Combinatorial semigroup theory and its applications

(1)有限半群是否是半群的合并基的可决性问题仍未得到解决。表示可拓性是半群成为半群合并基的必要条件。我们证明了可决性。有限半群是否具有表示可拓性的问题。利用Mathematica软件构造了一个有限正则半群，该半群具有表示可拓性，但不是半群的合并基。(2)研究了“m”是半群的合并基和“m”是有限半群的合并基的等价性。因此，我们得到了每一个作为有限半群的合并基的半群都具有表示可拓性。我们还确定了有限带的结构，这是有限半群的合并基。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，证明了一类广义*逆半群具有强合并性质。在逻辑规划的语义中证明了逻辑的可判定性。利用分配格对逻辑规划的完备性定理进行了刻画。我们还刻画了一个中心上有有限维的简单阿提宁环的Pr'fer域的素理想和原理想。(4)研究了纤维同伦，将纤维纤维纤维化和纤维纤维化引入映射范畴。在此基础上，我们进一步得到了映射类的绝对可缩性和绝对可缩性的几个结果。(5)设K为一个素数导体的三次循环场。给出了λ_3 (K) = μ_3 (K) = 0的简单充分条件，其中λ_3 (K)、μ_3 (K)分别是K的分环z_3扩展的Iwasawa 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.