scholarly journals The Unbounded Fuzzy Order Convergence in Fuzzy Riesz Spaces

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 971 ◽  
Author(s):  
Mobashir Iqbal ◽  
M. G. Abbas Malik ◽  
Yasir Bashir ◽  
Zia Bashir
Keyword(s):  
Weak Order ◽  
Order Convergence ◽  
Order Unit ◽  
Theoretical Concepts ◽  
Riesz Spaces ◽  
Fuzzy Order

The fuzzy order convergence in fuzzy Riesz spaces is defined only for fuzzy order bounded nets. The aim of this paper is to define and study unbounded fuzzy order convergence and some of its applications. Furthermore, some theoretical concepts like the fuzzy weak order unit and fuzzy ideals are studied in relation to unbounded fuzzy order convergence.

scholarly journals On the algebraic dimension of Riesz spaces

2021 ◽  
Vol 56 (1) ◽  
pp. 67-71
Author(s):  
N. M. Baziv ◽  
O. B. Hrybel
Keyword(s):  
Riesz Space ◽  
Weak Order ◽  
Order Unit ◽  
Projection Property ◽  
Linear Spaces ◽  
Infinite Dimensional ◽  
Algebraic Dimension ◽  
Riesz Spaces

We prove that the algebraic dimension of an infinite dimensional $C$-$\sigma$-complete Riesz space (in particular, of a Dedekind $\sigma$-complete and a laterally $\sigma$-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.

Topological Completeness of Order Intervals in Riesz Spaces

1984 ◽  
Vol 27 (3) ◽  
pp. 316-323
Author(s):  
P. G. Dodds
Keyword(s):  
Riesz Space ◽  
Weak Order ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Order Unit ◽  
Riesz Spaces ◽  
Necessary And Sufficient ◽  
Order Intervals

AbstractIt is shown that if L is a Dedekind complete Riesz space equipped with a locally solid topology T defined by strongly (A, 0) Riesz pseudonorms, then order intervals of L are T-complete. This is an extension of a well known theorem of Nakano. The second part of the paper gives a necessary and sufficient condition for topological completeness of order intervals in a Dedekind σ-complete Riesz space which has a weak order unit and which is equipped with a locally solid σ-Fatou topology.

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 Ideal structure and factorization properties of the regular kernel operators

Positivity ◽  
2019 ◽  
Vol 24 (5) ◽  
pp. 1211-1229
Author(s):  
A. Blanco
Keyword(s):  
Banach Lattice ◽  
Weak Order ◽  
Ideal Structure ◽  
Order Unit ◽  
Factorization Property ◽  
Kernel Operator ◽  
Regular Kernel ◽  
And Algebra ◽  
Kernel Operators ◽  
The Ideal

AbstractWe consider the structure of the lattice of (order and algebra) ideals of the band of regular kernel operators on $$L^p$$ L p -spaces. We show, in particular, that for any $$L^p(\mu )$$ L p ( μ ) space, with $$\mu $$ μ $$\sigma $$ σ -finite and $$1<p<\infty $$ 1 < p < ∞ , the norm-closure of the ideal of finite-rank operators on $$L^p(\mu )$$ L p ( μ ) , is the only non-trivial proper closed (order and algebra) ideal of this band. Key to our results in the $$L^p$$ L p setting is the fact that every regular kernel operator on an $$L^p(\mu )$$ L p ( μ ) space ($$\mu $$ μ and p as before) factors with regular factors through $$\ell _p$$ ℓ p . We show that a similar but weaker factorization property, where $$\ell _p$$ ℓ p is replaced by some reflexive purely atomic Banach lattice, characterizes the regular kernel operators from a reflexive Banach lattice with weak order unit to a KB-space with weak order unit.

Essential completeness of archimedean ℓ-groups with weak order unit

2013 ◽  
Vol 217 (5) ◽  
pp. 915-926 ◽  
Author(s):  
B. Banaschewski ◽  
A.W. Hager
Keyword(s):  
Weak Order ◽  
Order Unit

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 Certain extensions and factorizations of α-complete homomorphisms in archimedean lattice-ordered groups

1997 ◽  
Vol 62 (2) ◽  
pp. 239-258 ◽  
Author(s):  
Anthony W. Hager ◽  
Ann Kizanis
Keyword(s):  
Full Subcategory ◽  
Cardinal Number ◽  
Weak Order ◽  
Infinite Cardinal ◽  
Order Unit ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Epireflective Subcategory ◽  
Archimedean Lattice

AbstractAs a consequence of general principles, we add to the array of ‘hulls’ in the category Arch (of archimedean ℓ-groups with ℓ-homomorphisms) and in its non-full subcategory W (whose objects have distinguished weak order unit, whose morphisms preserve the unit). The following discussion refers to either Arch or W. Let α be an infinite cardinal number or ∞, let Homα; denote the class of α-complete homomorphisms, and let R be a full epireflective subcategory with reflections denoted rG: G → rG. Then for each G, there is rαG ∈ Homα (G, R) such that for each ϕ ∈ Homα (G, R), there is unique with . Moreover if every rG is an essential embedding, then, for every α and every G, rαG = rG, and every Homα. If and R consists of all epicomplete objects, then every Homw1. For α = ∞, and for any R, every Hom∞.

scholarly journals Order-Convergence for Filters in a Dedekind Complete Riesz Spaces( K-Spaces)

2019 ◽  
Vol 1294 ◽  
pp. 032002
Author(s):  
Shatha Abdul-Hussein Kadhum ◽  
Shaimaa Abdul-Hussein Kadhum
Keyword(s):  
Order Convergence ◽  
Riesz Spaces
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