Splitting stationary sets in Pκλ for λ with small cofinality

For a regular uncountable cardinal kappa and a cardinal lambda with cf(lambda) < kappa < lambda, we investigate the consistency strength of the existence of a stationary set in P-kappa lambda which cannot be split into lambda(+) 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-kappa lambda such that every stationary subset of S can be split into lambda(+) many pairwise disjoint stationary subsets.