Combinatorial semigroup theory and its applications

(1)有限半群是否为半群的合并基的可判断性问题尚未解决。表示扩张性质是半群成为半群的合并基的必要条件。我们证明了可决断性。关于有限半群是否具有表示扩张性质的问题。(2)研究了性质‘M是半群的合并基与性质’M是有限半群的合并基的等价性。由此，我们得到了作为有限半群的合并基的每个半群都具有表示扩张性质。我们还确定了有限半群的合并基--有限带的结构。通过将Neumann方法从群推广到半群，给出了OK‘nski和Putcha定理的半群理论证明。(3)引入了广义*-逆半群的表示，并证明了这类广义*-逆半群具有强合并性。在逻辑程序设计的语义中证明了DBL逻辑的可判断性。利用分配格刻画了逻辑规划的完备性定理。研究了单Artin环的中心有限维的Pr‘fer整环的素理想和准理想。(4)研究了单Artin环的纤维同伦，并将纤维纤维和余纤维引入映射范畴。进一步，我们得到了关于映射范畴的绝对收缩和绝对收缩的几个结果。(5)设K是具有素导体的三次循环域。给出了λ_3(K)=μ_3(K)=0的简单充分条件，其中λ_3(K)和μ_3(K)分别是K的分圆Z_3扩张的岩泽λ不变量和μ不变量。

(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.