Reduced Products of Collapsing Algebras

Abstract

$\mathop{\rm rp}\nolimits ({\mathbb B})$ denotes the reduced power ${\mathbb B}^\omega /\Phi$ of a Boolean algebra ${\mathbb B}$, where $\Phi$ is the Fr\'{e}chet filter $\Phi$ on $\omega$. 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 $\mathop{\rm rp}\nolimits ^n (\mathop{\rm Col}\nolimits (\lambda ,\kappa))$, $n\in {\mathbb N}$, up to isomorphism of their Boolean completions. In particular, assuming that SCH and ${\mathfrak h} =\omega _1$ hold, we show that for any cardinals $\lambda\geq \omega$ and $\kappa \geq 2$ such that $\kappa\lambda >\omega$ and $\mathop{\rm cf}\nolimits (\lambda )\leq {\mathfrak c}$ we have $\mathop{\rm ro} (\mathop{\rm rp}\nolimits ^n(\mathop{\rm Col}\nolimits (\lambda ,\kappa)))\cong \mathop{\rm Col}\nolimits (\omega _1, (\kappa ^{<\lambda })^\omega )$, for each $n\in {\mathbb N}$. If ${\mathfrak b} ={\mathfrak d}$ and $0^\sharp$ does not exist, then the same holds whenever $\mathop{\rm cf}\nolimits (\lambda )= \omega$.