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.