On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras

Walter Schachermayer

Displaying similar documents to “On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras”

Boolean algebras, splitting theorems, and $Δ_{2}^{0}$ sets

Michael Morley, Robert Soare (1975)

Fundamenta Mathematicae

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$\forall_{n}$ -theories of Boolean algebras

Jan Waszkiewicz (1974)

Colloquium Mathematicae

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On BPI Restricted to Boolean Algebras of Size Continuum

Eric Hall, Kyriakos Keremedis (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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(i) The statement P(ω) = “every partition of ℝ has size ≤ |ℝ|” is equivalent to the proposition R(ω) = “for every subspace Y of the Tychonoff product $2^{(ω)}$ the restriction |Y = Y ∩ B: B ∈ of the standard clopen base of $2^{(ω)}$ to Y has size ≤ |(ω)|”. (ii) In ZF, P(ω) does not imply “every partition of (ω) has a choice set”. (iii) Under P(ω) the following two statements are equivalent: (a) For every Boolean algebra of size ≤ |ℝ| every filter can be extended to an ultrafilter. (b) Every Boolean...

Generalised irredundance in graphs: Nordhaus-Gaddum bounds

Ernest J. Cockayne, Stephen Finbow (2004)

Discussiones Mathematicae Graph Theory

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For each vertex s of the vertex subset S of a simple graph G, we define Boolean variables p = p(s,S), q = q(s,S) and r = r(s,S) which measure existence of three kinds of S-private neighbours (S-pns) of s. A 3-variable Boolean function f = f(p,q,r) may be considered as a compound existence property of S-pns. The subset S is called an f-set of G if f = 1 for all s ∈ S and the class of f-sets of G is denoted by $Ω_{f} (G)$ . Only 64 Boolean functions f can produce different classes $Ω_{f} (G)$ , special cases...

On $K$ -Boolean Rings

W. B. Vasantha Kandasamy (1992)

Publications du Département de mathématiques (Lyon)

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Orthomodular lattices that are horizontal sums of Boolean algebras

Ivan Chajda, Helmut Länger (2020)

Commentationes Mathematicae Universitatis Carolinae

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The paper deals with orthomodular lattices which are so-called horizontal sums of Boolean algebras. It is elementary that every such orthomodular lattice is simple and its blocks are just these Boolean algebras. Hence, the commutativity relation plays a key role and enables us to classify these orthomodular lattices. Moreover, this relation is closely related to the binary commutator which is a term function. Using the class $ℋ$ of horizontal sums of Boolean algebras, we establish an identity...

The lattice of subvarieties of the biregularization of the variety of Boolean algebras

Jerzy Płonka (2001)

Discussiones Mathematicae - General Algebra and Applications

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Let τ: F → N be a type of algebras, where F is a set of fundamental operation symbols and N is the set of all positive integers. An identity φ ≈ ψ is called biregular if it has the same variables in each of it sides and it has the same fundamental operation symbols in each of it sides. For a variety V of type τ we denote by $V_{b}$ the biregularization of V, i.e. the variety of type τ defined by all biregular identities from Id(V). Let B be the variety of Boolean algebras of type $τ_{b} : +, \cdot, ´ \to N$ , where...

Laslett’s transform for the Boolean model in $ℝ^{d}$

Rostislav Černý (2006)

Kybernetika

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Consider a stationary Boolean model $X$ with convex grains in $ℝ^{d}$ and let any exposed lower tangent point of $X$ be shifted towards the hyperplane $N_{0} = {x \in ℝ^{d} : x_{1} = 0}$ by the length of the part of the segment between the point and its projection onto the $N_{0}$ covered by $X$ . The resulting point process in the halfspace (the Laslett’s transform of $X$ ) is known to be stationary Poisson and of the same intensity as the original Boolean model. This result was first formulated for the planar Boolean model (see N. Cressie...

Partitions of $k$ -branching trees and the reaping number of Boolean algebras

Claude Laflamme (1993)

Commentationes Mathematicae Universitatis Carolinae

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The reaping number $𝔯_{m, n} (𝔹)$ of a Boolean algebra $𝔹$ is defined as the minimum size of a subset $𝒜 \subseteq 𝔹 ∖ {𝐎}$ such that for each $m$ -partition $𝒫$ of unity, some member of $𝒜$ meets less than $n$ elements of $𝒫$ . We show that for each $𝔹$ , $𝔯_{m, n} (𝔹) = 𝔯_{⌈ \frac{m}{n - 1} ⌉, 2} (𝔹)$ as conjectured by Dow, Steprāns and Watson. The proof relies on a partition theorem for finite trees; namely that every $k$ -branching tree whose maximal nodes are coloured with $ℓ$ colours contains an $m$ -branching subtree using at most $n$ colours if and only if $⌈ \frac{ℓ}{n} ⌉ < ⌈ \frac{k}{m - 1} ⌉$ .

Minimal generics from subvarieties of the clone extension of the variety of Boolean algebras

Jerzy Płonka (2008)

Colloquium Mathematicae

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Let τ be a type of algebras without nullary fundamental operation symbols. We call an identity φ ≈ ψ of type τ clone compatible if φ and ψ are the same variable or the sets of fundamental operation symbols in φ and ψ are nonempty and identical. For a variety of type τ we denote by $^{c}$ the variety of type τ defined by all clone compatible identities from Id(). We call $^{c}$ the clone extension of . In this paper we describe algebras and minimal generics of all subvarieties of $^{c}$ , where is the...

FKN Theorem on the biased cube

Piotr Nayar (2014)

Colloquium Mathematicae

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We consider Boolean functions defined on the discrete cube $- γ, γ^{- 1} ⁿ$ equipped with a product probability measure $μ^{\otimes n}$ , where $μ = β δ_{- γ} + α δ_{γ^{- 1}}$ and γ = √(α/β). This normalization ensures that the coordinate functions ${(x_{i})}_{i = 1, . . ., n}$ are orthonormal in $L ₂ (- γ, γ^{- 1} ⁿ, μ^{\otimes n})$ . We prove that if the spectrum of a Boolean function is concentrated on the first two Fourier levels, then the function is close to a certain function of one variable. Our theorem strengthens the non-symmetric FKN Theorem due to Jendrej, Oleszkiewicz and Wojtaszczyk. Moreover,...

Iterated Boolean random varieties and application to fracture statistics models

Dominique Jeulin (2016)

Applications of Mathematics

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Models of random sets and of point processes are introduced to simulate some specific clustering of points, namely on random lines in $ℝ^{2}$ and $ℝ^{3}$ and on random planes in $ℝ^{3}$ . The corresponding point processes are special cases of Cox processes. The generating distribution function of the probability distribution of the number of points in a convex set $K$ and the Choquet capacity $T (K)$ are given. A possible application is to model point defects in materials with some degree of alignment. Theoretical...

О структуре $A J W$ -алгебр типа $I_{2}$

А.Г. Кусраев (1998)

Sibirskij matematiceskij zurnal

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