COMBINATORIAL SEMIGROUP THEORY AND ITS APPLICATIONS

1. 有限半群是否具有某种性质(P)的决策问题已被许多数学家研究过，Spair和Guba证明了对于许多性质(P)的决策问题是不可判定的。针对这一问题，Shoji证明了存在一种判定有限半群是否具有表示可拓性的算法。进一步，shoji证明了以下结果(1)对于完全0-简单半群S，下列是等价的。(i) S是一种特殊的混合基。（ii） S要么左边绝对平，要么右边绝对平。（iii） S满足左湮灭子条件或右湮灭子条件。(2)对于有限可交换半群T，下列是等价的。(1) T是一种完全特殊的混合碱。（ii） T为完全合并碱。(3) 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研究了简单阿提宁环中的普鲁特阶。特别地，上田描述了普鲁特阶的支理想和非支理想。(3) Kondo给出了一个非线性4值逻辑的公理系统，该系统的Lindenbaum代数是带蕴涵的de Morgan代数。(4) Miwa获得了超紧空间的一个新的表征。Miwa还定义了新的覆盖属性，并研究了这些覆盖属性在不同映射下的不变性和逆不变性。(5) Kikkawa引入了李三重代数的投影性的代数概念，研究了李代数的投影性的性质。

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.