Investigations of Q(,kappa)lambda: Some notions of normality for filters of partitions

Mark Fuller, University of Massachusetts Amherst

Abstract

The space $P\sb{\kappa}$($\lambda$), the set of all subsets of $\lambda$ of size less than $\kappa$ ($\kappa, \lambda$ cardinals with $\lambda \ge \kappa$), has been the focus of much set theoretic research in the last twenty years, particularly because of its connection with supercompact cardinals. Let $Q\sb{\kappa}(\lambda$) be the set of all partitions of $\lambda$ into fewer than $\kappa$ pieces, partially ordered by refinement. We assume throughout that $\kappa$ is a strongly inaccessible cardinal. In view of both extending the parallels between $P\sb{\kappa}(\lambda$) and $Q\sb{\kappa}(\lambda$) and of finding a more manageable diagonal intersection, we define two diagonal intersections, $\Delta\sb1$ and $\Delta\sb2$, under which the club filter is closed, i.e. normal. This is motivated by the twofold characterization of normality of the club filter ${\cal C}$ on either a cardinal $\kappa$ or on $P\sb{\kappa}(\lambda$): ${\cal C}$ is closed under diagonal intersections if and only if every choice function is constant on a stationary set (Fodor's Theorem). Regarding the diagonal intersections as operators $\Delta$ on filters, we are interested in whether there exist any other filters which are closed under $\Delta$. To this end, we make use in each case of a coding set, whose presence in the basis of a particular filter guarantees that the latter is closed under the diagonal intersection operator.

Subject Area

Mathematics

Recommended Citation

Fuller, Mark, "Investigations of Q(,kappa)lambda: Some notions of normality for filters of partitions" (1991). Doctoral Dissertations Available from Proquest. AAI9132851.
https://scholarworks.umass.edu/dissertations/AAI9132851

COinS