Iterated reduced powers of collapsing algebras

Abstract

$\mathrm{rp}\left(B\right)$ denotes the reduced power $B^{\omega }/\Phi$ of a Boolean algebra $B$, where Φ is the Fréchet filter on ω. We investigate iterated reduced powers (${\mathrm{rp}}^0\left(B\right)=B$ and ${\mathrm{rp}}^{n+1}\left(B\right)=\mathrm{rp}\left({\mathrm{rp}}^n\right(B\left)\right)$) of collapsing algebras and our main intention is to classify the algebras ${\mathrm{rp}}^n\left(\mathrm{Col}\right(\lambda ,\kappa \left)\right)$, $n\ge 1$, up to isomorphism of their Boolean completions. In particular, assuming that SCH and $h={\omega }_1$ hold, we show that for any cardinals $\lambda \ge \omega$ and $\kappa \ge 2$ such that $\kappa \lambda >\omega$ and $\mathrm{cf}\left(\lambda \right)\le c$ we have $\mathrm{ro}\left({\mathrm{rp}}^n\right(\mathrm{Col}(\lambda ,\kappa )\left)\right)\cong \mathrm{Col}({\omega }_1,{\left({\kappa }^{<\lambda }\right)}^{\omega })$, for each $n\ge 1$; more precisely,$\mathrm{ro}\left({\mathrm{rp}}^n\right(\mathrm{Col}(\lambda ,\kappa )\left)\right)\cong \{\begin{array}{cc}\mathrm{Col}({\omega }_1,c),& \text{ if }{\kappa }^{<\lambda }\le c;\\ \mathrm{Col}({\omega }_1,{\kappa }^{<\lambda }),& \text{ if }{\kappa }^{<\lambda }>c\wedge \mathrm{cf}\left({\kappa }^{<\lambda }\right)>\omega ;\\ \mathrm{Col}({\omega }_1,{\left({\kappa }^{<\lambda }\right)}^{+}),& \text{ if }{\kappa }^{<\lambda }>c\wedge \mathrm{cf}\left({\kappa }^{<\lambda }\right)=\omega .\end{array}$ If $b=d$ and $0^{♯}$ does not exist, then the same holds whenever $\mathrm{cf}\left(\lambda \right)=\omega$.