scholarly journals Truncated Vector Lattices

10.29007/mtcs ◽  
2018 ◽  
Author(s):  
Richard Ball
Keyword(s):  
Vector Lattice ◽  
Representation Theorems ◽  
Vector Lattices ◽  
Direct Generalization ◽  
Underlying Space ◽  
Essential Properties ◽  
Sheaf Representation ◽  
Yosida Representation ◽  
The Difference ◽  
Lebesgue Integration

In analysis, truncation is the operation of replacing a nonnegative real-valued function a (x) by its pointwise meet a (x) ∧ 1 with the constant $1$ function. A vector lattice A is said to be closed under truncation if a ∧ 1 ∈ A for all a ∈ A+. Note that A need notcontain 1 itself.Truncation is fundamental to analysis. To give only one example, Lebesgue integration generalizes beautifully to any vector lattice of real-valued functions on a set X, provided the vector lattice is closed under truncation. But vector lattices lacking this property may have integrals which cannot be represented by any measure on X. Nevertheless, when the integral is formulated in a context broader than RX, for example in pointfree analysis, the question oftruncation inevitably arises.What is truncation, or more properly, what are its essential properties? In this paper we answer this question by providing the appropriate axiomatization, and then go on to present several representation theorems. The first is adirect generalization of the classical Yosida representation of an archimedean vector lattice with order unit. The second is a direct generalization of Madden's pointfree representation of archimedean vector lattices. If time permits, we briefly discuss a third sheaf representation which has no direct antecedent in the literature.However, in all three representations the lack of a unit forces a crucial distinction from the corresponding unital representation theorem. The universal object in each case is some sort of family of continuous real-valued functions. The difference is that these functions must vanish at a specified point of the underlying space or locale or sheaf space. With that adjustment, the generalization from units to truncations goes remarkably smoothly.

scholarly journals Minimal vector lattice covers

1971 ◽  
Vol 5 (3) ◽  
pp. 331-335 ◽  
Author(s):  
Roger D. Bleier
Keyword(s):  
Vector Lattice ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Vector Lattices ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Archimedean Lattice

We show that each archimedean lattice-ordered group is contained in a unique (up to isomorphism) minimal archimedean vector lattice. This improves a result of Paul F. Conrad appearing previously in this Bulletin. Moreover, we show that this relationship between archimedean lattice-ordered groups and archimedean vector lattices is functorial.

Separation and Approximation in Topological Vector Lattices

1959 ◽  
Vol 11 ◽  
pp. 286-296 ◽  
Author(s):  
Solomon Leader
Keyword(s):  
Spectral Theory ◽  
Vector Lattice ◽  
Weierstrass Theorem ◽  
Fundamental Lemma ◽  
Vector Lattices ◽  
Integrable Functions ◽  
Nikodym Theorem

Spectral theory in its lattice-theoretic setting proves abstractly that the indicators of measurable sets generate the space L of Lebesgue-integrable functions on an interval. We are concerned here with abstractions suggested by the fact that indicators of intervals suffice to generate L. Our results show that the approximation of arbitrary elements of a topological vector lattice rests upon the ability to separate disjoint elements/ and g by an operation that behaves in the limit like a projection annihilating/ and leaving g invariant.The introduction of this concept of separation together with the notion of limit unit leads (via the Fundamental Lemma) to abstract generalizations of the Radon-Nikodym Theorem (Theorem 1) and the Stone-Weierstrass Theorem (Theorem 3).

Orthomorphisms of archimedean vector lattices

1981 ◽  
Vol 89 (1) ◽  
pp. 119-128 ◽  
Author(s):  
S. J. Bernau
Keyword(s):  
Linear Operator ◽  
Vector Lattice ◽  
Jordan Decomposition ◽  
Duality Results ◽  
Vector Lattices ◽  
Elementary Methods ◽  
Disjointness Preserving ◽  
Order Boundedness

AbstractA linear operator T on a vector lattice L preserves disjointness if Tx ⊥ y whenever x ⊥ y. If such a T is positive it is automatically order bounded. An ortho-morphism is an order bounded disjointness preserving linear operator on L. In this note we show that the theory of orthomorphisms on archimedean vector lattices admits a totally elementary exposition. Elementary methods are also effective in duality considerations when the order dual separates points of L. For the Jordan decomposition T = T+ − T− with T+x = (Tx+)+ − (Tx−)+ we can dtrop the order boundedness assumption if we assume either that T preserves ideals or that L is normed and T is continuous. Alternatively we may keep order boundedness and assume only |Tx| ⊥ |Ty| whenever x ⊥ y. The main duality results show: T preserves ideals if and only if T** does; T is an orthomorphism if and only if T* is; T is central (|T| is bounded by a multiple of the identity) if and only if T* is central if and only if T and T* preserve ideals.

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.

A PROBABILISTIC SEMANTICS FOR COUNTERFACTUALS. PART A

2011 ◽  
Vol 5 (1) ◽  
pp. 26-84 ◽  
Author(s):  
HANNES LEITGEB
Keyword(s):  
Probability Measure ◽  
David Lewis ◽  
Truth Condition ◽  
The Other ◽  
Representation Theorems ◽  
Probabilistic Semantics ◽  
Truth Conditional ◽  
The Difference ◽  
Probabilistic Nature

This is part A of a paper in which we defend a semantics for counterfactuals which is probabilistic in the sense that the truth condition for counterfactuals refers to a probability measure. Because of its probabilistic nature, it allows a counterfactual ‘if A then B’ to be true even in the presence of relevant ‘A and not B’-worlds, as long such exceptions are not too widely spread. The semantics is made precise and studied in different versions which are related to each other by representation theorems. Despite its probabilistic nature, we show that the semantics and the resulting system of logic may be regarded as a naturalistically vindicated variant of David Lewis’ truth-conditional semantics and logic of counterfactuals. At the same time, the semantics overlaps in various ways with the non-truth-conditional suppositional theory for conditionals that derives from Ernest Adams’ work. We argue that counterfactuals have two kinds of pragmatic meanings and come attached with two types of degrees of acceptability or belief, one being suppositional, the other one being truth based as determined by our probabilistic semantics; these degrees could not always coincide due to a new triviality result for counterfactuals, and they should not be identified in the light of their different interpretation and pragmatic purpose. However, for plain assertability the difference between them does not matter. Hence, if the suppositional theory of counterfactuals is formulated with sufficient care, our truth-conditional theory of counterfactuals is consistent with it. The results of our investigation are used to assess a claim considered by Hawthorne and Hájek, that is, the thesis that most ordinary counterfactuals are false.

scholarly journals Repair, resilience and asymmetric segregation of damage in the context of replicative ageing: it is a balancing act

2018 ◽  
Author(s):  
Johannes Borgqvist ◽  
Niek Welkenhuysen ◽  
Marija Cvijovic
Keyword(s):  
Cell Division ◽  
Mammalian Cells ◽  
Asymmetric Cell Division ◽  
Replicative Lifespan ◽  
Damage Repair ◽  
Proposed Model ◽  
Essential Properties ◽  
High Age ◽  
The Difference ◽  
Eukaryotic Organisms

AbstractAccumulation of damaged proteins is a hallmark of ageing, occurring in organisms ranging from bacteria and yeast to mammalian cells. During cell division in Saccharomyces cerevisiae, damaged proteins are retained within the mother cell, resulting in a new daughter cell with full replicative potential and an ageing mother with a reduced replicative lifespan (RLS). The cell-specific features determining the lifespan remain elusive. It has been suggested that the RLS is dependent on the ability of the cell to repair and retain pre-existing damage. To deepen the understanding of how these factors influence the life span of individual cells, we developed and experimentally validated a dynamic model of damage accumulation accounting for replicative ageing. The model includes five essential properties: cell growth, damage formation, damage repair, cell division and cell death, represented in a theoretical framework describing the conditions allowing for replicative ageing, starvation, immortality or clonal senescence. We introduce the resilience to damage, which can be interpreted as the difference in volume between an old and a young cell. We show that the capacity to retain damage deteriorates with high age, that asymmetric division allows for retention of damage, and that there is a trade-off between retention and the resilience property. Finally, we derive the maximal degree of asymmetry as a function of resilience, proposing that asymmetric cell division is beneficial with respect to replicative ageing as it increases the RLS of a given organism. The proposed model contributes to a deeper understanding of the ageing process in eukaryotic organisms.

Free Vector Lattices

1968 ◽  
Vol 20 ◽  
pp. 58-66 ◽  
Author(s):  
Kirby A. Baker
Keyword(s):  
Universal Algebra ◽  
Vector Lattice ◽  
Piecewise Linear ◽  
Continuous Functions ◽  
Real Numbers ◽  
Vector Lattices ◽  
Explicit Characterization

This note presents a useful explicit characterization of the free vector lattice FVL(ℵ) on ℵ generators as a vector lattice of piecewise linear, continuous functions on Rℵ, where ℵ is any cardinal and R is the set of real numbers. A transfinite construction of FVL(ℵ) has been given by Weinberg (14) and simplified by Holland (13, § 5). Weinberg's construction yields the fact that FVL(ℵ) is semi-simple; the present characterization is obtained by combining this fact with a theorem from universal algebra due to Garrett Birkhoff.

A Martingale Convergence Theorem in Vector Lattices

1966 ◽  
Vol 18 ◽  
pp. 424-432 ◽  
Author(s):  
Ralph DeMarr
Keyword(s):  
Convergence Theorem ◽  
Vector Lattice ◽  
Random Variables ◽  
Order Convergence ◽  
Arbitrary Vector ◽  
Martingale Convergence Theorem ◽  
Convergence Almost Everywhere ◽  
Vector Lattices ◽  
Almost Everywhere ◽  
Martingale Convergence

The martingale convergence theorem was first proved by Doob (3) who considered a sequence of real-valued random variables. Since various collections of real-valued random variables can be regarded as vector lattices, it seems of interest to prove the martingale convergence theorem in an arbitrary vector lattice. In doing so we use the concept of order convergence that is related to convergence almost everywhere, the type of convergence used in Doob's theorem.

scholarly journals A note on vector lattices

1967 ◽  
Vol 7 (1) ◽  
pp. 32-38 ◽  
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Vector Lattice ◽  
Vector Lattices ◽  
Image Position

Let E be a vector lattice in the sense of Birkhoff [1]. We use the following notations:

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