Normality of a filter over a space of partitions

Abstract

In this paper we investigate the closure of certain filters under different definitions of diagonal intersection. The space of partitions over which filters concern us is Qκ(λ), the set of partitions of λ into fewer than κ pieces, invented by Henle and Zwicker [4] in the spirit of Pκ(λ). Various notions of normality for filters over Qκ(λ) have been introduced in [4] and [7]. Our objective is to find a notion of normality in terms of a tractable diagonal intersection which also in some sense reflects the construction of a partition. Extending the parallels between Pκ(λ) and Qκ(λ) we define two diagonal intersections, Δ1 and Δ2, under which the club filter is closed.