Remarks on measurable Boolean algebras and sequential cardinals

G. Plebauek

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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 $μ^{ℶ_{ω}} \leq λ = c f (λ) \leq 2^{μ}$ then $𝔹$ satisfies the λ-Knaster condition (using the “revised GCH theorem”).

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It is well known that if a nontrivial ideal ℑ on κ is normal, its quotient Boolean algebra P(κ)/ℑ is $κ^{+}$ -complete. It is also known that such completeness of the quotient does not characterize normality, since P(κ)/ℑ turns out to be $κ^{+}$ -complete whenever ℑ is prenormal, i.e. whenever there exists a minimal ℑ-measurable function in $^{κ} κ$ . Recently, it has been established by Zrotowski (see [Z1], [CWZ] and [Z2]) that for non-Mahlo κ, not only is the above condition sufficient but also necessary...

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