Splitting stationary sets in P_κλ for λ with small cofinality

For a regular uncountable cardinal _κ and a cardinal λ with cf (λ) < κ < λ, we investigate the consistency strength of the existence of a stationary set in P _κλ which cannot be split into λ⁺ many pairwise disjoint stationary subsets. To do this, we introduce a new notion for ideals, which is a variation of normality of ideals. We also prove that there is a stationary set S in P_κλ such that every stationary subset of S can be split into λ⁺ many pairwise disjoint stationary subsets.