Some Characterizations of The Projection Property in Archimedean Riesz Spaces

1972 ◽  
Vol 24 (2) ◽  
pp. 306-311 ◽  
Author(s):  
K. K. Kutty ◽  
J. Quinn
Keyword(s):  
Band Structure ◽  
Prime Ideal ◽  
Vector Lattice ◽  
Strong Relationship ◽  
Ideal Structure ◽  
Projection Property ◽  
Dedekind Completion ◽  
Riesz Spaces

In this paper we give some new characterizations of the projection property in Archimedean Riesz spaces. Our approach primarily explores the interrelationships between such things as the band structure or the prime ideal structure of an Archimedean vector lattice and corresponding structures of its Dedekind completion. Our results show that, in general, there is a ‘strong“ relationship if and only if the original vector lattice has the projection property. The main result of this paper is Theorem 2.6 which both summarizes and extends all of the results we obtain prior to it.

scholarly journals On the prime ideal structure of symbolic Rees algebras

2012 ◽  
Vol 4 (3) ◽  
pp. 327-343
Author(s):  
S. Bouchiba ◽  
S. Kabbaj
Keyword(s):  
Prime Ideal ◽  
Ideal Structure ◽  
Rees Algebras

scholarly journals The prime ideal structure of the Minkowski ring of polytopes

1992 ◽  
Vol 78 (3) ◽  
pp. 239-251 ◽  
Author(s):  
Klaus G. Fischer ◽  
Jay Shapiro
Keyword(s):  
Prime Ideal ◽  
Ideal Structure

Relatively uniform convergences in archimedean lattice ordered groups

2010 ◽  
Vol 60 (4) ◽  
Author(s):  
Ján Jakubík ◽  
Štefan Černák
Keyword(s):  
Uniform Convergence ◽  
Vector Lattice ◽  
Radical Class ◽  
Ordered Group ◽  
Dedekind Completion ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Divisible Hull ◽  
Archimedean Lattice ◽  
Relatively Uniform Convergence

AbstractFor an archimedean lattice ordered group G let G d and G∧ be the divisible hull or the Dedekind completion of G, respectively. Put G d∧ = X. Then X is a vector lattice. In the present paper we deal with the relations between the relatively uniform convergence on X and the relatively uniform convergence on G. We also consider the relations between the o-convergence and the relatively uniform convergence on G. For any nonempty class τ of lattice ordered groups we introduce the notion of τ-radical class; we apply this notion by investigating relative uniform convergences.

scholarly journals Vector lattice covers of ideals and bands in Pre-Riesz spaces

2018 ◽  
Vol 42 (7) ◽  
pp. 919-937 ◽  
Author(s):  
Anke Kalauch ◽  
Helena Malinowski
Keyword(s):  
Vector Lattice ◽  
Riesz Spaces

The tensor product of Archimedean ordered vector spaces

1988 ◽  
Vol 104 (2) ◽  
pp. 331-345 ◽  
Author(s):  
J. J. Grobler ◽  
C. C. A. Labuschagne
Keyword(s):  
Tensor Product ◽  
Ordered Set ◽  
Riesz Space ◽  
Vector Spaces ◽  
Projection Property ◽  
Simple Method ◽  
Riesz Spaces ◽  
Space Tensor ◽  
Ordered Vector Spaces ◽  
Representation Techniques

A Riesz space tensor product of Archimedean Riesz spaces was introduced by D. H. Fremlin[2, 3]. His construction as well as a subsequent simplified version by H. H. Schaefer[10] depended on representation techniques and it is our aim to find a more direct way to prove the existence of the tensor product and to derive its properties. This tensor product proved to be extremely useful in the theory of positive operators on Banach lattices (see [3] and [10]) and should be considered as one of the basic constructions in the theory of Riesz spaces. It is therefore of interest to construct it in an intrinsic way. The problem to do this was already posed by Fremlin in [2]. In this paper we shall present two different approaches, the first of which is analogous to the formation of a free lattice generated by a given partially ordered set. (See [5], p. 41.) In the second one we first assume the Riesz spaces involved to have the principal projection property. In this case a simple method of construction by step-elements is available and the tensor product of arbitrary Archimedean Riesz spaces can then be obtained by embedding the spaces into their Dedekind completions. To complete the latter step we need results on the extension of Riesz bimorphisms which will be proved in §1. Both our approaches hinge on results about the tensor product of ordered vector spaces. It turns out that a unique tensor product for ordered vector spaces exists and is contained in the Riesz space tensor product. This is investigated in §2.

scholarly journals The Riesz Hull of a Semisimple MV-Algebra

2015 ◽  
Vol 65 (4) ◽  
Author(s):  
D. Diaconescu ◽  
I. Leuștean
Keyword(s):  
Vector Lattice ◽  
Strong Unit ◽  
Riesz Spaces ◽  
Vector Lattices ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Mv Algebra ◽  
Standard Construction ◽  
Abelian Lattice ◽  
Mv Algebras

AbstractMV-algebras and Riesz MV-algebras are categorically equivalent to abelian lattice-ordered groups with strong unit and, respectively, with Riesz spaces (vector-lattices) with strong unit. A standard construction in the literature of lattice-ordered groups is the vector-lattice hull of an archimedean latticeordered group. Following a similar approach, in this paper we define the Riesz hull of a semisimple MV-algebra.

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.

scholarly journals On the prime ideal structure of tensor products of algebras

2002 ◽  
Vol 176 (2-3) ◽  
pp. 89-112 ◽  
Author(s):  
Samir Bouchiba ◽  
David E. Dobbs ◽  
Salah-Eddine Kabbaj
Keyword(s):  
Prime Ideal ◽  
Tensor Products ◽  
Ideal Structure
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