On semiproperness of Namba forcings and ideals in Prikry extensions

We study some variations of Namba forcing $\mathrm {Nm}(\kappa ,\lambda )$ over $\mathcal {P}_{\kappa }\lambda $ and show that its semiproperness implies $\mathrm {SSR}([\lambda ]^{\omega },{ \lt }\kappa )$.

In particular, Prikry forcing at $\mu $ forces that Namba forcing $\mathrm {Nm}(\mu ^{+})$ is not semiproper. This shows that there are no semiproper saturated ideals in extensions by Prikry-type forcings. We also show that $[\lambda ^{+}]^{\mu ^{+}}$ cannot carry a $\lambda ^{+}$-saturated semiproper ideal if $\mu $ is singular with countable cofinality.