关于模糊关系R生成的模糊R-半群的若干研究

In this program, by overcoming the shortcoming which a fuzzed classical binary operation can not keep the process up, the definition of fuzzy R-semigroup generated by fuzzy relationship R is introduced. Meanwhile, the structure and properties of fuzzy R-semigroup are explored by means of combining classical research methods of semigroup with some tools of fuzzy mathematics. Furthermore, fuzzy subsemigroups generated by an element are considered, and conditions of a fuzzy subsemigroup forming a fuzzy group are described，which vastly enrich the theory of fuzzy power group. At last, by applying the above achievement to information and communication field, the characters of the fuzzy information channel and encoding \decoding methods in source and channel coding technology are studied. The research of this scheme will enrich the contents of fuzzy mathematics, algebra theory of semigroup and communications theory, set up some new research methods and accelerate the crossing and mutual development of fuzzy mathematics and algebra theory of semigroup and communication theory.

本项目在克服目前代数运算模糊化后不能进行连续运算的缺点后，引入了由模糊R-关系生成的模糊R-半群概念。同时，利用经典半群研究手段和模糊化研究工具相结合的方法，探讨模糊R-半群的性质和结构。进一步，考虑单个元素生成模糊R-子半群的性质，刻画模糊R-子半群成群的条件。这将是对模糊幂群理论的极大丰富和补充。最后，考虑将已获得的结论应用于信息通信领域，研究信源和信道上的编码、译码方式的改进及模糊信道的相关特性。本项目的研究将丰富模糊数学、半群代数理论和通信理论的研究内容，开辟新的研究途径，促进各学科交叉和共同发展。

本项目是代数理论和模糊集理论的交叉。在克服目前代数运算模糊化后不能进行连续运算的缺点后，给出了模糊幂半群（一类特殊模糊R-半群）中OP极限的概念和得到了OP极限存在的充分条件，确定了有限循环群生成的模糊幂半群中元素的OP极限的有限分层结构；讨论模糊性质时，得到并讨论了表现隶属函数的概念、性质及有关应用；利用可逆元、逆断面与特殊幂等元研究了正则半群和正则半群的推广—富足半群；进一步探讨有关的实际应用问题。给出了一类LDPC码的编码方法，运用表现隶属函数优化了评委打分和图形图像处理问题。本项目的研究促进了模糊集理论和代数理论的结合和发展。