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In this paper, we introduce the class of demi-strongly order bounded operators on a Riesz space generalization of strongly order bounded operators. Let M be a Riesz space, an operator H from M into M is said to be a demi-strongly order bounded operator if for every net {uα} in M+ whenever 0 ≤ uα ↑ ≤ u ′′, u ′′ in M∼∼ and {uα −H(uα)} is order bounded in M, then {uα} is order bounded in M. We obtain a characterization of the b-property by the term of demi-strongly order bounded operators. In addition, we study the relationship between strongly order bounded operators and demi-strongly order bounded operators. Finally, we also investigate some properties of the class of demi-strongly order bounded operators. [ABSTRACT FROM AUTHOR]

The Yosida representation for an Archimedean vector lattice A with weak unit u, denoted (A, u), reveals similarities between the ideas of the title, FST and SMP. If A is Archimedean, the conclusion of the FST means exactly that for each 0 < e ∈ A , the Yosida space for (e dd , e) , denoted Y e , has a base of clopen sets. This yields a short “Yosida based" proof of FST. On the other hand, SMP implies that each Y e has a π -base of clopen sets. The converse fails, but holds if A has a strong unit (and in a somewhat more general situation). [ABSTRACT FROM AUTHOR]

The contraction condition in the Banach contraction principle forces a function to be continuous. Many authors overcome this obligation and weaken the hypotheses via metric spaces endowed with a partial order. In this paper, we present some coupled fixed point theorems for the functions having mixed monotone properties on ordered vector metric spaces, which are more general spaces than partially ordered metric spaces. We also define the double monotone property and investigate the previous results with this property. In the last section, we prove the uniqueness of a coupled fixed point for non-monotone functions. In addition, we present some illustrative examples to emphasize that our results are more general than the ones in the literature. [ABSTRACT FROM AUTHOR]

Let E and F be two Archimedean Riesz spaces. An operator T : E → F is said to be unbounded order continuous (uo-continuous), if u α → uo 0 in E implies T u α → uo 0 in F. In this study, our main aim is to give the solution to two open problems which are posed by Bahramnezhad and Azar. Using this, we obtain that the space L uo (E , F) of order bounded unbounded order continuous operators is an ideal in L b (E , F) for Dedekind complete Riesz space F. In general, by giving an example that the space L uo (E , F) of order bounded unbounded order continuous operators is not a band in L b (E , F) , we obtain the conditions on E or F for the space L uo (E , F) to be a band in L b (E , F) . Then, we give the extension theorem for uo-continuous operators similar to Veksler’s theorem for order continuous operators. [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]

The statistically unbounded p-convergence is an abstraction of the statistical order, unbounded order, and p-convergences. We investigate the concept of the statistically unbounded convergence on lattice-normed Riesz spaces with respect to statistical p-decreasing sequences. Also, we get some relations between this concept and the other kinds of statistical convergences on Riesz spaces. [ABSTRACT FROM AUTHOR]

We study orthogonally additive operators on Riesz spaces. Our first result gives necessary and sufficient conditions on a pair of Riesz spaces (E, F) for which every orthogonally additive operator from E to F is laterally-to-order bounded. Second result extends an analogue of Pitt’s compactness theorem obtained by the second and third named authors for narrow linear operators to the setting of orthogonally additive operators. Third result provides sufficient conditions on a pair of orthogonally additive operators S and T to have S ∨ T , as well as to have S ∧ T without any assumption on the domain and range spaces. Finally we prove an analogue of Meyer’s theorem on the existence of modules of disjointness preserving operator for orthogonally additive operators. [ABSTRACT FROM AUTHOR]

A new class of the Itô integral for Brownian motion is defined and studied in the framework of Riesz spaces. The stochastic process with respect to this stochastic integral is non-adapted and it is a motivitation to construct near-martingales in Riesz spaces. Furthermore, we state Doob–Meyer decomposition theorem for near-submartingales in Riesz spaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate that under which conditions the inverse of an invertible ideal operator becomes an ideal operator. To determine conditions, in the first case, we consider the center of a Riesz space and achieve the solution by using some properties of it. In the second case, we first give an example of an ideal operator which is not order bounded. Then, we research when an ideal operator is order bounded. Thus, we conclude that if an invertible operator is order bounded then being an ideal operator, a band operator, and a disjointness preserving operator are equivalent and the same situation holds for the inverses of these three operators. [ABSTRACT FROM AUTHOR]

A vector-b-metric space is a more general space than a b-metric space because it is Riesz space valued. The study of ordered vector spaces is important in economic analysis. In this paper, we prove some common fixed point results for three and four mappings on vector-b-metric space with a graph. Our results extend many known results for three and four mappings on vector metric spaces. [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]

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. [ABSTRACT FROM AUTHOR]

Let U and V be two Archimedean Riesz spaces. An operator S : U → V is said to be unbounded order continuous (uo-continuous), if rα uo → 0 in U implies Srα uo → 0 in V . In this paper, we give some properties of the uo-continuous dual Uuo~ of U. We show that a nonzero linear functional f on U is uo-continuous if and only if f is a linear combination of finitely many order continuous lattice homomorphisms. The result allows us to characterize the uo-continuous dual Uuo~. In general, by giving an example that the uo-continuous dual U~ uo is not a band in U~, we obtain the conditions for the uo-continuous dual of a Banach lattice U to be a band in U~. Then, we examine the properties of uo-continuous operators. We show that S is an order continuous operator if and only if S is an unbounded order continuous operator when S is a lattice homomorphism between two Riesz spaces U and V . Finally, we proved that if an order bounded operator S : U → V between Archimedean Riesz space U and atomic Dedekind complete Riesz space V is uo-continuous, then /S/ is uo-continuous. [ABSTRACT FROM AUTHOR]

In this note we prove that positive multi-valued operators on Riesz spaces which are everywhere defined are always continuous. Further, we prove that the sum of two positive relations having the same null part is itself positive. This result allows us to study the domination problem in the class of positive relations. [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 $\sigma$-complete then for every $e \in E^+$ every positive finitely additive $F$-valued measure defined on the Boolean algebra $\mathfrak{F}_e$ of fragments of $e$ has a unique positive linear extension to the ideal $E_e$ of $E$ generated by $e$. If, moreover, the measure is $\tau$-continuous then the linear extension is order continuous.

The Generalized Relative Entropy inequality is a ubiquitous property in mathematical models applied in physics or biology. In spite of its importance, it is currently proved on a case-by-case basis in the literature. Here, we show that GRE is actually a generic consequence of Łoskot-Rudnicki’s inequality that is reminiscent of Jensen’s inequality. [ABSTRACT FROM AUTHOR]

In this work we review some common fixed point theorems for four non-continuous mappings in ε-metric spaces, where the metric is Riesz space valued. These results cover famous comparable results in the existing literature by considering fewer conditions. [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]

In this paper, we will study on some topologies induced by order convergence in a Riesz space. We will investigate the relationships of them. [ABSTRACT FROM AUTHOR]

Let be a unital -module over an -algebra . With the help of Arens extension theory, a module structure on can be defined. The paper deals mainly with properties of the Arens homomorphism , which is defined by the module structure on . Necessary and sufficient conditions for an submodule of to be an order ideal are obtained. [ABSTRACT FROM AUTHOR]

We generalize some maximal probability inequalities, proven for a class of random variables, to the measure-free setting of Riesz spaces. We prove generalizations of the Kolmogorov inequality, Hájek–Rényi inequality, Lévy’s inequality and Etemadi’s inequality. [ABSTRACT FROM AUTHOR]

We study measurable elements of a Riesz space $E$, i.e. elements $e \in E \setminus \{0\}$ for which the Boolean algebra $\mathfrak{F}_e$ of fragments of $e$ is measurable. In particular, we prove that the set $E_{\rm meas}$ of all measurable elements of a Riesz space $E$ with the principal projection property together with zero is a $\sigma$-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 $\mathcal{U}$ of bands of $E$ is measurable. (2) $E_{\rm meas} = E$ and $E$ satisfies the countable chain condition. (3) $E$ can be embedded as an order dense subspace of $L_0(\mu)$ for some probability measure $\mu$.

Rectangular metric space was first defined by Branciari [3] in 2000. In this paper we prove some fixed point theorems on vector valued rectangular metric space. Our results generalize some fixed point results for scalar valued. [ABSTRACT FROM AUTHOR]

We introduce the -statistical monotone and the -statistical order convergent sequences in Riesz spaces. We also give some relations between the lattice operations and the -statistical convergence in Riesz spaces, and also, some relations between the order convergence and -statistical order convergence. [ABSTRACT FROM AUTHOR]

In this paper we present a fixed point theorem for order-preserving Prešić type operators on ordered vector metric spaces. This result extends many results in the literature obtained for Prešić type operators both on metric spaces and partially ordered metric spaces. We also emphasize the relationships between our work and the previous ones in the literature. Finally we give an example showing the fact that neither results for Prešić type contractions on metric spaces nor the results for ordered Prešić type contractions on ordered metric space is applicable to it. [ABSTRACT FROM AUTHOR]

A projection lateral band $G$ in a Riesz space $E$ is defined to be a lateral band which is the image of an orthogonally additive projection $Q: E \to E$ possessing the property that $Q(x)$ is a fragment of $x$ for all $x \in E$, called a lateral retraction of $E$ onto $G$ (which is then proved to be unique). We investigate properties of lateral retracts, that are, images of lateral retractions, and describe lateral retractions onto principal projection lateral bands (i.e. lateral bands generated by single elements) in a Riesz space with the principal projection property. Moreover, we prove that every lateral retract is a lateral band, and every lateral band in a Dedekind complete Riesz space is a projection lateral band.

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 present the algebraic structures behind the approaches used to work with pairwise comparison matrices and, in general, the representation of preferences. We obtain a general definition of consistency and a universal decomposition in the space of PCMs, which allow us to define a consistency index. Also Arrow's theorem, which is presented in a general form, is relevant. All the presented results can be seen in the main formulations of PCMs, i.e., multiplicative, additive and fuzzy approach, by the fact that each of them is a particular interpretation of the more general algebraic structure needed to deal with these theories. [ABSTRACT FROM AUTHOR]

Statistical convergence is an active area, and it appears in a wide variety of topics. However, it has not been studied extensively in Riesz spaces. There are a few studies about the statistical convergence on Riesz spaces, but they only focus on the relationship between statistical and order convergences of sequences in Riesz spaces. In this paper, we introduce the notion of (V, λ)-order summable by using the concept of λ-statistical monotone and the λ-statistical order convergent sequences in Riesz spaces. Moreover, we give some relations between (V, λ)-order summable and λ-statistical order convergence. [ABSTRACT FROM AUTHOR]

Recently, the different types of unbounded convergences (u o , u n , u a w , u a w ∗) in Banach lattices were studied. In this paper, we study the continuous functionals with respect to unbounded convergences. We first characterize the continuity of linear functionals for these convergences. Then we define the corresponding unbounded dual spaces and get their exact form. Based on these results, we discuss order continuity and reflexivity of Banach lattices. Some related results are obtained as well. [ABSTRACT FROM AUTHOR]

P.K. Das and G.C. Nayak [3] introduced the concept of generalized variational like inequalities in H-spaces in the presence of T-η-invex function. There the existence theorems of mixed generalized variational like inequalities are studied in H-spaces and also in Riesz spaces in the presence of T-η-invex function. In the present note, we extend their results to partial KKM spaces, which contain H-spaces as very particular subclass.

We prove that the algebraic dimension of an infinite dimensional C-σ-complete Riesz space (in particular, of a Dedekind σ-complete and a laterally σ-complete Riesz space) with the principal projection property which either has a weak order unit or is not purely atomic, is at least continuum. A similar (incomparable to ours) result for complete metric linear spaces is well known. [ABSTRACT FROM AUTHOR]

A mixed lattice group is a generalization of a lattice ordered group. The theory of mixed lattice semigroups dates back to the 1970s, but the corresponding theory for groups and vector spaces has been relatively unexplored. In this paper we investigate the basic structure of mixed lattice groups, and study how some of the fundamental concepts in Riesz spaces and lattice ordered groups, such as the absolute value and other related ideas, can be extended to mixed lattice groups and mixed lattice vector spaces. We also investigate ideals and study the properties of mixed lattice group homomorphisms and quotient groups. Most of the results in this paper have their analogues in the theory of Riesz spaces. [ABSTRACT FROM AUTHOR]

The aim of this article is to describe a class of ∗ -algebras that allows to treat well-behaved algebras of unbounded operators independently of a representation. To this end, Archimedean ordered ∗ -algebras ( ∗ -algebras whose real linear subspace of Hermitian elements are an Archimedean ordered vector space with rather weak compatibilities with the algebraic structure) are examined. The order induces a translation-invariant uniform metric which comes from a C ∗ -norm in the bounded case. It will then be shown that uniformly complete Archimedean ordered ∗ -algebras have good order properties (like existence of infima, suprema or absolute values) if and only if they have good algebraic properties (like existence of inverses or square roots). This suggests the definition of Su ∗ -algebras as uniformly complete Archimedean ordered ∗ -algebras which have all these equivalent properties. All methods used are completely elementary and do not require any representation theory and not even any assumptions of boundedness, so Su ∗ -algebras generalize some important properties of C ∗ -algebras to algebras of unbounded operators. Similarly, they generalize some properties of Φ -algebras (certain lattice-ordered commutative real algebras) to non-commutative ordered ∗ -algebras. As an example, Su ∗ -algebras of unbounded operators on a Hilbert space are constructed. They arise e.g. as ∗ -algebras of symmetries of a self-adjoint (not necessarily bounded) Hamiltonian operator of a quantum mechanical system. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new concept for Riesz almost lacunary $\Gamma^{3}$ sequence spaces strong $P-$ convergent to zero with respect to an Musielak-Orlicz function and examine some properties of the resulting sequence spaces. We also introduce and study statistical convergence of Riesz almost lacunary $\Gamma^{3}$ sequence spaces and also some inclusion theorems are discussed.

In this paper we introduce the backward operator is $\nabla$ and study the notion of $\nabla-$ statistical convergence and $\nabla-$ statistical Cauchy sequence using by almost $\left(\lambda_{m}\mu_{n}\right)$ convergence in $\chi^{2}-$Riesz space and also some inclusion theorems are discussed.

The full lattice convergence on a locally solid Riesz space is an abstraction of the topological, order, and relatively uniform convergences. We investigate four modifications of a full convergence c on a Riesz space. The first one produces a sequential convergence sc. The second makes an absolute c -convergence and generalizes the absolute weak convergence. The third modification makes an unbounded c -convergence and generalizes various unbounded convergences recently studied in the literature. The last one is applicable whenever c is a full convergence on a commutative l -algebra and produces the multiplicative modification mc of c. We study general properties of full lattice convergence with emphasis on universally complete Riesz spaces and on Archimedean f -algebras. The technique and results in this paper unify and extend those which were developed and obtained in recent literature on unbounded convergences. [ABSTRACT FROM AUTHOR]

We say that a Riesz space E has the horizontal Egorov property if for every net (ƒα) in E, order convergent to ƒ ∈ E with |ƒα| + |ƒ| ≤ e ∈ E+ for all α, there exists a net (eβ) of fragments of e laterally convergent to e such that for every nβ, the net (|ƒ − ƒα | ∧ eβ)α e-uniformly tends to zero. Our main result asserts that every Dedekind complete Riesz space which satisfies the weak distributive law possesses the horizontal Egorov property. A Riesz space E is said to satisfy the weak distributive law if for every e ∈ E+ \ 0 the Boolean algebra Fe of fragments of e satisfies the weak distributive law; that is, whenever (Πn)n ∈ N is a sequence of partitions of Fe, there is a partition Π of Fe such that every element of Π is finitely covered by each of Πn (e.g., every measurable Boolean algebra is so). Using a new technical tool, we show that for every net (ƒα) order convergent to ƒ in a Riesz space with the horizontal Egorov property there are a horizontally vanishing net (vβ) and a net (uα,β)(α,β) ∈ A × B, which uniformly tends to zero for every fixed β such that |ƒ − ƒα| ≤ uα, β + vβ for all α, β. [ABSTRACT FROM AUTHOR]

In this paper, the Gâteaux and Frêchet differentiations of functions on Riesz space are introduced without topological structure. Thus, we aim to study Gâteaux and Frêchet differentiability functions in vector lattice by developing topology-free techniques, and also, we give some relations with other kinds of operators. [ABSTRACT FROM AUTHOR]

The multiplicative convergence on Riesz algebras introduced and investigated with respect to norm and order convergences. If X is a Riesz space and E is a Riesz algebra then the vector norm μ: XE can be considered. Then (X, μ, E) is called algebraic lattice normed spaces. A net (sα)α A in an (X, μ, E) is said to be multiplicative μ-convergent to x if μ(xα-x) holds for all. In this paper, the general properties of this convergence are studied. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new concept for generalized almost (λ_{m_{i}}μ_{n_{ℓ}}γ_{k_{j}}) convergence in χ_{R_{λ_{m_{i}}μ_{n_{ℓ}}γ_{k_{j}}}}³-Riesz spaces strong P- convergent to zero with respect to an Orlicz function and examine some properties of the resulting sequence spaces. We also introduce and study statistical convergence of generalized almost (λ_{m_{i}}μ_{n_{ℓ}}γ_{k_{j}}) convergence in χ_{R_{λ_{m_{i}}μ_{n_{ℓ}}γ_{k_{j}}}}³-Riesz space and also some inclusion theorems are discussed.

This paper deals with a study on classes of non linear operators. Let $SL(X,Y)$ be the set of all sublinear operators between two Riesz spaces $X$ and $Y$. It is a convex cone of the space $H(X,Y)$ of all positively homogeneous operators. In this paper we study some spaces generated by this cone, therefore we study several properties, which are well known in the theory of Riesz spaces, like order continuity, order boundedness etc. Finally, we try to generalise the concept of adjoint operator. First, by using the analytic form of Hahn-Banach theorem, we adapt the notion of adjoint operator to the category of positively homogeneous operators. Then we apply it to the class of operators generated by the sublinear operators.

We study multiorthomorphisms on Riesz spaces. As an application, we introduce and we describe a new class of order bounded orthogonally additive homogeneous polynomials. [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]

A sequence (аn) in a Riesz space E is called uo-convergent (or unbounded order convergent) to a ∈ E if ∣an - a∣ ^ u →° 0 for all u ∈ E+ and unbounded order Cauchy (uo-Cauchy) if ∣an - an+p∣ is uo-convergent to 0. In the first part of this study we define ud,E-convergence (or unbounded vectorial convergence) in vector metric spaces, which is more general than usual metric spaces, and examine relations between unbounded order convergence, unbounded vectorial convergence, vectorial convergence and order convergence. In the last part we construct the unbounded Cauchy completion of vector metric spaces by the motivation of the fact that every metric space has Cauchy completion. In this way, we have obtained a more general completion of vector metric spaces. [ABSTRACT FROM AUTHOR]