Reduced Products of Collapsing Algebras

Abstract

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

26 pages