2000 Fiscal Year Final Research Report Summary
Combinatorial semigroup theory and its applications
| Project/Area Number |
11640028
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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, Interdisiplinary Faculty of Sicence and Engineering, Professor, 総合理工学部, 教授 (50093646)
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| Co-Investigator(Kenkyū-buntansha) |
KONDO Michiro Shimane University, Interdisiplinary Faculty of Sicence and Engineering, Asociate Professor, 総合理工学部, 助教授 (40211916)
MIWA Takuo Shimane University, Interdisiplinary Faculty of Sicence and Engineering, Professor, 総合理工学部, 教授 (60032455)
IMAOKA Teruo Shimane University, Interdisiplinary Faculty of Sicence and Engineering, Professor, 総合理工学部, 教授 (60032603)
OZAKI Manabu Shimane University, Interdisiplinary Faculty of Sicence and Engineering, Lecturer, 総合理工学部, 講師 (80287961)
UEDA Akira Shimane University, Interdisiplinary Faculty of Sicence and Engineering, Asociate Professor, 総合理工学部, 助教授 (70213345)
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| Project Period (FY) |
1999 – 2000
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| Keywords | semigroup / amalgamation / generalized *-semigroup / Fiber / homotopy / Logic programming / Retract / Iwasawa invariant |
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| Research Abstract |
(1) The decidability problem of whether a finite semigroup is an amalgamation bases for semigroups or not remains unsolved. Representation extension property is a necessary condition for a semigroup to be an amalgamation base for semigroups. We proved decidability. of the problem of whether a finite semigroup has representation extension property or not. By using Software "Mathematica" we constructed a finite regular semigroup which has representation extension property but is not an amalgamation base for semigroups.
(2) We studied about the equivalence of the property 'Mbeing an amalgamation base for semigroups and the property 'Mbeing an amalgamation base for finite semigroups. Consequently, we obtained that every semigroup which is an amalgamation base for finite semigroups has representation extension property. Also we determined the strucure of finite bands which is an amalgamation base for finite semigroups. We gave semigroup-theoretical proof of Ok'nski and Putcha's theorem by entending Neumann's method from groups to semigroups.
(3) We introduced representation of generalized *-inverse semigroups and proved that the class of generalized *-inverse semigroups has strong amalgamation property. We proved decidability of DBLlogics in sematice of Logical Programming. We characterized completeness theorem for Logical Programming by making use of distributive lattices. Also we characterized prime and primary ideals of Pr'fer domain in a simple Artinian ring with finite dimension over its center.
(4) We studied Fiber homotopy and introduced fibrewise fibration and cofibration into the category of maps. Further, we obtained several results on absolute retraction and contractiblity of the category of maps.
(5) Let K be a cubic cyclic field with prime conductor. We gave simple sufficient conditions for λ_3 (K) = μ_3 (K) = 0, where λ_3 (K), μ_3 (K) are Iwasawa λ-invariant, μ-invariant of the cyclotomic Z_3-extension of K, respectively.
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