Reduced Products of Collapsing Algebras

Abstract

$\mathop{\rm rp}\nolimits ({\mathbb B})$ denotes the reduced power ${\mathbb B}^ω/Φ$ of a Boolean algebra ${\mathbb B}$, where $Φ$ is the Fréchet filter $Φ$ on $ω$. We investigate iterated reduced powers ($\mathop{\rm rp}\nolimits ^0 ({\mathbb B})={\mathbb B}$ and $\mathop{\rm rp}\nolimits ^{n+1} ({\mathbb B} )=\mathop{\rm rp}\nolimits (\mathop{\rm rp}\nolimits ^n ({\mathbb B}))$) of collapsing algebras and our main intention is to classify the algebras $\mathop{\rm rp}\nolimits ^n (\mathop{\rm Col}\nolimits (λ,κ))$, $n\in {\mathbb N}$, up to isomorphism of their Boolean completions. In particular, assuming that SCH and ${\mathfrak h} =ω_1$ hold, we show that for any cardinals $λ\geq ω$ and $κ\geq 2$ such that $κλ>ω$ and $\mathop{\rm cf}\nolimits (λ)\leq {\mathfrak c}$ we have $\mathop{\rm ro} (\mathop{\rm rp}\nolimits ^n(\mathop{\rm Col}\nolimits (λ,κ)))\cong \mathop{\rm Col}\nolimits (ω_1, (κ^{<λ})^ω)$, for each $n\in {\mathbb N}$. If ${\mathfrak b} ={\mathfrak d}$ and $0^\sharp$ does not exist, then the same holds whenever $\mathop{\rm cf}\nolimits (λ)= ω$.

Theorems & Definitions (20)

• Theorem 3.1
• Corollary 3.2
• Proposition 4.1
• Theorem 5.1
• Example 5.2
• Theorem 5.3
• Theorem 5.4
• Theorem 6.1
• Theorem 6.3
• Theorem 6.4