Multiplication in Vector Lattices

1968 ◽  
Vol 20 ◽  
pp. 1136-1149 ◽  
Author(s):  
Norman M. Rice
Keyword(s):  
Vector Lattice ◽  
Weak Order ◽  
Order Unit ◽  
Dedekind Complete Vector Lattice ◽  
Vector Lattices ◽  
Complete Vector Lattice ◽  
Complete Vector

B. Z. Vulih has shown (13) how an essentially unique intrinsic multiplication can be defined in a Dedekind complete vector lattice L having a weak order unit. Since this work is available only in Russian, a brief outline is given in § 2 (cf. also the review by E. Hewitt (4), and for details, consult (13) or (11)).

scholarly journals Epicompletetion of archimedean l–groups and vector lattices with weak unit

1990 ◽  
Vol 48 (1) ◽  
pp. 25-56 ◽  
Author(s):  
Richard N. Ball ◽  
Anthony W. Hager
Keyword(s):  
Vector Lattice ◽  
Boolean Algebras ◽  
Weak Order ◽  
Minimal Extension ◽  
Order Unit ◽  
Weak Unit ◽  
Vector Lattices ◽  
Measurable Functions ◽  
Baire Functions

AbstractIn the category W of archimedean l–groups with distinguished weak order unit, with unitpreserving l–homorphism, let B be the class of W-objects of the form D(X), with X basically disconnected, or, what is the same thing (we show), the W-objects of the M/N, where M is a vector lattice of measurable functions and N is an abstract ideal of null functions. In earlier work, we have characterized the epimorphisms in W, and shown that an object G is epicomplete (that is, has no proper epic extension) if and only if G ∈ B. This describes the epicompletetions of a give G (that is, epicomplete objects epically containing G). First, we note that an epicompletion of G is just a “B-completion”, that is, a minimal extension of G by a B–object, that is, by a vector lattice of measurable functions modulo null functions. (C[0, 1] has 2c non-eqivalent such extensions.) Then (we show) the B–completions, or epicompletions, of G are exactly the quotients of the l–group B(Y(G)) of real-valued Baire functions on the Yosida space Y(G) of G, by σ-ideals I for which G embeds naturally in B(Y(G))/I. There is a smallest I, called N(G), and over the embedding G ≦ B(Y(G))/N(G) lifts any homorphism from G to a B–object. (The existence, though not the nature, of such a “reflective” epicompletion was first shown by Madden and Vermeer, using locales, then verified by us using properties of the class B.) There is a unique maximal (not maximum) such I, called M(Y(G)), and B(Y(G))/M(Y(G)) is the unique essentialBcompletion. There is an intermediate σ -ideal, called Z(Y(G)), and the embedding G ≦ B(y(G))/Z(Y(G)) is a σ-embedding, and functorial for σ -homomorphisms. The sistuation stands in strong analogy to the theory in Boolean algebras of free σ -algebras and σ -extensions, though there are crucial differences.

scholarly journals Convergence structures and locally solid topologies on vector lattices of operators

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Yang Deng ◽  
Marcel de Jeu
Keyword(s):  
Vector Lattice ◽  
Strong Operator Topology ◽  
Order Convergence ◽  
Bounded Operators ◽  
Vector Lattices ◽  
Complete Vector Lattice ◽  
Convergence Structure ◽  
General Order ◽  
Complete Vector ◽  
Order Boundedness

AbstractFor vector lattices E and F, where F is Dedekind complete and supplied with a locally solid topology, we introduce the corresponding locally solid absolute strong operator topology on the order bounded operators $${\mathscr{L}}_{\mathrm{ob}}(E,F)$$ L ob ( E , F ) from E into F. Using this, it follows that $${\mathscr{L}}_{\mathrm{ob}}(E,F)$$ L ob ( E , F ) admits a Hausdorff uo-Lebesgue topology whenever F does. For each of order convergence, unbounded order convergence, and—when applicable—convergence in the Hausdorff uo-Lebesgue topology, there are both a uniform and a strong convergence structure on $${\mathscr{L}}_{\mathrm{ob}}(E,F)$$ L ob ( E , F ) . Of the six conceivable inclusions within these three pairs, only one is generally valid. On the orthomorphisms of a Dedekind complete vector lattice, however, five are generally valid, and the sixth is valid for order bounded nets. The latter condition is redundant in the case of sequences of orthomorphisms, as a consequence of a uniform order boundedness principle for orthomorphisms that we establish. We furthermore show that, in contrast to general order bounded operators, orthomorphisms preserve not only order convergence of nets, but unbounded order convergence and—when applicable—convergence in the Hausdorff uo-Lebesgue topology as well.

scholarly journals A Note on Commutative l-Groups

1971 ◽  
Vol 12 (1) ◽  
pp. 69-74 ◽  
Author(s):  
T. P. Speed ◽  
E. Strzelecki
Keyword(s):  
Vector Lattice ◽  
Sufficient Conditions ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Vector Lattices ◽  
Complete Vector Lattice ◽  
Lattice Ideal ◽  
Complete Vector ◽  
Necessary And Sufficient ◽  
Wide Family

Let G be a commutative lattice ordered group. Theorem 1 gives necessary and sufficient conditions under which a⊥ with a∈G is a maximal l-ideal. A wide family of, l-groups G having the property that the orthogonal complement of each atom is a maximal l-ideal is described. Conditionally σ-complete and hence conditionally complete vector lattices belong to the family.It follows immediately that if a is an atom in a conditionally complete vector lattice then a⊥ is a maximal vector lattice ideal. This theorem has been proved in [7] by Yamamuro. Theorem 2 generalizes another result contained in [7]. Namely we prove that if M is a closed maximal l-ideal of an archimedean l-group G then there exists an atom a ∈ G such that M = a⊥.

scholarly journals Продолжение умножения почти f-алгебры

2012 ◽  
Author(s):  
E. Chil
Keyword(s):  
Vector Lattice ◽  
Dedekind Complete Vector Lattice ◽  
Complete Vector Lattice ◽  
Complete Vector

It is proved that an almost f-algebra multiplication and a d-algebra multiplication defined on a majorizing vector sublattice of a Dedekind complete vector lattice can be extended to the whole vector lattice by using purely algebraic and order theoretical means.

scholarly journals Atomic Operators in Vector Lattices

2020 ◽  
Vol 17 (5) ◽  
Author(s):  
Ralph Chill ◽  
Marat Pliev
Keyword(s):  
Nonlinear Operator ◽  
Vector Lattice ◽  
Projection Property ◽  
New Class ◽  
Vector Lattices ◽  
Complete Vector Lattice ◽  
Whole Space ◽  
Complete Vector ◽  
Orthogonally Additive ◽  
Orthogonally Additive Operators

Abstract In this paper, we introduce a new class of operators on vector lattices. We say that a linear or nonlinear operator T from a vector lattice E to a vector lattice F is atomic if there exists a Boolean homomorphism $$\Phi $$ Φ from the Boolean algebra $${\mathfrak {B}}(E)$$ B ( E ) of all order projections on E to $${\mathfrak {B}}(F)$$ B ( F ) such that $$T\pi =\Phi (\pi )T$$ T π = Φ ( π ) T for every order projection $$\pi \in {\mathfrak {B}}(E)$$ π ∈ B ( E ) . We show that the set of all atomic operators defined on a vector lattice E with the principal projection property and taking values in a Dedekind complete vector lattice F is a band in the vector lattice of all regular orthogonally additive operators from E to F. We give the formula for the order projection onto this band, and we obtain an analytic representation for atomic operators between spaces of measurable functions. Finally, we consider the procedure of the extension of an atomic map from a lateral ideal to the whole space.

scholarly journals Orthogonally Biadditive Operators

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Nonna Dzhusoeva ◽  
Ruslan Kulaev ◽  
Marat Pliev
Keyword(s):  
Vector Lattice ◽  
Cartesian Product ◽  
New Class ◽  
Mild Conditions ◽  
Vector Lattices ◽  
Complete Vector Lattice ◽  
Measurable Functions ◽  
Complete Vector ◽  
Orthogonally Additive ◽  
Order Continuous

In this article, we introduce and study a new class of operators defined on a Cartesian product of ideal spaces of measurable functions. We use the general approach of the theory of vector lattices. We say that an operator T : E × F ⟶ W defined on a Cartesian product of vector lattices E and F and taking values in a vector lattice W is orthogonally biadditive if all partial operators T y : E ⟶ W and T x : F ⟶ W are orthogonally additive. In the first part of the article, we prove that, under some mild conditions, a vector space of all regular orthogonally biadditive operators O B A r E , F ; W is a Dedekind complete vector lattice. We show that the set of all horizontally-to-order continuous regular orthogonally biadditive operators is a projection band in O B A r E , F ; W . In the last section of the paper, we investigate orthogonally biadditive operators on a Cartesian product of ideal spaces of measurable functions. We show that an integral Uryson operator which depends on two functional variables is orthogonally biadditive and obtain a criterion of the regularity of an orthogonally biadditive Uryson operator.

scholarly journals Nonsmooth analysis and optimization on partially ordered vector spaces

1992 ◽  
Vol 15 (1) ◽  
pp. 65-81 ◽  
Author(s):  
Thomas W. Reiland
Keyword(s):  
Vector Lattice ◽  
Vector Spaces ◽  
Range Space ◽  
Topological Vector Spaces ◽  
Complete Vector Lattice ◽  
Lipschitz Mappings ◽  
Ordered Vector Spaces ◽  
Partially Ordered Vector Spaces ◽  
Complete Vector ◽  
Locally Convex

Interval-Lipschitz mappings between topological vector spaces are defined and compared with other Lipschitz-type operators. A theory of generalized gradients is presented when both spaces are locally convex and the range space is an order complete vector lattice. Sample applications to the theory of nonsmooth optimization are given.

CONTINUOUS GENERALIZED (θ, ϕ)-SEPARATING DERIVATIONS ON ARCHIMEDEAN ALMOST f-ALGEBRAS

2012 ◽  
Vol 05 (03) ◽  
pp. 1250045 ◽  
Author(s):  
Mohamed Ali Toumi
Keyword(s):  
Vector Lattice ◽  
Continuous Mapping ◽  
Additive Mapping ◽  
Generalized Derivations ◽  
Bilinear Maps ◽  
Complete Vector Lattice ◽  
Positive Mapping ◽  
Ordered Algebra ◽  
Complete Vector ◽  
Archimedean Lattice

Let A be an ℓ-algebra and let θ and ϕ be two endomorphisms of A. The couple (θ, ϕ) is called to be separating if xy = 0 implies θ(x)ϕ(y) = 0. If in addition θ and ϕ are ring endomorphisms of A, then the couple (θ, ϕ) is said to be ring-separating. An additive mapping δ : A → A is called (θ, ϕ)-separating derivation on A if there exists a (θ, ϕ)-separating couple with δ(xy) = δ(x)θ(y) + ϕ(x)δ(y), holds for all x, y ∈ A. If an addition θ, ϕ and δ are continuous, then δ is called a continuous (θ, ϕ)-ring-separating derivation. If in addition the couple (θ, ϕ) is ring-separating then δ is called a continuous (θ, ϕ)-ring-separating derivation. An additive mapping F : A → A is called a continuous generalized (θ, ϕ)-separating derivation on A if F is continuous mapping and if there exists a derivation d : A → A such that θ and ϕ are continuous, (θ, ϕ) is a separating couple and F(xy) = F(x)θ(y) + ϕ(x)d(y), holds for all x, y ∈ A. In this paper, we give a description of continuous (θ, ϕ)-ring-separating derivations on some ℓ-algebras. This generalizes a well-known theorem by Colville, Davis, and Keimel [Positive derivations on f-rings, J. Austral. Math. Soc23 (1977) 371–375] and generalizes the results of Boulabiar in [Positive derivations on almost f-rings, Order19 (2002) 385–395], Ben Amor [On orthosymmetric bilinear maps, Positivity14(1) (2010) 123–130] and Toumi et al. in [Order bounded derivations on Archimedean almost f-algebras, Positivity14(2) (2010) 239–245]. Moreover, inspiring from [Toumi, Order-bounded generalized derivations on Archimedean almost f-algebras, Commun. Algebra38(1) (2010) 154–164], it is shown that the notion of continuous generalized (θ, ϕ)-separating derivation on an archimedean almost f-algebra A is the concept of generalized θ-multiplier, that is an additive mapping satisfying F(xyz) = F(x)θ(yz), for all x, y, z ∈ A. In the case where A is an archimedean f-algebra, the situation improves. Indeed, the collection of all continuous generalized (θ, ϕ)-separating derivation on A coincides with the concept of θ-multiplier, that is an additive mapping satisfying F(xy) = F(x)θ(y), for all x, y ∈ A. If in addition A is a Dedekind complete vector lattice and θ is a positive mapping, then the set of all order bounded generalized of the form (θ, ϕ)-separating derivations on A, under composition, is an archimedean lattice-ordered algebra.

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