STRUCTURAL PROPERTIES OF IDEALS OVER 𝒫κλ II

Abstract

Continuing the paper “Structural properties of ideals over ${𝒫}_{\kappa }\lambda$ I” we first show that for every ideal $I$ over ${𝒫}_{\kappa }\lambda$ there is a function $f$ such that $f_{*}\left(I\right)=\{X\subset {𝒫}_{\kappa }\lambda :f^{-1}[X]\in I\}$ and ${\mathrm{J}}_{\kappa ,\lambda }$ are coherent. On the other hand, ${\mathrm{J}}_{\kappa ,\lambda }\not\subset g_{*}({\mathrm{I}}_{\kappa ,\lambda }\upharpoonright X)$ for any $g$ and $X\in {\mathrm{I}}_{\kappa ,\lambda }^{+}$ under some conditions, e.g. .

We also study coherence and rigidity for ideals over ${𝒫}_{\kappa }\lambda$, and get characterizations of these properties similar to those by Baumgartner, Taylor and Wagon for ideals on $\kappa$.