2002 Fiscal Year Final Research Report Summary
Ideals with large cardinal properties
| Project/Area Number |
13640113
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Nagoya University |
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Principal Investigator |
MATSUBARA Yo Nagoya univ., grad.school of human info., asso.prof., 大学院・人間情報学研究科, 助教授 (30242788)
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| Co-Investigator(Kenkyū-buntansha) |
SHIOYA Masahiro Tsukuba Univ., inst.of math., assi.prof., 数学系, 講師 (30251028)
ABE Yoshihiro Kanagawa Univ., school of eng., asso.prof., 工学部, 助教授 (10159452)
YOSHINOBU Yasuo Nagoya univ., grad.school of human info., assi., 大学院・人間情報学研究科, 助手 (90281063)
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| Project Period (FY) |
2001 – 2002
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| Keywords | set theory / large cardinals |
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| Research Abstract |
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.
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