Cellularity of free products of Boolean algebras (or topologies)

Saharon Shelah¹

¹

Fundamenta Mathematicae, Tome 166 (2000) no. 1, pp. 153-208.

Résumé

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").

Détail

DOI : 10.4064/fm-166-1-2-153-208

Keywords: set theory, pcf, Boolean algebras, cellularity, product, colourings