Cellularity of free products of Boolean algebras (or topologies)

Saharon Shelah

Fundamenta Mathematicae (2000)

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

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, θ = ( 2 c f ( μ ) ) + and 2 μ = μ + then there are Boolean algebras 𝔹 1 , 𝔹 2 such that c ( 𝔹 1 ) = μ , c ( 𝔹 2 ) < θ b u t c ( 𝔹 1 * 𝔹 2 ) = μ + . 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 μ ℶ ω ≤ λ = c f ( λ ) ≤ 2 μ 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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