On semiproperness of Namba forcings and Ideals in Prikry extensions

Abstract

In this paper, we study some variations of Namba forcing $\mathrm{Nm}(\kappa ,\lambda )$ over ${\mathcal{P}}_{\kappa }\lambda$ and show that its semiproperness implies $\mathrm{SSR}([\lambda {]}^{\omega },<\kappa )$. In particular, Prikry forcing at $\mu$ forces Namba forcing $\mathrm{Nm}({\mu }^{+})$ is not semiproper. This shows that there is no semiproper saturated ideal in the 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.