scholarly journals Sheaf representations and locality of Riesz spaces with order unit

2021 ◽  
Vol 13 ◽  
Author(s):  
Antonio Di Nola ◽  
Giacomo Lenzi ◽  
Luca Spada
Keyword(s):  
Hausdorff Space ◽  
Spectral Space ◽  
Real Interval ◽  
Variety Of Algebras ◽  
Order Unit ◽  
Categorical Equivalence ◽  
Riesz Spaces ◽  
Algebraic Properties ◽  
Sheaf Representations ◽  
Strong Order

We present an algebraic study of Riesz spaces (=real vector lattices) with a (strong) order unit.  We exploit a categorical equivalence between those structures and a variety of algebras called RMV-algebras.  We prove two different sheaf representations for Riesz spaces with order unit: the first represents them as sheaves of linearly ordered Riesz spaces over a spectral space, the second represent them as sheaves of "local" Riesz spaces over a compact Hausdorff space.  Motivated by the latter representation we study the class of local RMV-algebras.  We study the algebraic properties of local RMV-algebra and provide a characterisation of them as special retracts of the real interval [0,1]. Finally, we prove that the category of local RMV-algebras is equivalent to the category of all Riesz spaces. 

Lifting components in clean abelian ℓ-groups

2018 ◽  
Vol 68 (2) ◽  
pp. 299-310
Author(s):  
Karim Boulabiar ◽  
Samir Smiti
Keyword(s):  
Vector Lattice ◽  
General Setting ◽  
Ring Theory ◽  
Natural Analogue ◽  
Order Unit ◽  
Step Functions ◽  
Strong Order

Abstract Let G be an abelian ℓ-group with a strong order unit u > 0. We call G u-clean after Hager, Kimber, and McGovern if every element of G can be written as a sum of a strong order unit of G and a u-component of G. We prove that G is u-clean if and only if u-components of G can be lifted modulo any ℓ-ideal of G. Moreover, we introduce a notion of u-suitable ℓ-groups (as a natural analogue of the corresponding notion in Ring Theory) and we prove that the ℓ-group G is u-clean when and only when it is u-suitable. Also, we show that if E is a vector lattice, then E is u-clean if and only if the space of all u-step functions of E is u-uniformly dense in E. As applications, we will generalize a result by Banaschewski on maximal ℓ-ideals of an archimedean bounded f-algebras to the non-archimedean case. We also extend a result by Miers on polynomially ideal C(X)-type algebras to the more general setting of bounded f-algebras.

scholarly journals On the upper dual Zariski topology

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 483-489
Author(s):  
Seçil Çeken
Keyword(s):  
Prime Ideal ◽  
Compact Space ◽  
Spectral Space ◽  
Zariski Topology ◽  
Algebraic Properties ◽  
Patch Topology ◽  
Totally Disconnected ◽  
Second Spectrum

Let R be a ring with identity and M be a left R-module. The set of all second submodules of M is called the second spectrum of M and denoted by Specs(M). For each prime ideal p of R we define Specsp(M) := {S? Specs(M) : annR(S) = p}. A second submodule Q of M is called an upper second submodule if there exists a prime ideal p of R such that Specs p(M)? 0 and Q = ? S2Specsp(M)S. The set of all upper second submodules of M is called upper second spectrum of M and denoted by u.Specs(M). In this paper, we discuss the relationships between various algebraic properties of M and the topological conditions on u.Specs(M) with the dual Zarsiki topology. Also, we topologize u.Specs(M) with the patch topology and the finer patch topology. We show that for every left R-module M, u.Specs(M) with the finer patch topology is a Hausdorff, totally disconnected space and if M is Artinian then u.Specs(M) is a compact space with the patch and finer patch topology. Finally, by applying Hochster?s characterization of a spectral space, we show that if M is an Artinian left R-module, then u.Specs(M) with the dual Zariski topology is a spectral space.

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.

Hölder categories

2014 ◽  
Vol 64 (3) ◽  
Author(s):  
Anthony Hager ◽  
Jorge Martínez
Keyword(s):  
Order Unit ◽  
Real Numbers ◽  
Initial Element ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Terminal Object ◽  
Epireflective Hull ◽  
Strong Order ◽  
Archimedean Lattice ◽  
Initial Object

AbstractHölder categories are invented to provide an axiomatic foundation for the study of categories of archimedean lattice-ordered algebraic structures. The basis of such a study is Hölder’s Theorem (1908), stating that the archimedean totally ordered groups are precisely the subgroups of the additive real numbers ℝ with the usual addition and ordering, which remains the single most consequential result in the studies of lattice-ordered algebraic systems since Birkhoff and Fuchs to the present.This study originated with interest in W*, the category of all archimedean lattice-ordered groups with a designated strong order unit, and the ℓ-homomorphisms which preserve those units, and, more precisely, with interest in the epireflections on W*. In the course of this study, certain abstract notions jumped to the forefront. Two of these, in particular, seem to have been mostly overlooked; some notion of simplicity appears to be essential to any kind of categorical study of W*, as are the quasi-initial objects in a category. Once these two notions have been brought into the conversation, a Hölder category may then be defined as one which is complete, well powered, and in which(a) the initial object I is simple, and(b) there is a simple quasi-initial coseparator R.In this framework it is shown that the epireflective hull of R is the least monoreflective class. And, when I = R — that is, the initial element is simple and a coseparator — a theorem of Bezhanishvili, Morandi, and Olberding, for bounded archimedean f-algebras with identity, can be be generalized, as follows: for any Hölder category subject to the stipulation that the initial object is a simple coseparator, every uniformly nontrivial reflection — meaning that the reflection of each non-terminal object is non-terminal — is a monoreflection.Also shown here is the fact that the atoms in the class of epireflective classes are the epireflective hulls of the simple quasi-initial objects. From this observation one easily deduces a converse to the result of Bezhanishvili, Morandi, and Olberding: if in a Hölder category every epireflection is a monoreflection, then the initial object is a coseparator.

scholarly journals Operations that preserve integrability, and truncated Riesz spaces

2020 ◽  
Vol 32 (6) ◽  
pp. 1487-1513
Author(s):  
Marco Abbadini
Keyword(s):  
Real Number ◽  
Finite Measure ◽  
Variety Of Algebras ◽  
Riesz Spaces ◽  
Weak Unit ◽  
Measure Spaces ◽  
Concrete Model

AbstractFor any real number {p\in[1,+\infty)}, we characterise the operations {\mathbb{R}^{I}\to\mathbb{R}} that preserve p-integrability, i.e., the operations under which, for every measure μ, the set {\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 {\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, {\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.

DITKIN CONDITIONS

2017 ◽  
Vol 60 (1) ◽  
pp. 153-163
Author(s):  
AZADEH NIKOU ◽  
ANTHONY G. O'FARRELL
Keyword(s):  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Commutative Banach Algebra ◽  
Algebraic Properties ◽  
The Relationship

AbstractThis paper is about the connection between certain Banach-algebraic properties of a commutative Banach algebra E with unit and the associated commutative Banach algebra C(X, E) of all continuous functions from a compact Hausdorff space X into E. The properties concern Ditkin's condition and bounded relative units. We show that these properties are shared by E and C(X, E). We also consider the relationship between these properties in the algebras E, B and $\~{B}$ that appear in the so-called admissible quadruples (X, E, B, $\~{B}$).

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.

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.

Some Notes on Elimination Properties for The Theory of Riesz MV-Chains

2015 ◽  
Vol 65 (4) ◽  
Author(s):  
Enrico Marchioni
Keyword(s):  
Representation Theorem ◽  
Order Theory ◽  
Variety Of Algebras ◽  
Short Article ◽  
First Order ◽  
Riesz Spaces ◽  
First Order Theory ◽  
Skolem Functions ◽  
Strongly Connected ◽  
Mv Algebras

AbstractRiesz MV-algebras are a variety of algebras strongly connected to Riesz spaces. In this short article we investigate some elimination properties of the first-order theory RMV of linearly ordered Riesz MV-algebras and show that RMV admits elimination of quantifiers and uniform elimination of imaginary elements. In the process, we also prove several other results such as modelcompleteness, o-minimality, definability of Skolem functions, and a version of the Di Nola Representation Theorem for Riesz MV-algebras.

New torsion theory in unital abelian ℓ-groups

2011 ◽  
Vol 61 (3) ◽  
Author(s):  
Jorge Martínez
Keyword(s):  
Torsion Theory ◽  
Torsion Class ◽  
Finite Range ◽  
Order Unit ◽  
Concrete Category ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Abelian Lattice ◽  
Strong Order

AbstractThis paper introduces the notion of a functorial torsion class (FTC): in a concrete category $\mathfrak{C}$ which has image factorization, one considers monocoreflective subcategories which are closed under formation of subobjects.Here the interest is in FTCs in the category of abelian lattice-ordered groups with designated strong order unit. The FTCs $\mathfrak{T}$ consisting of archimedean latticeordered groups are characterized: for each subgroup A of the rationals with the identity 1, either $\mathfrak{T} = \mathfrak{S}\left( A \right)$, the class of all lattice-ordered groups of functions on a set X which have finite range in A, or $$\mathfrak{T} = \mathbb{T}\left( A \right)$$, the class of all subgroups of A with 1.As for FTCs possessing non-archimedean groups, it is shown that if $\mathfrak{T}$ is an FTC containing a subgroup A of the reals with 1, of rank two or greater, then $\mathfrak{T}$ contains all ℓ-groups of the form $A\vec \times G$, for all abelian lattice-ordered groups G. Finally, the least FTC that contains a non-archimedean group is the class of all $\mathbb{Z}\vec \times G$, for all abelian lattice-ordered groups G.

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