Combinatorial semigroup theory and its applications

组合半群理论及其应用

• 批准号: 13640023
• 负责人: SHOJI Kunitaka
• 金额: $ 2.24万
• 依托单位: Shimane University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640023/
• 关键词: semigroup; amalgam; algorithm; fibre; homotopy; rewriting system; caluculus; Iwasawa theory; レトラクト; 岩沢不変; 群; 融合問題; 語の問題; 自由融合積; メンバーズシップ問題; 不値環

(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.

(1)是否存在判定一个有限半群是所有半群的合并基的算法，这个问题至今没有解决。另一方面，已知作为所有半群的合并基的半群总是具有表示扩张性质。在这个项目中，我们证明了存在一个算法来决定有限半群是否具有表示扩张性质。另外，一个具有表示扩张性质的完全0-单半群是所有半群的合并基。然而，我们可以通过软件“Mathematic”构造一个具有表示扩张性质的有限正则半群的例子，而不是所有半群的合并基。此外，我们还证明了存在一个判定有限半群是否左绝对平坦的算法。结果将出现在即将发表的论文中。(2)我们证明了一个有限半群是所有有限半群的一个合并，它具有表示扩张性质。由此，我们确定了有限带的结构，它是所有有限半群的融合。(3)给出了有限逆半群上Okininski和Putcha定理的另一个证明，它是B.Neumann关于群的结果的自然推广. (4)我们证明了λ μ-模型的构造可以通过使用不动点算子和Godel-Gentzen平移来给出. (5)研究了纤维同伦、纤维纤维化和纤维上纤维化。(6)设p是一个奇素数。利用Iwasawa理论，构造了极大真实的子域具有任意大p秩的类群和仅有四个素因子的导体的循环域.

项目成果

Y.Hotta, T.Miwa: "A new approach to fibrewise fibration and cofibrations"Topology and its application. 122. 205-222 (2002)

Y.Hotta、T.Miwa：“纤维纤维化和共纤维化的新方法”拓扑及其应用。

K,Shoji.: "Commutative semigroups which are semigroup amalgamation bases"J. Algebra. 238. 1-50 (2001)

K，Shoji.：“作为半群合并基的交换半群”J.

M,Ozaki.: "An application of Iwasawa theory to constructing fields Q (ζ+ζ^<-1>) which have class group with large p-rank"Nagoya J. Math., To appear.

M, Ozaki.：“岩泽理论在构造具有大 p 秩的类群的域 Q (z+z^<-1>) 中的应用”Nagoya J. Math.，出现。

M.Ozaki: "Iwasawa λ_{3}-invariants of certain cubic fields"Acta Arithmetica. 97. 387-398 (2001)

M.Ozaki：“Iwasawa λ_{3}-某些三次域的不变量”Acta Arithmetica 97. 387-398 (2001)。

K.Shoji: "A proof of Okninski and Putcha's theorem"Proc.of the Third Inter. Colloq.on Words, languages and Combinatorics, Ed. by M. Ito, \ & T. Imaoka, World Scientific. (To appear).

K.Shoji：“Okninski 和 Putcha 定理的证明”Proc.of the Third Inter。