Assumptions for the next few posts are:
- $\kappa$ weakly compact
- $\mu>2^\kappa$ is singular of cofinality $\kappa$
- $A\subseteq\mu$ is a set of regular cardinals
- $A$ is unbounded in $\mu$ of cardinality $\kappa$
- $J$ is a $\kappa$-complete ideal on $A$ extending the bounded ideal
- $\lambda=\tcf(\prod A/ J)$
Since $J$ extends the bounded ideal, we may as well assume
Of course this implies $|A|<\min(A)$, and so without loss of generality $A$ is progressive.
Let $D$ be an ultrafilter on $A$ disjoint to $J$. Then $\cf(\prod A/D)=\lambda$ and hence $\lambda\in \pcf(A)$. Furthermore, if $B_\lambda[A]$ is a pcf generator for $\lambda$ then $B_\lambda[A]$ must be in $D$ and therefore unbounded in $A$.
It follows that $\tcf(\prod B_\lambda[A]/J)=\lambda$ as well, and so we may assume (by passing to $B_\lambda[A]$ if necessary) that
Our assumption that $2^\kappa<\min(A)$ tells us that
(1) $\pcf(\pcf(A))=\pcf(A)$, and
(2) $|\pcf(A)|\leq 2^\kappa<\min(A)=\min(\pcf(A))$
and so $\pcf(A)$ has a transitive set of generators, that is, a sequence $\langle B_\theta:\theta\in\pcf(A)\rangle$ such that
- $B_\theta\subseteq \pcf(A)$
- $B_\theta$ is a generator for $\theta$ in $\pcf(\pcf(A))=\pcf(A)$, and
- $\tau\in B_\theta\Longrightarrow B_\tau\subseteq B_\theta$.