模糊与粗糙代数系统中的若干问题研究
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摘要
模糊代数是模糊数学理论研究的最为活跃的研究方向之一。1971年，A.Rosenfeld将模糊集理论应用到群上，提出了模糊群的概念。随后许多作者致力于将代数学中的理论扩展到模糊的背景之中，并逐渐形成了模糊数学的一个重要分支——模糊代数学。先后出版了专著《L-Subfeld and L-Subspace》，《FuzzyCommutative Algebra》与《Fuzzy Group Theory》等。此外，相似于模糊理论的情形，随着粗糙集理论的发展，研究者也越来越感兴趣于粗糙代数的研究。基于这些实际，本文对群这个代数系统在模糊背景和粗糙背景下做了一些研究。全文共分八章，主要研究工作分六个部分：
第一部分研究了模糊子群的等价刻画。利用有限生成群的生成元，模糊子群的几个新的等价条件被构建。这些条件揭示了有限生成群的模糊子群对生成元的隶属度之间的联系。在这部份中，应用这些新的刻画我们构造出了一类交换群的模糊子群的具体情形。
第二部分利用模糊理论中的一个基本概念，即模糊阶去刻画普通的群。换句话说，就是一些普通的群可以由模糊阶完全刻画。特别地，解决了John Mordeson等于2005年在《Fuzzy group theory》中提出的一个问题（用模糊阶去刻画循环群）。这些研究建立了模糊群理论与经典群理论之间的联系。
第三部分研究了广义模糊陪集。将现有文献中模糊陪集的概念做了合理的拓展。并用这个更广义的概念给出了模糊正规子群的七个等价刻画。另外，借助于这个广义陪集的概念建立了模糊正规化子的概念，并做了相关研究。
第四部分研究了极大模糊子群。通过对经典群论中极大子群的分析，提出了极大模糊子群的概念。并研究了极大模糊子群的相关性质。通过研究发现，利用极大模糊子群这个概念，不但可以刻画幂零群而且可以研究模糊子群的分解问题。
第五部分研究了模糊置换群和模糊群作用。首先研究了经典群论中群作用的特征函数表示。然后利用这个表示很自然地将群作用这个概念引入到模糊背景当中，提出了模糊群作用这个概念。正是这个概念的正确引入，才使模糊置换群理论中的相关内容得以顺利研究，如，对模糊圈积和模糊本原群的研究。
第六部分研究了上、下粗糙群。先研究了以群为论域的上、下近似算子的一些性质。然后借助于经典群论中的陪集代表系的概念给出了与一个下粗糙子群下粗糙相等的最大粗糙子群（是指集合的势最大）的表达式，相应地，也给出了与一个上粗糙子群上粗糙相等的最小上粗糙子群（是指集合的势最小）的表达式。最后，提出了两个近似空间之间的同态的概念。利用这个概念建立了两个论域上的粗糙子群之间的联系。
The research on fuzzy algebra has become one of the most active topics in fuzzymathematics in recent years. Applying the concept of fuzzy sets of Zadeh to grouptheory, Rosenfeld introduced the notion of a fuzzy group as early as1971. Then manyauthors have been involved in extending the results of algebra to the fuzzy setting, andgradually form an important branch of modern mathematics——fuzzy algebra. Manyimportant monographs have been published, such as 《L-Subfeld and L-Subspace》,《Fuzzy Commutative Algebra》,《Fuzzy Group Theory》etc.. Similarly, with thedevelopment of rough set theory, the researches have also become more interested instudying rough algebra. Based on these facts, we do some researches on the fuzzyalgebra and rough algebra in this paper. The dissertation consists of six parts with eightchapters:
Part1is devoted to investigating the equivalent characterizations of fuzzysubgroups. Several equivalent conditions of fuzzy subgroups are presented by means ofthe generators of groups. These conditions reveal the relationships among membershipsof the generators. The structures of fuzzy subgroups of some groups can becharacterized clearly by these equivalent conditions.
Part2is devoted to characterize the groups by means of the notion of fuzzy order.That is to say, the structures of some groups can be completely characterized by fuzzyorders of their fuzzy subgroups. In particular, one open question proposed by J.N.Mordeson etc. is solved. These studies establish the relationship between the fuzzytheories and the classical theories.
Part3contributes the generalization of fuzzy cosets. A more general definition offuzzy coset is presented, that is, the concept of fuzzy coset of a fuzzy subgroup relativeto a fuzzy subgroup is introduced. Further, it is shown that a normal fuzzy subgroup of afuzzy subgroup can be characterized in terms of fuzzy cosets. In addition, the concept offuzzy normalizer is generalized by means of fuzzy cosets, and some researches on thebroad concept are done.
Part4is to study the maximal fuzzy subgroups. By analyzing the definition of the classical maximal subgroup, the notion of maximal fuzzy subgroup is defined and thetheory related to fuzzy maximal subgroups is explored. Further investigations indicatedthat the nilpotent groups can be characterized and decomposed the product of its fuzzysubgroups by means of maximal fuzzy subgroups.
Part5is about the studies of fuzzy permutation groups. In this paper, the authorfirst explores the issue of the characteristic function representation of the classical groupaction. Then the author introduces the notion of fuzzy group action on the basis ofwhich the author does some researches on fuzzy permutation groups.
Part6is the researches on rough groups. Some new propositions of the lower andthe upper approximations in a group are given. The concepts of minimal upper roughsubgroup and maximal lower rough subgroup are introduced and their equivalentcharacterizations are explored. Finally, the notion of homomorphisms of approximationspaces is introduced. By means of the notion the relationship between rough subgroupsbased on two uinverses is established.
第一部分研究了模糊子群的等价刻画。利用有限生成群的生成元，模糊子群的几个新的等价条件被构建。这些条件揭示了有限生成群的模糊子群对生成元的隶属度之间的联系。在这部份中，应用这些新的刻画我们构造出了一类交换群的模糊子群的具体情形。
第二部分利用模糊理论中的一个基本概念，即模糊阶去刻画普通的群。换句话说，就是一些普通的群可以由模糊阶完全刻画。特别地，解决了John Mordeson等于2005年在《Fuzzy group theory》中提出的一个问题（用模糊阶去刻画循环群）。这些研究建立了模糊群理论与经典群理论之间的联系。
第三部分研究了广义模糊陪集。将现有文献中模糊陪集的概念做了合理的拓展。并用这个更广义的概念给出了模糊正规子群的七个等价刻画。另外，借助于这个广义陪集的概念建立了模糊正规化子的概念，并做了相关研究。
第四部分研究了极大模糊子群。通过对经典群论中极大子群的分析，提出了极大模糊子群的概念。并研究了极大模糊子群的相关性质。通过研究发现，利用极大模糊子群这个概念，不但可以刻画幂零群而且可以研究模糊子群的分解问题。
第五部分研究了模糊置换群和模糊群作用。首先研究了经典群论中群作用的特征函数表示。然后利用这个表示很自然地将群作用这个概念引入到模糊背景当中，提出了模糊群作用这个概念。正是这个概念的正确引入，才使模糊置换群理论中的相关内容得以顺利研究，如，对模糊圈积和模糊本原群的研究。
第六部分研究了上、下粗糙群。先研究了以群为论域的上、下近似算子的一些性质。然后借助于经典群论中的陪集代表系的概念给出了与一个下粗糙子群下粗糙相等的最大粗糙子群（是指集合的势最大）的表达式，相应地，也给出了与一个上粗糙子群上粗糙相等的最小上粗糙子群（是指集合的势最小）的表达式。最后，提出了两个近似空间之间的同态的概念。利用这个概念建立了两个论域上的粗糙子群之间的联系。
The research on fuzzy algebra has become one of the most active topics in fuzzymathematics in recent years. Applying the concept of fuzzy sets of Zadeh to grouptheory, Rosenfeld introduced the notion of a fuzzy group as early as1971. Then manyauthors have been involved in extending the results of algebra to the fuzzy setting, andgradually form an important branch of modern mathematics——fuzzy algebra. Manyimportant monographs have been published, such as 《L-Subfeld and L-Subspace》,《Fuzzy Commutative Algebra》,《Fuzzy Group Theory》etc.. Similarly, with thedevelopment of rough set theory, the researches have also become more interested instudying rough algebra. Based on these facts, we do some researches on the fuzzyalgebra and rough algebra in this paper. The dissertation consists of six parts with eightchapters:
Part1is devoted to investigating the equivalent characterizations of fuzzysubgroups. Several equivalent conditions of fuzzy subgroups are presented by means ofthe generators of groups. These conditions reveal the relationships among membershipsof the generators. The structures of fuzzy subgroups of some groups can becharacterized clearly by these equivalent conditions.
Part2is devoted to characterize the groups by means of the notion of fuzzy order.That is to say, the structures of some groups can be completely characterized by fuzzyorders of their fuzzy subgroups. In particular, one open question proposed by J.N.Mordeson etc. is solved. These studies establish the relationship between the fuzzytheories and the classical theories.
Part3contributes the generalization of fuzzy cosets. A more general definition offuzzy coset is presented, that is, the concept of fuzzy coset of a fuzzy subgroup relativeto a fuzzy subgroup is introduced. Further, it is shown that a normal fuzzy subgroup of afuzzy subgroup can be characterized in terms of fuzzy cosets. In addition, the concept offuzzy normalizer is generalized by means of fuzzy cosets, and some researches on thebroad concept are done.
Part4is to study the maximal fuzzy subgroups. By analyzing the definition of the classical maximal subgroup, the notion of maximal fuzzy subgroup is defined and thetheory related to fuzzy maximal subgroups is explored. Further investigations indicatedthat the nilpotent groups can be characterized and decomposed the product of its fuzzysubgroups by means of maximal fuzzy subgroups.
Part5is about the studies of fuzzy permutation groups. In this paper, the authorfirst explores the issue of the characteristic function representation of the classical groupaction. Then the author introduces the notion of fuzzy group action on the basis ofwhich the author does some researches on fuzzy permutation groups.
Part6is the researches on rough groups. Some new propositions of the lower andthe upper approximations in a group are given. The concepts of minimal upper roughsubgroup and maximal lower rough subgroup are introduced and their equivalentcharacterizations are explored. Finally, the notion of homomorphisms of approximationspaces is introduced. By means of the notion the relationship between rough subgroupsbased on two uinverses is established.
引文
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[3] A. Rosenfeld. Fuzzy groups. J. Math.Anal.Appl.,1971,35:512-517
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