半群与Fuzzy半群及其粗糙集的代数性质

【作者】 陈铃；

【导师】 刘文奇；

【作者基本信息】 昆明理工大学 ， 系统理论， 2006， 硕士

【摘要】 半群的代数理论是在数学内部和外部双重条件下，从20世纪50年代到60年代发展起来的一个崭新的代数分支。1990年，Biswas R提出了反Fuzzy子群的定义；1995年沈正维提出了一个群的反Fuzzy子群和正规反Fuzzy子群的定义。在此基础上，本文首先以一个半群作为基本集，给出了半群的反Fuzzy半群及其补的定义，并给出了其充要条件；在同态映射的条件下，讨论了反Fuzzy半群与正规反Fuzzy半群的同态问题。然后，给出了反Fuzzy(左、右、双、内禀)理想的定义与其充要条件；在同态映射下，讨论了其同态问题。1965年，美国计算机与控制论专家Zadeh LA提出了Fuzzy集理论；1982年，波兰数学家pawlak Z提出了一种数据分析理论——粗糙集理论。二者是处理Fuzzy性和不确定性的数学工具，将二者结合起来研究，近年来越来越受到国际学术界的关注。本文以一个Fuzzy半群作为基本集，将粗糙集理论应用于Fuzzy半群中来，主要讨论了Fuzzy半群中两个非空Fuzzy子集积的粗理想的性质，进一步补充和完善了Fuzzy半群中粗糙集的数学结构。1994年，Biswas R与Nanda S提出了粗糙群和粗糙子群的定义；2005年谢祥云提出了粗糙半群的粗糙同态映射。在此基础上，本文进一步将粗糙集的思想引入到一个Fuzzy代数系统，导出了Fuzzy粗糙半群与Fuzzy粗糙商半群的定义，并讨论了Fuzzy粗糙同态与Fuzzy粗糙同构的问题。更多还原

【Abstract】 The semigroup algebra theory,which has a huge development under interior and exterior double Mathematical conditions form 1950’s to 1960’s,is a new branch of algebra theory.In 1990,R.Biswas gave definition of anti-fuzzy subgroup.In 1995,Shen gave definitions of anti-fuzzy subgroup and normal anti-fiizzy subgroup of group.Basing on this,the definition of anti-fiizzy semigroup and its complements based on semigroup are introduced, and the sufficient and necessary of the exsistence of the anti-fuzzy semigroup is given in this paper. The homomorphism of anti-fuzzy semigroups and the hononorphism of normal anti-fuzzy semigroups are discussed. At the same time, the correlative definitions anti-fuzzy ideals of semigroup and the sufficient and necessary of the exsistence of anti-fuzzy ideals are introducted,and the homomorphism of anti-fuzzy ideals are discussed.In 1965,L.A.Zadeh, a famous American electron engineer and cybernetics expert, put forward the theory of Fuzzy Sets. In 1982,Z.Pawlak, a famous Poland mathematician, put forward Rough Sets for the data analysis. Fuzzy sets and rough sets are means handling on fuzzy and indetermination. Today, fuzzy set theory, and rough set theory are combined in mathematics, and the combination is been adverted more and more by international academy. The properties of fuzzy rough ideals of two non-empty fuzzy subsets products in fuzzy semigroups are discussed in this paper,which based on a fuzzy semigroup. Thus the rough theory in fuzzy semigroups is completed and perfected.In 1994,the definitions of rough groups and rough subgroups were introduced by R.Biswas and S.Nanda. In 2005,Xie introduced a rough homomorphism mapping of rough semigroups.In this paper,by using the rough theory which is applied to the fuzzy algebra systems, the definitions of fuzzy rough semigroups and fuzzy rough quotient semigroups are educed. Moreover, the fuzzy rough homomorphism and the fuzzy rough isomorphism are discussed.更多还原