COMBINATORIAL SEMIGROUP THEORY AND ITS APPLICATIONS

1.有限半群是否具有某种性质(P)的判定问题已经被许多数学家研究过，Sair和Guba证明了对于许多性质(P)，判定问题是不可判定的。关于这个问题，Shoji证明了存在一个判定有限半群是否具有表示扩张性质的算法。此外，Shoji证明了如下结果：(1)对于完全0-单半群S，下列条件是等价的：(I)S是一种特殊的合并基；(Ii)S是左绝对平坦的或右绝对平坦的；(Iii)S满足左零化子条件或右零化子条件；(2)对于有限交换半群T，下列条件是等价的：(I)T是完全特殊的合并基。(Ii)T是完全合并基。(Iii)T是E-可分的。作为组合半群理论的应用，我们得到了以下结果：(1)Imaoka研究了广义逆*-半群的表示。(2)Ueda研究了单Artin环上的Prufer序。特别地，上田刻画了Prufer阶分枝和不分枝的素理想。(3)近藤给出了一个非线性四值逻辑的公理系统，它的Lindenbaum代数是具有蕴涵的de Morgan代数。(4)Miwa得到了超仿紧空间的一个新的刻画。Miwa还定义了新的覆盖性质，并研究了这些覆盖性质的各种映射下的不变性和逆不变性。(5)Kikkawa引入了李三代数的射影代数概念，研究了射影李代数的性质。

1. Decision problem whether or not a finite semigroup has a certain property (P) has been studied by many mathematicians, and Spair and Guba proved that for many properties (P) the decision problem is undecidable. Concerning this problem, Shoji proved that there exists an algorithm to decide whether or not a finite semigroup has the representation extension property. Furthermore, shoji proved the following results(1) For completely 0-simple semigroup S, the following are equivalent.(i) S is a special amalgamation base.(ii) S is either left absolutely flat or right absolutely flat.(iii) S satisfies either left annihilator condition or right annihilator condition.(2) For finite commutative semigroup T, the following are equivalent.(i) T is a completely special amalgamation base.(ii) T is completely amalgamation base.(iii) T is E-separable.2. As applications of combinatorial semigroup theory we obtained the following results.(1) Imaoka investigated about representations of generalized inverse *-semigroups.(2) Ueda studied about Prufer orders in simple Artinian rings. In particular, Ueda characterized branched and unbranched prime ideals of Prufer orders.(3) Kondo gave an axiom system of a non-linear 4-valued logic , whose Lindenbaum algebra is the de Morgan algebra with implication.(4) Miwa obtained a new characterization of superparacompact spaces. Miwa also defined new covering properties and studied invariance and inverse invariance under various maps of these covering properties.(5) Kikkawa introduced the algebraic concept of projectivity of a Lie triple algebra and investigated about properties of Lie algebra of projectivity.