves Mundici's MV-spectrum Problem for cardinalities up to ℵ 1. The bound ℵ 1 is sharp. Item (1) is extended to commutative diagrams indexed by forests in which every node has countable height. All our results are stated in terms of vector lattices over any countable totally ordered division ring. [ABSTRACT FROM AUTHOR]

We elaborate, strengthen, and generalize known representation theorems by different authors for regular operators on vector and Banach lattices. Our main result asserts, in particular, that every regular linear operator T acting from a vector lattice E with the principal projection property to a Dedekind complete vector lattice F, which is an ideal of some order continuous Banach lattice G, admits a unique representation T=Ta+Tc$T = T_a + T_c$, where Ta$T_a$ is the sum of an absolutely order summable family of disjointness preserving operators and Tc$T_c$ is an order narrow (= diffuse) operator. Our main contribution is waiver of the order continuity assumption on T. In proofs, we use new techniques that allow obtaining more general results for a wider class of orthogonally additive operators, which has somewhat different order structure than the linear subspace of linear operators. [ABSTRACT FROM AUTHOR]

In this note it is established that a finite family of positive linear operators acting from an Archimedean vector lattice into an Archimedean -algebra with unit is disjointness preserving if and only if the polynomial presented in the form of the product of powers of these operators is orthogonally additive. A similar statement is established for the sum of polynomials represented as products of powers of positive operators. [ABSTRACT FROM AUTHOR]

It is well known that the lattice Id c G of all principal ℓ -ideals of any Abelian ℓ -group G is a completely normal distributive 0-lattice; yet not every completely normal distributive 0-lattice is a homomorphic image of some Id c G , via a counterexample of cardinality ℵ 2 . We prove that every completely normal distributive 0-lattice with at most ℵ 1 elements is a homomorphic image of some Id c G . By Stone duality, this means that every completely normal generalized spectral space with at most ℵ 1 compact open sets is homeomorphic to a spectral subspace of the ℓ -spectrum of some Abelian ℓ -group. [ABSTRACT FROM AUTHOR]

In this paper, we extend to the setting of positive orthogonally bi-additive operators several results from Aliprantis and Burkinshaw (Math Z 185: 245-257, 1983), de Pagter (Math Anal Appl 472: 238-245, 2019), Pliev and Popov (Siberian Math J 57: 552-557, 2016), Pliev and Ramdane (Mediter J Math 15(2): 55, 2018). First, we show that a positive order bounded orthogonally bi-additive map T : I → W defined on a lateral ideal of a Cartesian product of vector lattices E and F and taking values in a Dedekind complete vector lattice W can be extended to the whole space E × F . Then we prove that a positive orthogonally bi-additive operator T : E × F → W is laterally-to-order continuous if and only if the kernel of each S with 0 ≤ S ≤ T is laterally closed. Finally, we calculate the laterally-to-order continuous part of a positive orthogonally bi-additive operator T : E × F → W . [ABSTRACT FROM AUTHOR]

We establish the existence of a simultaneous extension from a majorizing sublattice in the classes of regular multilinear operators and regular homogeneous polynomials on vector lattices. By simultaneous extension from a sublattice we mean a right inverse of the restriction operator to this sublattice which is an order continuous lattice homomorphism. The main theorems generalize some earlier results for orthogonally additive polynomials and bilinear operators. The proofs base on linearization by Fremlin's tensor product and the existence of a right inverse of an order continuous operator with Levy and Maharam property. [ABSTRACT FROM AUTHOR]

Let E be a sublattice of a vector lattice F. A continuous operator T from E into a normed vector space X is said to be õrder-norm continuous if ... implies ... for every (xα)α∈A ⊆ E. This paper aims to investigate the properties of this new class of operators and explore their relationships with existing classifications of operators. We introduce a new class of operators called õrder weakly compact operators. A continuous operator T : E → X is considered õrder weakly compact if T(A) in X is a relatively weakly compact set for every Fo-bounded A ⊆ E. In this manuscript, we examine various properties of this class of operators and explore their connections with õrder-norm continuous operators. [ABSTRACT FROM AUTHOR]

It is a subtle fact of the theory of regular operators on Banach lattices that every linear operator T \colon L_{1}(\mu)\to L_1(\nu) is norm-bounded if and only if it is regular. We generalize this result to the setting of operators taking values in a Lebesgue–Bochner space. Our main result asserts that every linear operator T \colon L_{1}(\mu)\to L_{1}(\nu,X) is norm-bounded if and only if it is dominated. We show that this result is no longer true for Lebesgue–Bochner domain spaces. As a consequence of the main theorem, we obtain a generalized version of the Grothendieck inequality for linear norm-bounded operators from L_{1}(\mu) to a Lebesgue–Bochner space L_{1}(\nu,X). [ABSTRACT FROM AUTHOR]

In this paper, we characterize when, for any infinite cardinal α , the Fremlin tensor product of two Archimedean Riesz spaces (see Fremlin in Am J Math 94:777–798, 1972) is Dedekind α -complete. We also provide an example of an ideal I in an Archimedean Riesz space E such that the Fremlin tensor product of I with itself is not an ideal in the Fremlin tensor product of E with itself. [ABSTRACT FROM AUTHOR]

A lattice pseudonormed lattice is a triplet (X, ρ, E) where X and E are vector lattices and ρ: X → E+ is an E-valued pseudonorm on X. By abstracting order theoretical properties of the norm of an AM-space with a strong norm unit to the case of lattice pseudonormed lattices, we obtain various distinct and special classes of lattice valued pseudonorms on X. By using these classes, we study properties of pseudonorm compact, pseudo-semicompact, order-AM-compact, and pseudo-AM-compact operators defined between lattice pseudonormed lattices. [ABSTRACT FROM AUTHOR]

Let M and N be Archimedean vector lattices. We introduce orthogonally additive band operators and orthogonally additive inverse band operators from M to N and examine their properties. We investigate the relationship between orthogonally additive band operators and orthogonally additive disjointness preserving operators and show that under some assumptions on vector lattices M or N, these two classes are the same. By using this relation, we show that if μ is a bijective orthogonally additive band operator (resp. orthogonally additive disjointness preserving operator) from M into N then μ-1:N →M is an orthogonally additive band operator (resp. orthogonally additive disjointness preserving operator). [ABSTRACT FROM AUTHOR]

Conditions under which the relatively uniform convergence in vector lattices is complete, topological, and agrees with the order convergence are presented. r-Completion of an Archimedean vector lattice is investigated. Basic properties of the Archimedization of a vector lattice are established. [ABSTRACT FROM AUTHOR]

We show that the Carathéodory space of place functions on the free product of two Boolean algebras is Riesz isomorphic with Fremlin's Archimedean Riesz space tensor product of their respective Carathéodory spaces of place functions. We provide a solution to Fremlin's problem 315Y(f) [ Measure Theory , Torres Fremlin, Colchester, 2004] concerning completeness in the free product of Boolean algebras by applying our results on the Archimedean Riesz space tensor product to Carathéodory spaces of place functions. [ABSTRACT FROM AUTHOR]

In this paper, we study a new class of locally dominated orthogonally additive operators on lattice-normed spaces (LNS). In the first part of the paper, sufficient conditions for the existence of a local exact majorant of a locally dominated operator and formulas for its calculation are given. The second part shows that the -compactness of a dominated orthogonally additive operator acting from a decomposable lattice-normed space to a Banach space with mixed norm implies the -compactness of its exact majorant. [ABSTRACT FROM AUTHOR]

A "Bochner-type" integral for vector lattice-valued functions with respect to (possibly infinite) vector lattice-valued measures is presented with respect to abstract convergences, satisfying suitable axioms, and some fundamental properties are studied. Moreover, by means of this integral, some convergence results on operators in vector lattice-valued modulars are proved. Some applications are given to moment kernels and to the Brownian motion. [ABSTRACT FROM AUTHOR]

Some kinds of statistical convergence have been studied and investigated on Riesz spaces with respect to order convergence recently. In this paper we introduce the concept of statistically continuous and bounded operators with statistically order convergent sequences on Riesz spaces. Moreover, we give some relations with other kinds of operators. [ABSTRACT FROM AUTHOR]

We provide two new characterizations of bounded orthogonally additive polynomials from a uniformly complete vector lattice into a convex bornological space using separately two polynomial identities of Kusraeva involving the root mean power and the geometric mean. Furthermore, it is shown that a polynomial on a vector lattice is orthogonally additive whenever it is orthogonally additive on the positive cone. These results improve recent characterizations of bounded orthogonally additive polynomials by G. Buskes and the author. [ABSTRACT FROM AUTHOR]

The classical Kantorovich theorem asserts the existence and uniqueness of a linear extension of a positive additive mapping, defined on the positive cone E+ of a Riesz space E taking values in an Archimedean Riesz space F, to the entire space E. We prove that, if E has the principal projection property and F is Dedekind σ-complete then for every e∈E+ every positive finitely additive F-valued measure defined on the Boolean algebra Fe of fragments of e has a unique positive linear extension to the ideal Ee of E generated by e. If, moreover, the measure is τ-continuous then the linear extension is order continuous. [ABSTRACT FROM AUTHOR]

Let E be a sublattice of a vector lattice F. (xa) Ç E is said to be order convergent to a vector x (in symbols xa --~C x), whenever there exists another net (ya) in F with the same index set satisfying ya 4 0 in F and |xa -- x| < ya for all indexes a. If F = E~~, this convergence is called b-order convergence and we write xa --C x. In this manuscript, first we study some properties of order convergence nets and we extend same results to the general case. In the second part, we introduce b-order continuous operators and we investigate some properties of this new concept. An operator T between two vector lattices E and F is said to be b-order continuous, if xa --+ 0 in E implies Txa --G 0 in F. [ABSTRACT FROM AUTHOR]

S-metric space was introduced by Sedghi et al. [5] in 2012. We derive some common fixed point results for self-mappings on vector valued complete S-metric space. In support of our results, we also give some examples. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We define bidual bounded uo-convergence in vector lattices and investigate relations between this convergence and b-property. We prove that for a regular Riesz dual system ⟨ X , X ∼ ⟩ , X has b-property if and only if the order convergence in X agrees with the order convergence in X ∼ ∼ . [ABSTRACT FROM AUTHOR]

An operator T from vector lattice E into topological vector space (F , τ) is said to be order-to-topology continuous whenever x α → o 0 implies T x α → τ 0 for each (x α) α ⊂ E . The collection of all order-to-topology continuous operators will be denoted by L o τ (E , F) . In this paper, we will study some properties of this new class of operators. We will investigate the relationships between order-to-topology continuous operators and others classes of operators such as order continuous, order weakly compact and b-weakly compact operators. Under some sufficient and necessary conditions we show that the adjoint of order-to-norm continuous operators is also order-to-norm continuous and vice verse. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem on the existence of the upper (lower) envelope of a convex cone or, more generally, a convex set for functions on the projective limit of vector lattices with values in the completion of the Kantorovich space or on the extended real line. We propose vectorial, ordinal, and topological dual interpretations of the existence conditions for such envelopes and the method of constructing it. Applications to the problem on the existence of a nontrivial (pluri)subharmonic and/or (pluri)harmonic minorant for functions in domains of a finite-dimensional real or complex space are considered. We also propose general approaches to problems on the nontriviality of weight classes of holomorphic functions, describing zero (sub)sets for such classes of holomorphic functions, and to the problem of representing a meromorphic function as a ratio of holomorphic function from a given weight class. [ABSTRACT FROM AUTHOR]

We study b -property of a sublattice (or an order ideal) F of a vector lattice E . In particular, b -property of E in E δ, the Dedekind completion of E, b -property of E in E u, the universal completion of E, and b -property of E in Eˆ(ˆτ ), the completion of E. [ABSTRACT FROM AUTHOR]

The paper aims to examine the problem of characterization and representation of order bounded multilinear operators between vector lattices which may be expressed as sums of disjointness preserving multilinear operators. [ABSTRACT FROM AUTHOR]

We investigate properties of order compact, unbounded order compact and relatively uniformly compact operators acting on vector lattices. An operator is said to be order compact if it maps an arbitrary order bounded net to a net with an order convergent subnet. Analogously, an operator is said to be unbounded order compact if it maps an arbitrary order bounded net to a net with uo-convergent subnet. After exposing the relationships between order compact, unbounded order compact, semicompact and GAM -compact operators; we study those operators mapping an arbitrary order bounded net to a net with a relatively uniformly convergent subnet. By using the nontopological concepts of order and unbounded order convergences, we derive new results related to these classes of operators. [ABSTRACT FROM AUTHOR]

We study measurable elements of a Riesz space E, i.e. elements e ∈ E ∖ {0} for which the Boolean algebra ξe of fragments of e is measurable. In particular, we prove that the set E meas of all measurable elements of a Riesz space E with the principal projection property together with zero is a σ -ideal of E . Another result asserts that, for a Riesz space E with the principal projection property the following assertions are equivalent. (1) The Boolean algebra U of bands of E is measurable. (2) E meas = E and E satisfies the countable chain condition. (3) E can be embedded as an order dense subspace of L0 (μ) for some probability measure μ . [ABSTRACT FROM AUTHOR]

For any real number p ∈ [ 1 , + ∞) {p\in[1,+\infty)} , we characterise the operations ℝ I → ℝ {\mathbb{R}^{I}\to\mathbb{R}} that preserve p-integrability, i.e., the operations under which, for every measure μ, the set ℒ p ⁢ (μ) {\mathcal{L}^{p}(\mu)} is closed. We investigate the infinitary variety of algebras whose operations are exactly such functions. It turns out that this variety coincides with the category of Dedekind σ-complete truncated Riesz spaces, where truncation is meant in the sense of R. N. Ball. We also prove that ℝ {\mathbb{R}} generates this variety. From this, we exhibit a concrete model of the free Dedekind σ-complete truncated Riesz spaces. Analogous results are obtained for operations that preserve p-integrability over finite measure spaces: the corresponding variety is shown to coincide with the much studied category of Dedekind σ-complete Riesz spaces with weak unit, ℝ {\mathbb{R}} is proved to generate this variety, and a concrete model of the free Dedekind σ-complete Riesz spaces with weak unit is exhibited. [ABSTRACT FROM AUTHOR]

We prove that the category of Dedekind σ -complete Riesz spaces is an infinitary variety, and we provide an explicit equational axiomatization. In fact, we show that finitely many axioms suffice over the usual equational axiomatization of Riesz spaces. Our main result is that R , regarded as a Dedekind σ -complete Riesz space, generates this category as a variety; further, we use this fact to obtain the even stronger result that R generates this category as a quasi-variety. Analogous results are established for the categories of (i) Dedekind σ -complete Riesz spaces with a weak order unit, (ii) Dedekind σ -complete lattice-ordered groups, and (iii) Dedekind σ -complete lattice-ordered groups with a weak order unit. [ABSTRACT FROM AUTHOR]

Unlike the uniform completion, the Dedekind completion of a vector lattice is not functorial. In order to repair the lack of functoriality of Dedekind completions, we enrich the signature of vector lattices with a proximity relation, thus arriving at the category pdv of proximity Dedekind vector lattices. We prove that the Dedekind completion induces a functor from the category bav of bounded archimedean vector lattices to pdv, which in fact is an equivalence. We utilize the results of Dilworth [14] to show that every proximity Dedekind vector lattice D is represented as the normal real-valued functions on the compact Hausdorff space associated with D. This yields a contravariant adjunction between pdv and the category KHaus of compact Hausdorff spaces, which restricts to a dual equivalence between KHaus and the proper subcategory of pdv consisting of those proximity Dedekind vector lattices in which the proximity is uniformly closed. We show how to derive the classic Yosida Representation [40], Kakutani-Krein Duality [24, 26], Stone-Gelfand-Naimark Duality [35, 16], and Stone- Nakano Theorem [35, 32] from our approach. [ABSTRACT FROM AUTHOR]

We show that the order convergence in a vector lattice X is not topological unless dimX < ∞. We also discuss conditions under which the uo− and uru−convergences are topological. [ABSTRACT FROM AUTHOR]

We construct a linear operator R, between two Banach lattices, such that R has a modulus which does not satisfy the Riesz–Kantorovich formula, thereby answering a long-standing question in the theory of regular operators. [ABSTRACT FROM AUTHOR]

In this paper we formulate and proof Girsanov's theorem in vector lattices. To reach this goal, we develop the theory of cross-variation processes, derive the cross-variation formula and the Kunita–Watanabe inequality. Also needed and derived are properties of exponential processes, Itô's rule for multi-dimensional processes and the integration by parts formula for martingales. After proving Girsanov's theorem for the one-dimensional case, we also discuss the multi-dimensional case. [ABSTRACT FROM AUTHOR]

In this paper, as a generalization of the concept of pseudo-almost f-algebra, we define a new concept of almost orthosymmetric bilinear map on a vector lattice and prove that the Arens triadjoint of a positive almost orthosymmetric bilinear map is positive almost orthosymmetric. This also extends results on the order bidual of pseudo-almost f-algebras. [ABSTRACT FROM AUTHOR]

Necessary and sufficient conditions are found under which the sum of N order bounded disjointness preserving operators is n-disjoint with n and N naturals. It is shown that the decomposition of an order bounded n-disjoint operator into a sum of disjointness preserving operators is unique up to "Boolean permutation," the meaning of which is clarified in the course of the presentation. [ABSTRACT FROM AUTHOR]

Let (xα) be a net in a locally solid vector lattice (X,τ); we say that (xα) is unbounded τ-convergent to a vector x∈X if |xα-x|∧w→τ0 for all w∈X+. In this paper, we study general properties of unbounded τ-convergence (shortly uτ-convergence). uτ-convergence generalizes unbounded norm convergence and unbounded absolute weak convergence in normed lattices that have been investigated recently. We introduce uτ-topology and briefly study metrizability and completeness of this topology. [ABSTRACT FROM AUTHOR]

Recently measure-free versions of the Brezis-Lieb lemma were proved for unbounded order convergence in vector lattices. In this article, we extend these versions to convergence vector lattices. [ABSTRACT FROM AUTHOR]

Let M={mλ}λ∈Λ be a separating family of lattice seminorms on a vector lattice X, then (X,M) is called a multi-normed vector lattice (or MNVL). We write xα→mx if mλ(xα-x)→0 for all λ∈Λ. A net xα in an MNVL X=(X,M) is said to be unbounded m-convergent (or um-convergent) to x if |xα-x|∧u→m0 for all u∈X+. um-Convergence generalizes un-convergence (Deng et al. in Positivity 21:963-974, 2017; Kandić et al. in J Math Anal Appl 451:259-279, 2017) and uaw-convergence (Zabeti in Positivity, 2017. doi:10.1007/s11117-017-0524-7), and specializes up-convergence (Aydın et al. in Unbounded p-convergence in lattice-normed vector lattices. arXiv:1609.05301) and uτ-convergence (Dabboorasad et al. in uτ-Convergence in locally solid vector lattices. arXiv:1706.02006v3). um-Convergence is always topological, whose corresponding topology is called unbounded m-topology (or um-topology). We show that, for an m-complete metrizable MNVL (X,M), the um-topology is metrizable iff X has a countable topological orthogonal system. In terms of um-completeness, we present a characterization of MNVLs possessing both Lebesgue’s and Levi’s properties. Then, we characterize MNVLs possessing simultaneously the σ-Lebesgue and σ-Levi properties in terms of sequential um-completeness. Finally, we prove that every m-bounded and um-closed set is um-compact iff the space is atomic and has Lebesgue’s and Levi’s properties. [ABSTRACT FROM AUTHOR]

We prove that the function and lattice definitions of a narrow operator defined on a Köthe Banach space E on a finite atomless measure space (Ω,Σ,μ) are equivalent if and only if the set of all simple functions is dense in E. This answers Problem 10.3 from Popov and Randrianantoanina (Narrow operators on function spaces and vector lattices, De Gruyter studies in mathematics 45, De Gruyter, Berlin, 2013). [ABSTRACT FROM AUTHOR]

Nonstandard hulls of a vector lattice were introduced and studied in many papers. Recently, these notions were extended to ordered vector spaces. In the present paper, following the construction of associated Banach-Kantorovich space due to Emelyanov, we describe and investigate the nonstandard hull of a lattice-normed space, which is the foregoing generalization of Luxemburg's nonstandard hull of a normed space. [ABSTRACT FROM AUTHOR]

A synaptic algebra A is a generalization of the self-adjoint part of a von Neumann algebra. We study a linear subspace V of A in regard to the question of when V is a vector lattice. Our main theorem states that if V contains the identity element of A and is closed under the formation of both the absolute value and the carrier of its elements, then V is a vector lattice if and only if the elements of V commute pairwise. [ABSTRACT FROM AUTHOR]

We show that an Urysohn lattice pre-homomorphism defined on a normal sublattice D of a vector lattice E can be extended to the whole space E and the extended operator is an Urysohn lattice homomorphism. We introduce a new class of nonlinear operators which called φ -operators and describe some of their properties. Finally we investigate a structure of positive orthogonally additive operators dominated by an Urysohn lattice homomorphism defined on a vector lattice E and taking value in Dedekind complete vector lattice F . [ABSTRACT FROM AUTHOR]

The aim of this work is to introduce a purely order-based Kantorovich–Wright type integration of scalar functions with respect to a vector measure defined on a δ -ring and taking values in a Dedekind σ -complete vector lattice and use it for obtaining general representation theorems for Dedekind complete vector lattices. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new concept of a d-bimorphism on a vector lattice and prove that, for vector lattices A and B, the Arens triadjoint T : (A')'n ×(A')'n → (B')'n of a d-bimorphism T : A×A→B is a d-bimorphism. This generalizes the concept of d-algebra and some results on the order bidual of d-algebras. [ABSTRACT FROM AUTHOR]