# Cellularity of free products of Boolean algebras (or topologies)

Fundamenta Mathematicae (2000)

• Volume: 166, Issue: 1-2, page 153-208
• ISSN: 0016-2736

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## Abstract

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The aim this paper is to present an answer to Problem 1 of Monk [10], [11]. We do this by proving in particular that if μ is a strong limit singular cardinal, $\theta ={\left({2}^{cf\left(\mu \right)}\right)}^{+}$ and ${2}^{\mu }={\mu }^{+}$ then there are Boolean algebras ${𝔹}_{1},{𝔹}_{2}$ such that $c\left({𝔹}_{1}\right)=\mu ,c\left({𝔹}_{2}\right)<\theta butc\left({𝔹}_{1}*{𝔹}_{2}\right)={\mu }^{+}$. Further we improve this result, deal with the method and the necessity of the assumptions. In particular we prove that if $𝔹$ is a ccc Boolean algebra and ${\mu }^{{\beth }_{\omega }}\le \lambda =cf\left(\lambda \right)\le {2}^{\mu }$ then $𝔹$ satisfies the λ-Knaster condition (using the “revised GCH theorem”).

## How to cite

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Shelah, Saharon. "Cellularity of free products of Boolean algebras (or topologies)." Fundamenta Mathematicae 166.1-2 (2000): 153-208. <http://eudml.org/doc/212474>.

@article{Shelah2000,
abstract = {The aim this paper is to present an answer to Problem 1 of Monk [10], [11]. We do this by proving in particular that if μ is a strong limit singular cardinal, $θ = (2^\{cf(μ)\})^+$ and $2^μ = μ^+$ then there are Boolean algebras $\mathbb \{B\}_1,\mathbb \{B\}_2$ such that $c(\mathbb \{B\}_1) = μ, c(\mathbb \{B\}_2) < θ but c(\mathbb \{B\}_1*\mathbb \{B\}_2)=μ^+$. Further we improve this result, deal with the method and the necessity of the assumptions. In particular we prove that if $\mathbb \{B\}$ is a ccc Boolean algebra and $μ^\{ℶ_ω\} ≤ λ = cf(λ) ≤ 2^μ$ then $\mathbb \{B\}$ satisfies the λ-Knaster condition (using the “revised GCH theorem”).},
author = {Shelah, Saharon},
journal = {Fundamenta Mathematicae},
keywords = {set theory; pcf; Boolean algebras; cellularity; product; colourings},
language = {eng},
number = {1-2},
pages = {153-208},
title = {Cellularity of free products of Boolean algebras (or topologies)},
url = {http://eudml.org/doc/212474},
volume = {166},
year = {2000},
}

TY - JOUR
AU - Shelah, Saharon
TI - Cellularity of free products of Boolean algebras (or topologies)
JO - Fundamenta Mathematicae
PY - 2000
VL - 166
IS - 1-2
SP - 153
EP - 208
AB - The aim this paper is to present an answer to Problem 1 of Monk [10], [11]. We do this by proving in particular that if μ is a strong limit singular cardinal, $θ = (2^{cf(μ)})^+$ and $2^μ = μ^+$ then there are Boolean algebras $\mathbb {B}_1,\mathbb {B}_2$ such that $c(\mathbb {B}_1) = μ, c(\mathbb {B}_2) < θ but c(\mathbb {B}_1*\mathbb {B}_2)=μ^+$. Further we improve this result, deal with the method and the necessity of the assumptions. In particular we prove that if $\mathbb {B}$ is a ccc Boolean algebra and $μ^{ℶ_ω} ≤ λ = cf(λ) ≤ 2^μ$ then $\mathbb {B}$ satisfies the λ-Knaster condition (using the “revised GCH theorem”).
LA - eng
KW - set theory; pcf; Boolean algebras; cellularity; product; colourings
UR - http://eudml.org/doc/212474
ER -

## References

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25. [25] S. Shelah, The Generalized Continuum Hypothesis revisited, Israel J. Math. 116 (2000), 285-321; math.LO/9809200. Zbl0955.03054
26. [26] R. M.Solovay, Strongly compact cardinals and the GCH, in: Proc. Tarski Symposium (Berkeley, 1971), Proc. Sympos. Pure Math. 25, Amer. Math. Soc., 1974, 365-372.

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