Square and non-reflection in the context of ${\mathcal{P}}_{\kappa }\lambda$

Abstract

We define ${\square }_{{\mathcal{P}}_{\kappa }\lambda }$, a square principle in the context of ${\mathcal{P}}_{\kappa }\lambda$, and prove its consistency relative to ZFC by a directed-closed forcing and hence that it is consistent to have ${\square }_{{\mathcal{P}}_{\kappa }\lambda }$ hold when $\kappa$ is supercompact, whereas ${\square }_{\kappa }$ is known to fail under this condition. The new principle is then extended to produce a principle with a non-reflection property. Another variation on ${\square }_{{\mathcal{P}}_{\kappa }\lambda }$ is also considered, this one based on a family of club subsets of ${\mathcal{P}}_{{\kappa }_x}\left(x\right)$. Finally, a new square principle for cardinals, denoted ${\square }_{\kappa }^f$, is introduced. This principle is proved consistent with $\kappa$ being supercompact. It is shown to yield a non-reflection result similar to that given by ${\square }_{\kappa }$.