2002 Fiscal Year Final Research Report Summary
Combinatorial semigroup theory and its applications
| Project/Area Number |
13640023
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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 |
Algebra
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| Research Institution | Shimane University |
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Principal Investigator |
SHOJI Kunitaka Shimane University, Mathematics, Professor, 総合理工学部, 教授 (50093646)
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| Co-Investigator(Kenkyū-buntansha) |
FUJITA Kenetsu Shimane University, Information, Assistant Professor, 総合理工学部, 助教授 (30228994)
MIWA Takuo Shimane University, Mathematics, Professor, 総合理工学部, 教授 (60032455)
IMAOKA Teruo Shimane University, Mathematics, Professor, 総合理工学部, 教授 (60032603)
OZAKI Manabu Shimane University, Mathematics, Assistant Professor, 総合理工学部, 助教授 (80287961)
UEDA Akia Shimane University, Mathematics, Assistant Professor, 総合理工学部, 助教授 (70213345)
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| Project Period (FY) |
2001 – 2002
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| Keywords | semigroup / amalgam / algorithm / fibre / homotopy / rewriting system / caluculus / Iwasawa theory |
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| Research Abstract |
(1) There is no solution of the problem whether or not there exists an algorithm to decide if a finite semigroup is an amalgamation base for all semigroups. On the other hand, it is known that a semigroup which is an amalgamation base for all semigroups always has the representation extension property. In this project, we prove that there exists an algorithm to decide if a finite semigroup has the representation extension property. Also, it is known that a completely 0-simplesemigroup with the representation extension property is an amalgamation base for all semigroups. However, we can construct by the software "Mathematic" an example of a finite regular semigroup with the representation extension property and without being an amalgamation base for all semigroups. Moreover, we prove that there exists an algorithm to decide if a finite semigroup is left absolutely flat. The result will appear in a forthcoming paper. (2) We prove that a finite semigroup which is an amalgamation for all finite semigroups has the representation extension property. As a consequence, we decide the structure of finite bands which is an amalgamation for all finite semigroups. (3) We give another proof of Okininski and Putcha's theorem on finite inverse semigroup, which is a natural extension of B.Neumann's result on group. (4) We show that the construction of λ μ-models can be given by the use of a fixed point operator and the Godel-Gentzen translation. (5) We study the fibrewise homotopy, fibrewise fibration and fibrewise cofibration. (6) Let p be an odd prime number. By using Iwasawa theory we construct cyclotopic fields whose maximal real subfields have class group with arbitrarily large p-rank and a conductor with only four prime factors.
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