Generalized free products

J. D. Monk

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Maximal free sequences in a Boolean algebra

J. D. Monk (2011)

Commentationes Mathematicae Universitatis Carolinae

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We study free sequences and related notions on Boolean algebras. A free sequence on a BA $A$ is a sequence $〈 a_{ξ} : ξ < α 〉$ of elements of $A$ , with $α$ an ordinal, such that for all $F, G \in {[α]}^{< ω}$ with $F < G$ we have $\prod_{ξ \in F} a_{ξ} \cdot \prod_{ξ \in G} - a_{ξ} \neq 0$ . A free sequence of length $α$ exists iff the Stone space $Ult (A)$ has a free sequence of length $α$ in the topological sense. A free sequence is maximal iff it cannot be extended at the end to a longer free sequence. The main notions studied here are the spectrum function $𝔣_{sp} (A) = {| α | : A has an infinite maximal free sequence of length α}$ and the associated min-max function $𝔣 (A) = min (𝔣_{sp} (A)) .$ Among...

Ideal independence, free sequences, and the ultrafilter number

Kevin Selker (2015)

Commentationes Mathematicae Universitatis Carolinae

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We make use of a forcing technique for extending Boolean algebras. The same type of forcing was employed in Baumgartner J.E., Komjáth P., Boolean algebras in which every chain and antichain is countable, Fund. Math. 111 (1981), 125–133, Koszmider P., Forcing minimal extensions of Boolean algebras, Trans. Amer. Math. Soc. 351 (1999), no. 8, 3073–3117, and elsewhere. Using and modifying a lemma of Koszmider, and using CH, we obtain an atomless BA, $A$ such that $𝔣 (A) = s_{mm} (A) < 𝔲 (A)$ , answering questions raised...

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...

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

Jan Waszkiewicz (1974)

Colloquium Mathematicae

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

Michael Morley, Robert Soare (1975)

Fundamenta Mathematicae

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On graphs G for which both g and G̅ are claw-free

Shinya Fujita (2005)

Discussiones Mathematicae Graph Theory

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Let G be a graph with |V(G)| ≥ 10. We prove that if both G and G̅ are claw-free, then minΔ(G), Δ(G̅) ≤ 2. As a generalization of this result in the case where |V(G)| is sufficiently large, we also prove that if both G and G̅ are $K_{1, t}$ -free, then minΔ(G),Δ(G̅) ≤ r(t- 1,t)-1 where r(t-1,t) is the Ramsey number.

Multiplicative free square of the free Poisson measure and examples of free symmetrization

Melanie Hinz, Wojciech Młotkowski (2010)

Colloquium Mathematicae

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We compute the moments and free cumulants of the measure $ρ_{t} : = π_{t} ⊠ π_{t}$ , where $π_{t}$ denotes the free Poisson law with parameter t > 0. We also compute free cumulants of the symmetrization of $ρ_{t}$ . Finally, we introduce the free symmetrization of a probability measure on ℝ and provide some examples.

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...

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...

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

А.Г. Кусраев (1998)

Sibirskij matematiceskij zurnal

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

The square-free kernel of $x^{2^{n}} - a^{2^{n}}$

Paulo Ribenboim (2002)

Acta Arithmetica

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$ℵ_{k}$ -free separable groups with prescribed endomorphism ring

Daniel Herden, Héctor Gabriel Salazar Pedroza (2015)

Fundamenta Mathematicae

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We will consider unital rings A with free additive group, and want to construct (in ZFC) for each natural number k a family of $ℵ_{k}$ -free A-modules G which are separable as abelian groups with special decompositions. Recall that an A-module G is $ℵ_{k}$ -free if every subset of size $< ℵ_{k}$ is contained in a free submodule (we will refine this in Definition 3.2); and it is separable as an abelian group if any finite subset of G is contained in a free direct summand of G. Despite the fact that such a...