【作者】 张鹏鸽；

【导师】 刘三阳；

【作者基本信息】 西安电子科技大学 ， 应用数学， 2006， 博士

【摘要】 “半群代数理论”在计算机科学、信息科学的推动下，经过六十余年的系统研究，已成为“代数学”中一个独具特色的学科分支。它与“群论”的关系类似于“环论”与“域论”的关系。这一地位的确立不仅在于一批系统的研究成果的出现，更在于一套独特的系统研究思路和方法的形成。 半群理论家说过：半群同余理论是半群代数理论中最深刻和最精彩的部分。特别是在Zadeh引入了模糊集的概念后，模糊关系也随之产生了。如同在研究一般半群的结构理论一样，我们可从半群的模糊理论出发研究其模糊性质，包括讨论半群上的模糊同余关系及其由模糊同余关系所确定的商半群等理论，基于这样的思想Kuroki将半群的同态基本定理推广到更一般的、内涵更丰富的同态定理。一般半群上模糊同余的深入研究并非易事，近年来人们主要考虑了逆半群模糊同余的刻画以及正则半群的模糊同余性质等。 本文主要研究几类富足半群的结构性质，并通过模糊同余关系研究了正则半群上的模糊结构理论。具体工作如下： 1．由于郭小江教授在文献[27]中证明了任何一个IC拟适当半群均是型-W半群。自然地，在广义正则半群的意义下，借助于Hall’s半群W_B、幂等元带B及两个逆半群T和W_B/γ之间的同态ψ构造出的纯整半群H(B，W_B/γ，ψ)，文中证明了织积S=(T，W_B/δ，ψ)是一个型-W半群，其中ψ是从型-A半群T到W_B/δ的幂等元分离好同余，且每个型-W半群均可如此构造。接着给出了两个型-W半群同构的充分必要条件。最后利用好同余证明了在富足半群意义下的满足正则性条件幂等元提升引理。这一引理正是Lallement’s引理的自然推广。 2．定义了一类F-富足半群，即对x∈S，满足|U~*(x)|=1的一类IC-拟适当半群，称之为u-IC拟适当半群，结合富足半群上的自然偏序关系给出了这类半群的诸多性质。利用半群S上的自同构幺半群End(S)定义了半直积S×ΦT，其中Φ是从T到End(S)的幺同态，S是半群，T是幺半群。接下来证明了当B是一个含恒等元i的带，M是一个消去幺半群，则半直积B×ΦM是一个u-IC拟适当半群，其中Φ是从M到Aut(B)的幺同态。最后得到关于u-IC拟适当半群的结构定理。即如上构造的半直积是u-IC拟适当半群；反之，任何一个u-IC拟适当半群均可如此构造。 3．给出了比u-IC拟适当半群更广泛的一类IC富足半群，称之为α-IC拟更多还原

【Abstract】 " Semigroup Algebra Theory " which is implelled by computer science and information science and researched systemically for more than sixty years, has been become a distinctive and special subject embranchment of " algebra ". As is showed in the relation of " Semigroup theory " and " Group theory ", it is genereral accepted that of " Ring theory " and " Field theory ". This status are established not only the appearance of a passel of system research production, but also the form of a suit of particular system research thoughtway.Semigroup theoretician said: semigroup congruence theory is the part of most profound and splendid in semigroup algebra theory. In particular,after Zadeh introduced the concept of fuzzy set, fuzzy relations are appeared. Such as studying the structure theory of general semigroup, we can research the semigroup fuzzy properties by setting up semigroup fuzzy theory, including the theory of fuzzy congruence on semigroup and quotient semigroup theory which confirmed by fuzzy congruence relation. Kuroki generalized the semigroup homomorphism theory that is more ecumenical and content ample based this ideas. The gerenal semigroup congruence theory can not be researched easily. Recently, the researcher study the fuzzy congruence on inverse semogroup and some fuzzy properties on regular semigroup.This thesis is devoted to study the structure theory of abundant semigroup and fuzzy structure theory of regular semigroups by fuzzy congruence relation. Details are as follows:1. Because the Professor Xiaojiang Guo proved that any IC quasi-adequate semigroup is a type-W semigroup. Naturally, under the sense of the gerenlized regular semigroups, the structure of orthdox semigroup H(B, W_B/γ,Ψ) was established, depended heavily on the Hall’s semigroup W_B with respect to a band B and the homomorphism γ between of inverse更多还原