Order closed ideals in pre-Riesz spaces and their relationship to bands

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.     

A closed graph theorem for order bounded operators

The closed graph theorem is one of the cornerstones of linear functional analysis in Fréchet spaces, and the extension of this result to more general topological vector spaces is a di?cult problem comprising a great deal of technical difficulty. However, the theory of convergence vector spaces provides a natural framework for closed graph theorems. In this paper we use techniques from convergence vector space theory to prove a version of the closed graph theorem for order bounded operators on Archimedean vector lattices. This illustrates the usefulness of convergence spaces in dealing with problems in vector lattice theory, problems that may fail to be amenable to the usual Hausdorff-Kuratowski-Bourbaki concept of topology.     

Tensor products of Archimedean partially ordered vector spaces

We study the tensor product of two directed Archimedean partially ordered vector spaces X and Y by means of Riesz completions. With the aid of the Fremlin tensor product of the Riesz completions of X and Y we show that the projective cone in X ? Y is contained in an Archimedean cone. The smallest Archimedean cone containing the projective cone satisfies an appropriate universal mapping property.     

Laterally closed lattice homomorphisms

Let A and B be two Archimedean vector lattices and let be a lattice homomorphism. We call that T is laterally closed if T(D) is a maximal orthogonal system in the band generated by T(A) in B, for each maximal orthogonal system D of A. In this paper we prove that any laterally closed lattice homomorphism T of an Archimedean vector lattice A with universal completion Au into a universally complete vector lattice B can be extended to a lattice homomorphism of Au into B, which is an improvement of a result of M. Duhoux and M. Meyer [M. Duhoux and M. Meyer, Extended orthomorphisms and lateral completion of Archimedean Riesz spaces, Ann. Soc. Sci. Bruxelles 98 (1984) 3-18], who established it for the order continuous lattice homomorphism case. Moreover, if in addition Au and B are with point separating order duals ^′(Au) and B^′ respectively, then the laterally closedness property becomes a necessary and sufficient condition for any lattice homomorphism to have a similar extension to the whole Au. As an application, we give a new representation theorem for laterally closed d-algebras from which we infer the existence of d-algebra multiplications on the universal completions of d-algebras.     

Order bounded orthosymmetric bilinear operator

It is proved by an order theoretical and purely algebraic method that any order bounded orthosymmetric bilinear operator b: E×E → F where E and F are Archimedean vector lattices is symmetric. This leads to a new and short proof of the commutativity of Archimedean almost f-algebras.     

Jensen type inequalities for positive bilinear operators

A general Jensen type inequality for positive bilinear operators between uniformly complete vector lattice is proved. Then some new inequalities for linear and bilinear operators and an interpolation result for positive bilinear operators between Calderón–Lozanovskiĭ spaces are deduced. The proof of the main result relies upon homogeneous functional calculus on vector lattices and the Fremlin tensor product of Archimedean vector lattices.     

Ideals and bands in pre-Riesz spaces

In a vector lattice, ideals and bands are well-investigated subjects. We study similar notions in a pre-Riesz space. The pre-Riesz spaces are exactly the order dense linear subspaces of vector lattices. Restriction and extension properties of ideals, solvex ideals and bands are investigated. Since every Archimedean directed partially ordered vector space is pre-Riesz, we establish properties of ideals and bands in such spaces.      

The Archimedean ℓ-group tensor product

We introduce a construction (inZ F-set theory) for the Archimedean -group tensor product. We relate this tensor product to the existing ones in the theory of Archimedean vector lattices and -groups.     

The Loomis–Sikorski Theorem revisited

We prove, constructively, that the Loomis–Sikorski Theorem for σ-complete Boolean algebras follows from a representation theorem for Archimedean vector lattices and a constructive representation of Boolean algebras as spaces of Carathéodory place functions. We also prove a constructive subdirect product representation theorem for arbitrary partially ordered vector spaces. Received August 10, 2006; accepted in final form May 30, 2007.     

Whales and the universal completion

In this paper we introduce whales in the collection of subsets of the Boolean algebra of bands in a Dedekind complete Riesz space and employ them to give a short (and constructive) proof of the existence of universal completions for Archimedean Riesz spaces.

     

MacNeille completion of centers and centers of MacNeille completions of lattice effect algebras: Generic scheme behind

We present an example of a natural class of atomic Archimedean sharply dominating lattice effect algebras with non-bifull and atomic centers. Further, we extend and clarify known results about MacNeille completion of centers and centers of MacNeille completions of lattice effect algebras.     

MacNeille completions of FL-algebras

We show that a large number of equations are preserved by Dedekind-MacNeille completions when applied to subdirectly irreducible FL-algebras/residuated lattices. These equations are identified in a systematic way, based on proof-theoretic ideas and techniques in substructural logics. It follows that many varieties of Heyting algebras and FL-algebras admit completions.     

On convergences of measurable functions in Archimedean vector lattices

For Archimedean vector lattices X, Y and the positive cone \mathbbL{\mathbb{L}} of all regular linear operators L : X → Y, a theory of sequential convergences of functions connected with an \mathbbL{\mathbb{L}} -valued measure is introduced and investigated.     

Standard completions for quasiordered sets

A standard completion for a quasiordered set Q is a closure system whose point closures are the principal ideals of Q. We characterize the following types of standard completions by means of their closure operators:

1. V-distributive completions,
2. Completely distributive completions,
3. A-completions (i.e. standard completions which are completely distributive algebraic lattices),
4. Boolean completions.

Moreover, completely distributive completions are described by certain idempotent relations, and the A-completions are shown to be in one-to-one correspondence with the join-dense subsets of Q. If a pseudocomplemented meet-semilattice Q has a Boolean completion ?, then Q must be a Boolean lattice and ? its MacNeille completion.     

Algebraic Order Bounded Disjointness Preserving Operators and Strongly Diagonal Operators

Let T be an order bounded disjointness preserving operator on an Archimedean vector lattice. The main result in this paper shows that T is algebraic if and only if there exist natural numbers m and n such that n ≥ m, and T^n!, when restricted to the vector sublattice generated by the range of T^m, is an algebraic orthomorphism. Moreover, n (respectively, m) can be chosen as the degree (respectively, the multiplicity of 0 as a root) of the minimal polynomial of T. In the process of proving this result, we define strongly diagonal operators and study algebraic order bounded disjointness preserving operators and locally algebraic orthomorphisms. In addition, we introduce a type of completeness on Archimedean vector lattices that is necessary and sufficient for locally algebraic orthomorphisms to coincide with algebraic orthomorphisms.     

Order completions of Archimedean Riesz spaces andl-groups

It is well-known that every Archimedean Riesz space (vector lattice) can be embedded in a certain minimal Dedekind complete Riesz space (itsDedekind completion) and that this space is essentially unique. There are other nice properties that a Riesz space can enjoy besides Dedekind completeness; for example, the projection property, the principal projection property,-Dedekind completeness, and (relative) uniform completeness. It is shown that every Archimedean Riesz space has an essentially unique completion with respect to each of these properties. These completions can be viewed as universal objects in appropriate categories. As such, their uniqueness is obvious (universal objects are always unique), and their existence can be demonstrated very simply by working within the Dedekind completion. This approach is free of clutter since all it needs is theexistence of the Dedekind completion, and not its particular form (which can be quite complicated). By using the same methods within the universal completion, we can isolate further order completions; in a sense, every possible order completion can be obtained in this way, since the universal completion is the largest Riesz space in which the original space is order dense. As an added bonus, all of our results apply equally well to Archimedeanl-groups.Presented by L. Fuchs.     

MacNeille completions of lattice expansions

There are two natural ways to extend an arbitrary map between (the carriers of) two lattices, to a map between their MacNeille completions. In this paper we investigate which properties of lattice maps are preserved under these constructions, and for which kind of maps the two extensions coincide. Our perspective involves a number of topologies on lattice completions, including the Scott topologies and topologies that are induced by the original lattice. We provide a characterization of the MacNeille completion in terms of these induced topologies. We then turn to expansions of lattices with additional operations, and address the question of which equational properties of such lattice expansions are preserved under various types of MacNeille completions that can be defined for these algebras. For a number of cases, including modal algebras and residuated (ortho)lattice expansions, we provide reasonably sharp sufficient conditions on the syntactic shape of equations that guarantee preservation. Generally, our results show that the more residuation properties the primitive operations satisfy, the more equations are preserved. Received August 21, 2005; accepted in final form October 17, 2006.     

On self-majorizing elements in Archimedean vector lattices

A finite element in an Archimedean vector lattice is called self-majorizing if its modulus is a majorant. Such elements exist in many vector lattices and naturally occur in different contexts. They are also known as semi-order units as the modulus of a self-majorizing element is an order unit in the band generated by the element. In this paper the properties of self-majorizing elements are studied systematically, and the relations between the sets of finite, totally finite and self-majorizing elements of a vector lattice are provided. In a Banach lattice an element \(\varphi \) is self-majorizing , if and only if the ideal and the band both generated by \(\varphi \) coincide.     

Riesz–Kantorovich formulas for operators on multi-wedged spaces

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.     

Precoherent quantale completions of partially ordered semigroups

Just as complete lattices can be viewed as the completions of posets, quantales can also be treated as the completions of partially ordered semigroups. Motivated by the study on the well-known Frink completions of posets, it is natural to consider the “Frink” completions for the case of partially ordered semigroups. For this purpose, we firstly introduce the notion of precoherent quantale completions of partially ordered semigroups, and construct the concrete forms of three types of precoherent quantale completions of a partially ordered semigroup. Moreover, we obtain a sufficient and necessary condition of the Frink completion on a partially ordered semigroup being a precoherent quantale completion. Finally, we investigate the injectivity in the category $$\mathbf {APoSgr}_{\le }$$ of algebraic partially ordered semigroups and their submultiplicative directed-supremum-preserving maps, and show that the $$\mathscr {E}_{\le }$$-injective objects of algebraic partially ordered semigroups are precisely the precoherent quantales, here $$\mathscr {E}_{\le }$$ denote the class of morphisms $$h:A\longrightarrow B$$ that preserve the compact elements and satisfy that $$h(a_1)\cdots h(a_n)\le h(b)$$ always implies $$a_1\cdots a_n\le b$$.