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A vector sublattice of the order bounded operators on a Dedekind complete vector lattice can be supplied with the convergence structures of order convergence, strong order convergence, unbounded order convergence, strong unbounded order convergence, and, when applicable, convergence with respect to a Hausdorff uo-Lebesgue topology and strong convergence with respect to such a topology. We determine the general validity of the implications between these six convergences on the order bounded operator and on the orthomorphisms. Furthermore, the continuity of left and right multiplications with respect to these convergence structures on the order bounded operators, on the order continuous operators, and on the orthomorphisms is investigated, as is their simultaneous continuity. A number of results are included on the equality of adherences of vector sublattices of the order bounded operators and of the orthomorphisms with respect to these convergence structures. These are consequences of more general results for vector sublattices of arbitrary Dedekind complete vector lattices. The special attention that is paid to vector sublattices of the orthomorphisms is motivated by explaining their relevance for representation theory on vector lattices. [ABSTRACT FROM AUTHOR]

In this article we present the construction of variable power (or variable concavification) of quasi-Banach lattices with a bounded variable exponent from ideal center. This construction generalizes the similar constructions with constant exponent. We also introduce the concepts of variable convexity and concavity of quasi-Banach lattices under the which the variable power of a quasi-Banach lattice is a Banach lattice. [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 provide a general framework for no-arbitrage concepts in topological vector lattices, which covers many of the well-known no-arbitrage concepts as particular cases. The main structural condition we impose is that the outcomes of trading strategies with initial wealth zero and those with positive initial wealth have the structure of a convex cone. As one consequence of our approach, the concepts NUPBR, NAA 1 and NA 1 may fail to be equivalent in our general setting. Furthermore, we derive abstract versions of the fundamental theorem of asset pricing (FTAP), including an abstract FTAP on Banach function spaces, and investigate when the FTAP is warranted in its classical form with a separating measure. We also consider a financial market with semimartingales which does not need to have a numéraire, and derive results which show the links between the no-arbitrage concepts by only using the theory of topological vector lattices and well-known results from stochastic analysis in a sequence of short proofs. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to give a vector lattice version of a Theorem by Burkholder about convergence of martingales. The proof is based on a vector lattice analogue of Austin's sample function theorem, proved recently by Grobler, Labuschagne and Marraffa and on a new characterization of elements of the sup-completion of a universally complete vector lattice do not belong to the space. [ABSTRACT FROM AUTHOR]

Since 2001, rough convergence has been discussed in research where the core factor degree of roughness has been treated only as a real number. Instead of analyzing different results based on the same definition, we start with the new flow of ideas to enlarge the spectrum of degree of roughness. As a consequence we invent a new approach to degree of roughness with approximation of neighborhood conception over locally solid Riesz spaces. While we start working with this new definition and scrutinize existing results, many of the earlier results are violated. Naturally our curiosity begins to know that under which condition(s) we may get back either some of the previous results or not? Apart from these, we consider various non-trivial results which enhance and signify the motivation of our paper. [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]

The aim of this note is to consider different notions for the Banach–Saks property in locally solid vector lattices as an extension for the known concepts of the Banach–Saks property in Banach lattices. We investigate relations between them; in particular, we shall characterize spaces in which, these notions agree. [ABSTRACT FROM AUTHOR]

We introduce the notion of (maximal) multi-truncations on a vector lattice as a generalization of the notion of truncations, an object of recent origin. We obtain a Johnson–Kist type representation of vector lattices with maximal multi-truncations as vector lattices of almost-finite extended-real continuous functions. The spectrum that allows such a representation is a particular set of prime ideals equipped with the Hull–Kernel topology. Various representations from the existing literature will appear as special cases of our general result. [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]

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 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]

In this paper we prove that the ℓ -group tensor product of two Archimedean f-rings is again an f-ring. We will use this result to characterize multiplicative ℓ -bimorphisms between unital f-rings. [ABSTRACT FROM AUTHOR]

The paper contains a systematic study of the lateral partial order ⊑ in a Riesz space (the relation x ⊑ y means that x is a fragment of y) with applications to nonlinear analysis of Riesz spaces. We introduce and study lateral fields, lateral ideals, lateral bands and consistent subsets and show the importance of these notions to the theory of orthogonally additive operators, like ideals and bands are important for linear operators. We prove the existence of a lateral band projection, provide an elegant formula for it and prove some properties of this orthogonally additive operator. One of our main results (Theorem 7.5) asserts that, if D is a lateral field in a Riesz space E with the intersection property, X a vector space and T 0 : D → X an orthogonally additive operator, then there exists an orthogonally additive extension T : E → X of T 0 . The intersection property of E means that every two-point subset of E has an infimum with respect to the lateral order. In particular, the principal projection property implies the intersection property. [ABSTRACT FROM AUTHOR]

In vector lattices, the concept of a projection band is a basic tool. We deal with projection bands in the more general setting of an Archimedean pre-Riesz space X. We relate them to projection bands in a vector lattice cover Y of X. If X is pervasive, then a projection band in X extends to a projection band in Y, whereas the restriction of a projection band B in Y is not a projection band in X, in general. We give conditions under which the restriction of B is a projection band in X. We introduce atoms and discrete elements in X and show that every atom is discrete. The converse implication is true, provided X is pervasive. In this setting, we link atoms in X to atoms in Y. If X contains an atom a > 0 , we show that the principal band generated by a is a projection band. Using atoms in a finite dimensional Archimedean pre-Riesz space X, we establish that X is pervasive if and only if it is a vector lattice. [ABSTRACT FROM AUTHOR]

For p ∈ [ 1 , ∞) , S. Kakutani and H.F. Bohnenblust have given characterizations of L p as a Banach lattice. We generalize that result to p ∈ (0 , ∞) . In particular, we show that a quasi-Banach lattice that satisfies ⌋ ⌋ u + v ⌊ ⌊ p = ⌋ ⌋ u ⌊ ⌊ p + ⌋ ⌋ v ⌊ ⌊ p if u ∧ v = 0 , is isometrically Riesz isomorphic to L p . [ABSTRACT FROM AUTHOR]

Truncated Riesz spaces was first introduced by Fremlin in the context of real-valued functions. An appropriate axiomatization of the concept was given by Ball. Keeping only the first Ball's Axiom (among three) as a definition of truncated Riesz spaces, the first named author and El Adeb proved that if E is truncated Riesz space then E ⊕ R can be equipped with a non-standard structure of Riesz space such that E becomes a Riesz subspace of E ⊕ R and the truncation of E is provided by meet with 1. In the present paper, we assume that the truncated Riesz space E has a lattice norm . and we give a necessary and sufficient condition for E ⊕ R to have a lattice norm extending . . Moreover, we show that under this condition, the set of all lattice norms on E ⊕ R extending . has essentially a largest element . 1 and a smallest element . 0 . Also, it turns out that any alternative lattice norm on E ⊕ R is either equivalent to . 1 or equals . 0 . As consequences, we show that E ⊕ R is a Banach lattice if and only if E is a Banach lattice and we get a representation's theorem sustained by the celebrate Kakutani's Representation Theorem. [ABSTRACT FROM AUTHOR]

Suppose that E is a vector lattice where the ordering and the lattice operations in E are defined pointwise by a countable family F = { f i | i ∈ N } of positive linear functional of E and Z is a sublattice of E. Based on algebraic and order properties of E we give necessary and sufficient conditions in order Z to be atomic. Especially we show the existence of a basic sequence { b n } of extremal points (atoms) of Z + so that for any x ∈ Z + a unique sequence (x ^ (n)) of real components of x with respect to { b n } exists so that x = sup { x ^ (n) b n | n ∈ N } and also x = s u p n ∑ i = 1 n x ^ (i) b i . These results give an answer to the problem of the existence of basic derivatives in financial markets. [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]

In a Dedekind complete Riesz space, E, we show that if (P n) is a sequence of band projections in E then lim sup n → ∞ P n - lim inf n → ∞ P n = lim sup n → ∞ P n (I - P n + 1). This identity is used to obtain conditional extensions in a Dedekind complete Riesz spaces with weak order unit and conditional expectation operator of the Barndorff-Nielsen and Balakrishnan–Stepanov generalizations of the first Borel–Cantelli theorem. [ABSTRACT FROM AUTHOR]

We explore inverses of disjointness preserving bijections in infinite dimensional normed pre-Riesz spaces by several methods. As in the case of Banach lattices, our aim is to show that such inverses are disjointness preserving. One method is extension of the operator to the Riesz completion, which works under suitable denseness and continuity conditions. Another method involves a condition on principle bands. Examples illustrate the differences to the Riesz space theory. [ABSTRACT FROM AUTHOR]

We investigate some properties of strongly order bounded operators. For example, we prove that if a Riesz space E is an ideal in E ∼ ∼ and F is a Dedekind complete Riesz space then for each ideal A of E, T is strongly order bounded on A if and only if T A is strongly order bounded. We show that the class of strongly order bounded operators satisfies the domination problem. On the other hand, we present two ways for decomposition of strongly order bounded operators, and we give some of their properties. Also, it is shown that E has order continuous norm or F has the b-property whenever each pre-regular operator form E into F is order bounded. [ABSTRACT FROM AUTHOR]

We prove, in particular, that if E is a Dedekind complete atomless Riesz space and X is a Banach space then the sum of a narrow and a C-compact laterally continuous orthogonally additive operators from E to X is narrow. This generalizes in several directions known results on narrowness of the sum of a narrow and a compact operators for the settings of linear and orthogonally additive operators defined on Köthe function spaces and Riesz spaces. [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]

We show that the free Banach lattice FBL (A) may be constructed as the completion of FVL (A) with respect to the maximal lattice seminorm ν on FVL (A) with ν (a) ⩽ 1 for all a ∈ A . We present a similar construction for the free Banach lattice FBL [ E ] generated by a Banach space E. [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]

Let G to be a torsion free abelian group. In this paper we introduce the following concepts: Algebraic line, algebraic line segment and thin convex subsets of G. Absorbing topological group, that is a generalization of topological vector space. A special subset of a mass lattice group (G, ≤ ) called a link in G which we can construct a locally solid topology on G by it. Some interesting results about unital lattice groups and Riesz spaces with the chief link topology on them have been presented. [ABSTRACT FROM AUTHOR]

Present paper deals with applications of asymptotic equivalence relations on operator nets. These relations are defined via unbounded convergences on vector lattices. Given two convergences c and d on a vector lattice, we study d -asymptotic properties of operator nets formed by c -continuous operators. Asymptotic equivalences are known to be useful and extremely important tools to study infinite behaviors of strongly convergent operator nets and continuous semigroups. After giving a general theory, paper focuses on d -martingale and d -Lotz–Räbiger nets. [ABSTRACT FROM AUTHOR]

In this paper we consider the relationship between order and topology in the vector lattice C(X) of all continuous functions on a Hausdorff space X. We prove that the restriction of f ∈ C (X) to a closed set A in the case when X ∈ T 3 1 2 induces an order continuous operator iff A = Int A ¯. This result enables us to easily characterize bands and projection bands in C 0 (X) , C b (X) and C(X). Our results serve us to provide a positive answer to the question on lifting un-convergence from closed ideals of C 0 (X) and C b (X) . [ABSTRACT FROM AUTHOR]

Let C(X) denotes the Riesz algebra of all real valued continuous functions on a Tychonoff space X, and let L be a vector subspace of C(X). A linear functional H : L → R is said to be 1 ¯ -evaluating if and only if H (f) ∈ f (X) ¯ for all f ∈ L . In this work, we investigate 1 ¯ -evaluating linear functionals on certain vector subspaces of C(X). Our main result in this direction asserts that, when L is a Riesz subalgebra of C(X) that contains idempotents, any linear functional H : L → R is 1 ¯ -evaluating if and only if it acts like a Riesz homomorphism on boundedly clean vector subspaces of L. [ABSTRACT FROM AUTHOR]

The Sandwich theorem (König in Archiv der Mathematik 23:500-1972, 1972) yields the existence of a linear functional sandwiched in between a given superlinear functional and a given sublinear functional. By combining an argument implicit in Fuchssteiner and Maitland Wright (Q J Math 28:155-162, 1977) with a theorem of Pataraia (in: Presented at the 65th peripatetic seminar on sheaves and logic, Aarhus, Denmark, November, 1997), we show that such a linear functional obtains as a common fixed point of a certain family of mappings. [ABSTRACT FROM AUTHOR]

We introduce and study new class of sets (almost L-limited sets). Also, we introduce new concept of property in Banach lattice (almost Gelfand-Phillips property) and we characterize this property using almost L-limited sets. On the other hand, we introduce the class of disjoint limited completely continuous operators which is a largest class than that of limited completely continuous operators, we characterize this class of operators and we study some of its properties. [ABSTRACT FROM AUTHOR]

In Archimedean vector lattices bands can be introduced via three different coinciding notions. First, they are order closed ideals. Second, they are precisely those ideals which equal their double disjoint complements. The third concept is that of an ideal which contains the supremum of any of its bounded subsets, provided the supremum exists in the vector lattice. We investigate these three notions and their relationships in the more general setting of Archimedean pre-Riesz spaces. We introduce the notion of a supremum closed ideal, which is related to the third aforementioned notion in vector lattices. We show that for a directed ideal I in a pervasive pre-Riesz space with the Riesz decomposition property these three concepts coincide, provided the double disjoint complement of I is directed. In pervasive pre-Riesz spaces every directed band is supremum closed and every supremum closed directed ideal I equals its double disjoint complement, provided the double disjoint complement of I is directed. In general, in Archimedean pre-Riesz spaces the three notions differ. For this we provide appropriate counterexamples. [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]

The first aim of this paper is to give a description of the (not necessarily linear) order isomorphisms C(X)→C(Y) where X, Y are compact Hausdorff spaces. For a simple case, suppose X is metrizable and T is such an order isomorphism. By a theorem of Kaplansky, T induces a homeomorphism τ:X→Y. We prove the existence of a homeomorphism X×R→Y×R that maps the graph of any f∈C(X) onto the graph of Tf. For nonmetrizable spaces the result is similar, although slightly more complicated. Secondly, we let X and Y be compact and extremally disconnected. The theory of the first part extends directly to order isomorphisms C∞(X)→C∞(Y). (Here C∞(X) is the space of all continuous functions X→[-∞,∞] that are finite on a dense set.) The third part of the paper considers order isomorphisms T between arbitrary Archimedean Riesz spaces E and F. We prove that such a T extends uniquely to an order isomorphism between their universal completions. (In the absence of linearity this is not obvious.) It follows, that there exist an extremally disconnected compact Hausdorff space X, Riesz isomorphisms ^ of E and F onto order dense Riesz subspaces of C∞(X) and an order isomorphism S:C∞(X)→C∞(X) such that Tf^=Sf^ (f∈E). [ABSTRACT FROM AUTHOR]

In this paper, using the concept of unbounded order convergence in Riesz spaces, we define new classes of operators, named unbounded order continuous (uo-continuous, for short) and boundedly unbounded order continuous operators. We give some conditions under which uo-continuity will be equivalent to order continuity of some operators on Riesz spaces. We show that the collection of all uo-continuous linear functionals on a Riesz space E is a band of E∼. [ABSTRACT FROM AUTHOR]

We study algebras generated by positive matrices, i.e., matrices with nonnegative entries. Some of our results hold in more general setting of vector lattices. We reprove and extend some theorems that have been recently shown by Kandić and Šivic. In particular, we give a more transparent proof of their result that the unital algebra generated by positive idempotent matrices E and F such that EF≥FE is equal to the linear span of the set {I,E,F,EF,FE,EFE,FEF,(EF)2,(FE)2}, and so its dimension is at most 9. We give examples of two positive idempotent matrices that generate unital algebra of dimension 2n if n is even, and of dimension (2n-1) if n is odd. We also prove that the algebra generated by positive matrices B1, B2,…,Bk is triangularizable if ABi≥BiA (i=1,2,…,k) for some positive matrix A with distinct eigenvalues. [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]

If L and M are partially ordered vector spaces, then one can consider regular linear maps from L to M, i.e. linear maps which can be written as the difference of two positive linear maps. If the space L is directed, then the space Lr(L,M) of all regular linear operators becomes a partially ordered vector space itself. We will mainly concern ourselves with the questions when the space Lr(L,M) is itself a Riesz space and how, even if it is not a Riesz space, its lattice operations work. The so-called Riesz-Kantorovich theorem gives sufficient conditions for which Lr(L,M) is a Riesz space and it also specifies the lattice operations by means of the Riesz-Kantorovich formula: if S,T∈Lr(L,M) and x∈L with x≥0 then the supremum S∨T in the point x is given by (S∨T)(x)=supS(y)+T(x-y):0≤y≤x.It is still an open problem if whenever in a more general setting the supremum of two regular operators exists in Lr(L,M), it automatically is given by the Riesz-Kantorovich formula. Our main result concerns the special case where L is a partially ordered vector space with a strong order unit and M is a (possibly infinite) product of copies of the real line, equipped with the lexicographic ordering. It will turn out that under some mild continuity and regularity conditions the lattice operations on Lr(L,M) are indeed given by the Riesz-Kantorovich formula, even though the space Lr(L,M) is not necessarily a Riesz space. [ABSTRACT FROM AUTHOR]

In the theory of operators on a Riesz space (vector lattice), an important result states that the Riesz homomorphisms (lattice homomorphisms) on C(X) are exactly the weighted composition operators. We extend this result to Riesz* homomorphisms on order dense subspaces of C(X). On those subspace we consider and compare various classes of operators that extend the notion of a Riesz homomorphism. Furthermore, using the weighted composition structure of Riesz* homomorphisms we obtain several results concerning bijective Riesz* homomorphisms. In particular, we characterize the automorphism group for order dense subspaces of C(X). Lastly, we develop a similar theory for Riesz* homomorphisms on subspace of C0(X), for a locally compact Hausdorff space X, and apply it to smooth manifolds and Sobolev spaces. [ABSTRACT FROM AUTHOR]

We introduce the notions of multi-suprema and multi-infima for vector spaces equipped with a collection of wedges, generalizing the notions of suprema and infima in ordered vector spaces. Multi-lattices are vector spaces that are closed under multi-suprema and multi-infima and are thus an abstraction of vector lattices. The Riesz decomposition property in the multi-wedged setting is also introduced, leading to Riesz-Kantorovich formulas for multi-suprema and multi-infima in certain spaces of operators. [ABSTRACT FROM AUTHOR]

Let C be an axiomatizable class of order continuous real or complex Banach lattices, that is, this class is closed under isometric vector lattice isomorphisms and ultraproducts, and the complementary class is closed under ultrapowers. We show that if linear isometric embeddings of members of C in their ultrapowers preserve disjointness, the class CB of Banach spaces obtained by forgetting the Banach lattice structure is still axiomatizable. Moreover if C coincides with its “script class” SC, so does CB with SCB. This allows us to give new examples of axiomatizable classes of Banach spaces, namely certain Musielak–Orlicz spaces, Nakano spaces, and mixed norm spaces. [ABSTRACT FROM AUTHOR]

We study the collection of finite elements Φ1(U(E,F)) in the vector lattice U(E,F) of orthogonally additive, order bounded (called abstract Uryson) operators between two vector lattices E and F, where F is Dedekind complete. In particular, for an atomic vector lattice E it is proved that for a finite element in φ∈U(E,R) there is only a finite set of mutually disjoint atoms, where φ does not vanish and, for an atomless vector lattice the zero-vector is the only finite element in the band of σ-laterally continuous abstract Uryson functionals. We also describe the ideal Φ1(U(Rn,Rm)) for n,m∈N and consider rank one operators to be finite elements in U(E,F). [ABSTRACT FROM AUTHOR]

The notion of almost everywhere convergence has been generalized to vector lattices as unbounded order convergence, which proves to be a very useful tool in the theory of vector and Banach lattices. In this short note, we establish some new results on unbounded order convergence that tie up some loose ends. In particular, we show that every norm bounded positive increasing net in an order continuous Banach lattice is uo-Cauchy and that every uo-Cauchy net in an order continuous Banach lattice has a uo-limit in the universal completion. [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]

Let E be a locally solid vector lattice. In this paper, we consider two particular vector subspaces of the space of all order bounded operators on E. With the aid of two appropriate topologies, we show that under some conditions, they establish both, locally solid vector lattices and topologically complete topological algebras. [ABSTRACT FROM AUTHOR]

We prove some properties of positive polynomial mappings between Riesz spaces, using finite difference calculus. We establish the polynomial analogue of the classical result that positive, additive mappings are linear. And we prove a polynomial version of the Kantorovich extension theorem. [ABSTRACT FROM AUTHOR]

A net $$(x_\alpha )$$ in a vector lattice X is unbounded order convergent to $$x \in X$$ if $$|x_\alpha - x| \wedge u$$ converges to 0 in order for all $$u\in X_+$$ . This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net $$(x_\alpha )$$ in a Banach lattice X is unbounded norm convergent to x if for all $$u\in X_+$$ . We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences. [ABSTRACT FROM AUTHOR]

We pose a number of questions and problems about Banach lattice algebras. These concern: What should the definition be? How to add an identity. Order theoretic properties of the multiplication. Order theoretic properties of the left regular representation. [ABSTRACT FROM AUTHOR]

We describe all two dimensional unital Riesz algebras and study representations of them in Riesz algebras of regular operators. Although our results are not complete, we do demonstrate that very varied behaviour can occur even though all these algebras can be given a Banach lattice algebra norm. [ABSTRACT FROM AUTHOR]