Lattice-ordered Rings and Modules

Steinberg, Stuart A.

doi:10.1007/978-1-4419-1721-8

Cited by 71 publications

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“…Now for the l-algebra M n R P with a positive cone P (see [1,6]), we denote P = A ∈ M n K kA ∈ P for some 0 < k ∈ R . Throughout this article, P denotes a general positive cone on M n R P denotes the positive cone on M n K extended from P P A = AM n R + denotes a positive cone on M n R with A ∈ M n R + and P D = DM n K + denotes a positive cone on M n K with D ∈ M n K + (see [3,5]).…”

Section: Lattice-ordered Matrix Algebras M N Rmentioning

confidence: 99%

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Lattice-Ordered Matrix Algebras Over Real GCD-Domains

Bai

Qiu

2013

Communications in Algebra

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Section: Lattice-ordered Matrix Algebras M N Rmentioning

confidence: 99%

“…Obviously, det A ∈ R × . Also it is well-known that for B ∈ M n R + BM n R + is the positive cone of a lattice order on M n R if and only if det B ∈ R × (see [6], p. 595). So we know that M 2 R P A is an l-algebra with positive cone P A = AM 2 R + .…”

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confidence: 99%

Lattice-Ordered Matrix Algebras Over Real GCD-Domains

Bai

Qiu

2013

Communications in Algebra

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“…It is easy to see that for each f ∈ P k , there exists g ∈ P k such that f g ∈ P k , and hence that P k is a minimal -prime -ideal by [5,Theorem 3.2.22]. It is obvious that {P k } is separating, and it is easy to check that each projection π k : L → L/P k , defined by π k (r) = r + P k , preserves infinite sups.…”

Section: The Examplesmentioning

confidence: 99%

Some properties of positive derivations on f-rings II

Redfield

2014

IJAMS

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Theorem 10(b)(i) in the article "Some properties of positive derivations on f -rings" by Henriksen and Smith asserts that if D is a positive derivation on a reduced f -ring and if x ∈ ker D, then {x} ⊥⊥ ⊆ ker D. A counterexample is provided to show that this assertion is false, and correct proofs are given for some results in the paper by Henriksen and Smith that use Theorem 10(b)(i) in their proofs.

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“…At first sight, it might seem that it is easy to establish the following remarkable property of orthosymmetric spaces. However, all proofs that can be found in the literature are quite involved and far from being trivial (see, e.g., Corollary 2 in [7] and Theorem 3.8.14 in [18]). By the way, it should be pointed out that this property is based on the fact that V is Archimedean.…”

Section: Introduction and First Propertiesmentioning

confidence: 99%

Orthosymmetric Archimedean-Valued Vector Lattices

Amor

Boulabiar

Jaber

2019

Trends in Mathematics

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We introduce and study the notion of orthosymmetric spaces over an Archimedean vector lattice as a generalization of finite-dimentional Euclidean inner spaces. A special attention has been paid to linear operators on these spaces. Example 1.1 As usual, R denotes the Archimedean vector lattice of all real numbers. Pick n ∈ N = {1, 2, ...} and suppose that the vector space R n is equipped with its usual structure of Euclidean space. In particular,f, e k g, e k for all f, g ∈ R n ,where (e 1 , ..., e n ) is the canonical (orthogonal) basis of R n . Simultaneously, R n is a vector lattice with respect to the coordinatewise ordering. That is,Therefore, f, g = 0 meaning that the inner product on R n is an R-valued orthosymmetric product. Thus, the Euclidean space R n is an orthosymmetric space over R.Beginning with the next lines, we shall impose the blanket assumption that all orthosymmetric spaces under consideration are taken over the fixed Archimedean vector lattice V (unless otherwise stated explicitly).The following property is useful for later purposes.Lemma 1.2 Let L be an orthosymmetric space. Then,Hence,This is the desired result.Example 1.6 Assume that the Euclidean plan R 2 is endowed with its lexicographic ordering. Hence, R 2 is a non-Archimedean vector lattice. Put f, g = x 1 y 1 for all f = (x 1 , x 2 ) , g = (y 1 , y 2 ) in R 2 .Since R 2 is totally ordered (i.e., linearly ordered), this formula defines an R-valued orthosymmetric product on R 2 . This means that R 2 is a non-Archimedean orthosymmetric space over R.The orthosymmetric space L is said to be definite if its neutral part is trivial. That is to say, L is definite if and only if f, f = 0 in V implies f = 0 in L.Definite orthosymmetric spaces have a better behavior as explained in the following.Proposition 1.7 Any definite orthosymmetric space is Archimedean.

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Lattice-ordered Rings and Modules

Cited by 71 publications

References 0 publications

Lattice-Ordered Matrix Algebras Over Real GCD-Domains

Lattice-Ordered Matrix Algebras Over Real GCD-Domains

Some properties of positive derivations on f-rings II

Orthosymmetric Archimedean-Valued Vector Lattices

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