Normality of a Filter over a Space of Partitions

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 Q κ ( λ ), the set of partitions of λ into fewer than κ pieces, invented by Henle and Zwicker [4] in the spirit of P κ ( λ ). Various notions of normality for filters over Q κ ( 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 P κ ( λ ) and Q κ ( λ ) we define two diagonal intersections, Δ 1 and Δ 2 , under which the club filter is closed.