Ideals with large cardinal properties

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Y.Matsubara and S.Shelah proved that the non-stationary ideal over P_κλ for every strong limit singular cardinal λ is nowhere precipitous. This implies that Menas' conjecture holds for strong limit singular cardinals λ.Y.Matsubara defined the notion of strategically closed ideals. Using a supercompact cardinal, he constructed a model with a strategically closed ideal over P_κλ where κ is a regular cardinal 【greater than or equal】 N_2 for every λ>κ. He also proved that the existence of such ideals implies the singular cardinal hypothesis and Rado's conjecture.If the poset P_I is proper, then we say that I is a proper ideal. In general if forcing with P_I preserves stationary subsets of P_κλ, then we say that I is a stationary preserving ideal. Matsubara proved that the existence of stationary preserving ideals is a large cardinal property. For example, he proved that if a proper ideal over P_<λ+>δ exists where δ is a cardinal sufficiently larger than λ, then the non-stationary ideal over P_<N1>λ is precipitous.In 1985 Gitik proved the consistency of the existence of stationary subsets X of P_κλ such that the non-stationary ideal restricted to X is κ^+-saturated using a supercompact cardinal. Shioya gave a new proof of this theorem using a weaker hypothesis. Shioya and S.Shelah pro ed the existence of a non-reflecting stationary subset of P_κκ^+.Yoshinobu investigated properties of partial orders using Banach-Mazur type games. He showed the hierarchical structure of strategically closed and strongly strategically closed partial orders. Yoshinobu together with M.Takahashi proved some basic properties of σ-short Boolean n algebras.