The study of filters on Pκλ started by Jech [5] as a natural generalization of that of filters on an uncountable regular cardinal κ. Several notions including weak normality have been generalized. However, there are two versions proposed as weak normality for filters on Pκλ. One is due to Abe [1] as a straightforward generalization of weak normality for filters on κ due to Kanamori [6] and the other is due to Mignone [8]. While Mignone's version is weaker than normality, Kanamori-Abe's version is not in general. In fact, Abe [2] has proved, generalizing Kanamori [6], that a filter is weakly normal in the sense of Abe iff it is weakly normal in the sense of Mignone and there exists no disjoint family of cfλ-many positive sets. Therefore Kanamori-Abe's version is essentially a large cardinal property and Mignone's version seems to be the most natural formulation of “weak” normality.In this paper, we study weak normality in the sense of Mignone. In [8], Mignone studies weak normality of canonically defined filters. We complement his chart and try to find the weakly normal closures of these filters (i.e., the minimal weakly normal filters extending them). Therefore our result is a natural refinement of Carr [4].It is now well known that combinatorics on Pκλ is not a naive generalization of that on κ. For example, Menas [7] showed that stationarity on Pκλ can be characterized by 2-dimensional regressive functions, but not by 1-dimensional ones when λ is strictly larger than κ. We show in terms of weak normality that combinatorics on Pκλ vary drastically with respect to cfλ.
SPACETIME SINGULARITIES: RECENT DEVELOPMENTS
