On BPI Restricted to Boolean Algebras of Size Continuum

Eric Hall; Kyriakos Keremedis

Displaying similar documents to “On BPI Restricted to Boolean Algebras of Size Continuum”

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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Tychonoff Products of Two-Element Sets and Some Weakenings of the Boolean Prime Ideal Theorem

Kyriakos Keremedis (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let X be an infinite set, and (X) the Boolean algebra of subsets of X. We consider the following statements: BPI(X): Every proper filter of (X) can be extended to an ultrafilter. UF(X): (X) has a free ultrafilter. We will show in ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice) that the following four statements are equivalent: (i) BPI(ω). (ii) The Tychonoff product $2^{ℝ}$ , where 2 is the discrete space 0,1, is compact. (iii) The Tychonoff product ${[0, 1]}^{ℝ}$ is compact. (iv) In a...

Remarks on the Stone Spaces of the Integers and the Reals without AC

Horst Herrlich, Kyriakos Keremedis, Eleftherios Tachtsis (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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In ZF, i.e., the Zermelo-Fraenkel set theory minus the Axiom of Choice AC, we investigate the relationship between the Tychonoff product $2^{(X)}$ , where 2 is 2 = 0,1 with the discrete topology, and the Stone space S(X) of the Boolean algebra of all subsets of X, where X = ω,ℝ. We also study the possible placement of well-known topological statements which concern the cited spaces in the hierarchy of weak choice principles.

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...

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 $⌈ \frac{ℓ}{n} ⌉ < ⌈ \frac{k}{m - 1} ⌉$ .

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...

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...

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...

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...

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,...

Symmetrization of probability measures, pushforward of order 2 and the Boolean convolution

Wojciech Młotkowski, Noriyoshi Sakuma (2011)

Banach Center Publications

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We study relations between the Boolean convolution and the symmetrization and the pushforward of order 2. In particular we prove that if μ₁,μ₂ are probability measures on [0,∞) then ${(μ ₁ ⨄ μ ₂)}^{s} = μ ₁^{s} ⨄ μ ₂^{s}$ and if ν₁,ν₂ are symmetric then ${(ν ₁ ⨄ ν ₂)}^{(2)} = ν ₁^{(2)} ⨄ ν ₂^{(2)}$ . Finally we investigate necessary and sufficient conditions under which the latter equality holds.

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...