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 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.

Unital topology on a unital l-group

2020 ◽  
Vol 70 (5) ◽  
pp. 1189-1196
Author(s):  
Mahmood Pourgholamhossein ◽  
Mohammad Ali Ranjbar
Keyword(s):  
Vector Lattice ◽  
Riesz Space ◽  
Lattice Ordered Group ◽  
Order Unit ◽  
Ordered Group ◽  
Strong Link ◽  
Fundamental Properties ◽  
Essential Properties ◽  
Unit Norm

AbstractIn this paper we investigate some fundamental properties of unital topology on a lattice ordered group with order unit. We show that some essential properties of order unit norm on a vector lattice with order unit, are valid for unital l-groups. For instance we show that for an Archimedean Riesz space G with order unit u, the unital topology and the strong link topology are the same.

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 Abstract integral spaces and minimal extensions

1971 ◽  
Vol 12 (2) ◽  
pp. 166-178 ◽  
Author(s):  
M. J. Maron
Keyword(s):  
Measurable Function ◽  
Linear Functional ◽  
Vector Lattice ◽  
Measure Theory ◽  
Elementary Step ◽  
Summable Function ◽  
Step Functions ◽  
Positive Linear Functional ◽  
Almost Everywhere ◽  
Pointwise Limit

The development of the theory of absolute integrals derives from certain key facts. Among them are:(I) An integral is a positive linear functional on a vector lattice, which is continuous in a certain sense.(II) A function equal almost everywhere to a summable function is itself summable.(III) Every measurable function is the pointwise limit of a sequence of elementary step functions.A device that often plays an important role in measure theory, but which has not beenfully exploited in the theory of abstract integrals is that of(IV) the smallest class containing a given class and having a certain property(such as being a σ-ring of sets). It is our purpose in this paper to examine the theory of abstract real-valued absolute integrals axiomatically, in such a way as to isolate and clarify the roles of (I) through (IV).

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 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. 

scholarly journals Going up and lying over in congruence-modular algebras

2019 ◽  
Vol 69 (2) ◽  
pp. 275-296
Author(s):  
George Georgescu ◽  
Claudia Mureşan
Keyword(s):  
General Setting ◽  
Ring Theory ◽  
Direct Products ◽  
Mv Algebras

Abstract In this paper, we extend properties Going Up and Lying Over from ring theory to the general setting of congruence-modular equational classes, using the notion of prime congruence defined through the commutator. We show how these two properties relate to each other, prove that they are preserved by finite direct products and quotients and provide algebraic and topological characterizations for them. We also point out many kinds of varieties in which these properties always hold, generalizing the results of Belluce on MV-algebras and Rasouli and Davvaz on BL-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.

FPF Ring Theory

1984 ◽  
Author(s):  
Carl Faith ◽  
Stanley Page
Keyword(s):  
Ring Theory
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