直觉L模糊子群的刻画

本文研究了一个群G上的直觉L模糊子群.借助于L模糊集的截集,反模糊子群,得到了群G上的直觉L模糊子群的等价刻画.在直觉L模糊子群和群G上的子群族之间可以建立起一一对应.     

(λ,μ)-反模糊子群的研究

基于(λ,μ)-反模糊子群概念及其基本性质,深入刻画了(λ,μ)-反模糊子群以及(λ,μ)-反模糊正规子群的结构.首先讨论了群G的(λ,μ)-反模糊子群在G的不同元素上隶属度的分布情况,其次研究了(λ,μ)-反模糊正规子群在G的不同元素上隶属度的分布情况,最后对循环群和阿贝尔群上(λ,μ)-反模糊子群及正规子群的结构进行详细讨论并给出了相应的结果.     

生成反模糊子群

引入生成反模糊子群的概念,给出生成反模糊子群的代数刻画式,讨论了生成反模糊子群的基本性质.     

直积群上几类直觉模糊子群及其投影

给出了直觉模糊子群的性质及其等价命题,并通过定义直觉模糊集的直积与投影,获得了直觉模糊直积群及直觉模糊投影子群,进而在直觉模糊标准子群的条件下,分别讨论了直积群上的直觉模糊正规子群、直觉模糊特征子群、直觉模糊共轭与其直觉模糊投影子群的关系.     

基于区间值模糊点和区间值模糊集邻属关系的区间值模糊子群

利用区间值模糊集的区间值水平截集的概念,给出了区间值模糊点与区间值模糊集邻属关系的定义,将这种邻属关系应用到区间值模糊代数的研究中,从而给出了(α,β)-区间值模糊子群的定义。通过研究16种(α,β)-区间值模糊子群,指出有意义的是(∈,∈)((∈,∈∨q),(∈∧q,∈))-区间值模糊子群。证明了群G的一个区间值模糊子集A为(∈,∈)((∈,∈∨q)或(∈∧q,∈))-区间值模糊子群的充要条件是对所有的λ=[a1,a2]≤[0.5,0.5],[0.5,0.5]μ=[b1,b2],其区间值水平截集Aλ和Aμ(Aλ或Aμ)为G的三值模糊子群。从而建立了基于区间值模糊点和区间值模糊集邻属关系的新的区间值模糊子群理论。     

群G的反模糊子群

给出一个群G的反模糊子群和正规反模糊子群的概念，这些定义不同于Rosenfeld和吴望名等的定义，本文还讨论了正规反模糊子群的一些性质及模糊商群等。     

具有极大循环子群的P-群的模糊子群构造北大核心

依据模糊子群与子群列之间的关系,通过分析具有极大循环子群的P-群的子群列的构造特点,给出了能够反映具有极大循环子群的P-群的模糊子群构造特征的模糊子群的阶、极大模糊子群和模糊子群的等价类数的计算公式.     

模糊粗糙子群

提出群中的模糊粗糙子群和模糊粗糙正规子群的概念。证明模糊子群的粗糙集是模糊子群，模糊正规子群的粗糙集是模糊正规子群。     

反模糊子群的运算

作为模糊代数的一个新领域,反模糊子群对模糊代数的研究至关重要.基于已有反模糊子群和反模糊正规子群的概念,对反模糊子群的相关性质进行了更为深入研究.首先给出了反模糊子群和反模糊正规子群的充要条件,其次给出了反模糊子群的余、逆、并、积、交等运算的定义,并讨论了这几种运算的性质,从而丰富了反模糊子群的研究.     

基于落影表现理论下的模糊子群

本文利用落影表现理论给出了一个群的模糊子群的两种定义，这两种模糊子群与现有的模糊子群有着十分密切的关系，从而为现有的模糊子群的定义提供了新的理论根据。通过本文的讨论便知：可以把落影表现理论应用到模糊代数的研究中去。     

有限群的Fuzzy次正规子群与Fuzzy极大子群

本文研究了有限群的Ｆ次正规子群，得出了一个Ｆ子群是Ｆ次正规子群的充要条件，讨论了Ｆ次正规子群的一些重要性质。另外，本文还引入了有限群的Ｆ极大子群的概念，给出了Ｆ子群是Ｆ极大群的充要条件。最后，给出了三个定理，讨论了有限群Ｇ可解、超可解、幂零与Ｇ的Ｆ次正规子群、Ｆ极大子群之间的联系。     

A characterization of fuzzy subgroups

An ordinary subgroup of a group G is (1) a subset of G, (2) closed under the group operation. In a fuzzy subgroup it is precisely these two notions that lose their deterministic character. A fuzzy subgroup μ of a group (G,·) associates with each group element a number, the larger the number the more certainly that element belongs to the fuzzy subgroup. The closure property is captured by the inequality μ(x · y)?T(μ(x), μ(y)). In A. Rosenfeld's original definition, T was the function ‘minimum’. However, any t-norm T provides a meaningful generalization of the closure property. Two classes of fuzzy subgroups are investigated. The fuzzy subgroups in one class are subgroup generated, those in the other are function generated. Each fuzzy subgroup in these classes satisfies the above inequality with T given by T(a, b) = max(a + b ?1, 0). While the two classes look different, each fuzzy subgroup in either is isomorphic to one in the other. It is shown that a fuzzy subgroup satisfies the above inequality with $\text{T =}\text{‘minimum’}$ if and only if it is subgroup generated of a very special type. Finally, these notions are applied to some abstract pattern recognition problems.     

Finite non-solvable groups in which the normalizer of every nonnormal cyclic subgroup is maximal

It is well known that the normality of subgroups plays an important part in the research of group theory. So it is reasonable to investigate the structure of a group by using normalizers of some kind of subgroups. In this paper we mainly investigate the structure of non-solvable groups in which the normalizer of every non-normal cyclic subgroup is a maximal subgroup, and finally the classification for this kind of groups is given.     

灰色数学的新分支——灰群

给出灰子半群、强灰子半群、灰子幺半群、强灰子幺半群、灰子群和强灰子群的定义和有关定理，在此基础上，讨论灰子群一模糊子群、一般子群的关系。并且研究了灰正规子群。     

Semidirect products OF fuzzy subgroups

The semidirect product of two groups is an important construction in group theory. In this paper we define the semidirect product of fuzzy subgroups and give conditions for it to be a fuzzy subgroup. This extends the work of H. Sherwood on direct products of fuzzy subgroups. We then give an example where a fuzzy subgroup of a semidirect product is a semidirect product of fuzzy subgroups.     

Bulletin

Anthony and Sherwood (1979, 1980) redefined the fuzzy subgroup, the idea introduced by Rosenfeld 1971. Foster 1979 introduced the concept of product fuzzy subgroups using Rosenfeld's definition. In this paper, the product fuzzy subgroups will be redefined according to Anthony and Sherwood's definition.     

Finite NPDM-groups

In this paper the classification is given for finite groups in which the normalizer of every non-normal cyclic subgroup of order divided by the minimal prime of|G|is a maximal subgroup.     

Fuzzy topology on fuzzy sets: Product fuzzy topology and fuzzy topological groups

The notion of product fuzzy topology in the case of fuzzy topology on fuzzy sets is introduced and the product invariance of fuzzy Hausdorffness, compactness, connectedness are examined. The product fuzzy topology is used to define fuzzy group topology on a fuzzy subgroup of a group G and some properties of fuzzy topological groups are obtained.