Combinatorial semigroup theory and its applications

(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一个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。研究结果将发表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.

(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.