Part(κ, λ) and Part*(κ, λ)

Abstract

We show that if κ is λ ^<κ-ineffable (respectively, almost λ ^<κ-ineffable) then $(NI_{\kappa ,\lambda }^{[\lambda ]^{ < \kappa } } |A)^ + \to ((NS_{\kappa ,\lambda }^{[\lambda ]^{ < \kappa } } )^ + )^2$ (respectively, $(NAI_{\kappa ,\lambda }^{[\lambda ]^{ < \kappa } } )^ + \to (I_{\kappa ,\lambda }^ + )^2 )$ for some A, where $NI_{\kappa ,\lambda }^{[\lambda ]^{ < \kappa } }$ (respectively, $NAI_{\kappa ,\lambda }^{[\lambda ]^{ < \kappa } }$) denotes the projection of the non-ineffable (respectively, non-almost ineffable) ideal on P _κ(λ ^<κ), and $NS_{\kappa ,\lambda }^{[\lambda ]^{ < \kappa } }$ the smallest strongly normal ideal on P _κ(λ).