2006 Fiscal Year Final Research Report Summary
Research of generic automorphisms of first order structures and its application to algebra
| Project/Area Number |
16540117
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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 | Kobe University (2005-2006)
Tokai University (2004) |
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Principal Investigator |
KIKYO Hirotaka Kobe University, Faculty of Engineering, Professor, 工学部, 教授 (80204824)
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| Co-Investigator(Kenkyū-buntansha) |
ITAI Masanori Tokai University, School of Science, Professor, 理学部, 教授 (80266361)
WATANABE Junzo Tokai University, School of Science, Professor, 理学部, 教授 (40022727)
TSUBOI Akito University of Tsukuba, Graduate School of Pure and Applied Sciences, Professor, 大学院数理物質科学研究科, 教授 (30180045)
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| Project Period (FY) |
2004 – 2006
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| Keywords | generic automorphism / generic predicate / generic structure / amalgamation property / Peano Arithmetic / definable model / Lefschetz Property / complete intersection |
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| Research Abstract |
1. The amalgamation property for automorphisms is preserved under the addition of generic predicates to a theory. Therefore, if we get an unstable theory by the addition of generic predicates to a stable theory, then the class of the generic automorphisms of the resulting theory is not elementary.
2. The omega-power of the additive group of the rational numbers equipped with a shift function is a quasi-minimal structure. Its theory can be axiomatized by sentences expressing a kind of genericity. Let K be a field. The class of generic automorphisms of an infinite K-vector space is elementary. Its theory can be axiomatized in the same manner as above. Z-power of a countable infinite K-vector space equipped with a shift function is a quasi-minimal model of this theory. It is omega-stable with Morley rank omega.
3. Consider a typical predimension function on the finite structures for a finite relational language and the class of finite structures in which the empty set is closed with respect to this predimension. Then the generic structure of this class has a universal-existential theory. This is a rather general solution to a problem of Baldwin-Shelah.
4. If a model N of PA is definable without parameters in an elementary extension M of the structure of natural numbers and N and M are elementarily equivalent, then N is definably isomorphic to M. But if we allow N to be definable with parameters then there are examples such that M and N are elementarily equivalent but non-isomorphic, or M and N are isomorphic but not definably isomorphic.
5. For graded Artinian K-algebras, we gave several characterizations of the strong and weak Lefschetz Properties. With these results, we found new classes of complete intersections with the strong Lefschetz Property.
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Research Products
(12 results)
