幂群所诱导的L－Fuzzy幂群

本文在幂群［１］的基础上，提出了幂群所诱导的Ｌ－Ｆｕｚｚｙ幂群、Ｌ－Ｆｕｚｚｙ幂群的λ－截、次幂群等—系列概念，并给出了幂群与其诱导的Ｌ－Ｆｕｚｚｙ幂群、λ截集群间的次同态、同态等关系。     

Fuzzy代数与Fuzzy商代数

本文更深入地讨论了Ｆｕｚｚｙ代数，Ｆｕｚｚｙ理想的性质，定义了Ｆｕｚｚｙ商代数，并证明了代数Ｙ关于代数Ｚ的同态ｆ－１［Ｏｚ］的Ｆｕｚｚｙ商代数与代数Ｚ同构等性质。     

幂群所诱导的L—Fuzzy幂群

本文在幂群［１］的基础上，提出了幂群所诱导的Ｌ－Ｆｕｚｚｙ幂群、Ｌ－Ｆｕｚｚｙ幂群的λ－截、次幂群等一系列概念，并给出了幂群与其诱导的Ｌ－Ｆｕｚｚｙ幂群、λ截集群间的次同态、同态等关系。     

关于“L—Fuzzy群同态与同构及其性质”一文的注记

文（１）给出了Ｌ－Ｆｕｚｚｙ群间的同态映射，满同态映射以及Ｌ－Ｆｕｚｚｙ群的Ｌ－正规子群等概念。本文指出以上三种定义的不合理性，并给出它们的合理性定义。     

Fuzzy子格与Fuzzy同态

每个代数结构一般都可引入相应的Ｆｕｚｚｙ子结构，为了进一步研究Ｆｕｚｚｙ子和以及Ｆｕｚｚｙ同态的各种性质，本文引入了亚格的概念，并建立了Ｆｕｚｚｙ子格与亚格的对应关系（定理１．７）；给出了Ｆｕｚｚｙ同态（同构）的分解定理（定理２．５和推论２．７）；最后讨论了同余关系商格的Ｆｕｚｚｙ子格问题，证明了在自然映射下，任一保序映射可以引导出商格中应的Ｆｕｚｚｙ子格，并由此引出了φ－Ｆｕｚｚｙ商子格的概念。     

L—Fuzzy群的同态与同构及其性质

本文在Ｌ－ｆｕｚｚｙ群的概念的基础上，用集合套概念定义了Ｌ－ｆｕｚｚｙ群间的同态与同构。进而给出了它们的一些性质。最后，我们给出了Ｌ－ｆｕｚｚｙ群的同态基本定理。     

Fuzzy幂群

文［１］讨论了代数结构的提升问题，提出了幂群的概念。模糊数学的发展要求各种数学结构不但要由论域向其幂集上提升，而且还要求向模糊幂集上提升，在更广泛的意义下讨论其结构问题，本文首先提出Ｆｕｚｚｙ幂群的概念，讨论了Ｆｕｚｚｙ幂群的结构及其同态问题。     

Fuzzy映射和乘积Fuzzy测度

从研究Ｆｕｚｚｙ集合的直积和Ｆｕｚｚｙ关系出发,讨论了Ｆｕｚｚｙ关系和Ｆｕｚｚｙ映射的一些性质,最后利用Ｆｕｚｚｙ关系的截影性质给出了乘积Ｈ型Ｆｕｚｚｙ测度的存在性定理。     

群的模糊同态与模糊商群的同构定理

利用模糊映射，给出群的模糊同态的概念，并得到模糊同态基本定理，同时建立模糊商群的同构定理。     

L-fuzzy商群的L-fuzzy同构定理

给出了L-fuzzy群的L-fuzzy同态基本定理。同时建立了L-fuzzy商群的L-fuzzy同态和L-fuzzy同构定理。     

交换群层的正合序列与同态定理

在交换群层（简称层）中给出了层的单（满）同态核与上核的泛性定理及正合交换图的一系列结论,进而证明了交换群层的同态（同构）定理。     

L-fuzzy子群的L-fuzzy同态

给出了L-fuzzy子幺半群的概念及刻画．给出了一种真正的L-fuzzy子幺半群间的Lfuzzy同态和L-fuzzy同构等概念．最后证明了L-fuzzy子群在L-fuzzy同态映射和L-fuzzy同构逆映射下不变．     

The structure and expression of power group

With the application of the topological upgrade, more and more experts are interested in the problems concerning the mathematical structure’s upgrade to its power set. This paper is a further study on the structure and interrelationship of a power group. By revealing the nature of relationship between the power group and a regular one, the present paper proves some theorems of the structure of the power group and constructs some theorems on its homomorphism and isomorphism as well, proposing an expression form and isomorphism replacement construct methods of the power group.     

HX环的存在性

自李洪兴１９９１年提出了ＨＸ环这一问题以来,一直有这么一个问题没有解决,即是否存在非平凡的ＨＸ环的例子？但至今既没有找到非平凡的ＨＸ环,也没有证明任一环Ｒ上仅存在平凡的ＨＸ环,钟育彬１９９５年证明了一类环上仅存在平凡的ＨＸ环,那么,是否所有环仅存在平凡的ＨＸ环呢？针对这一点,本文作进上步研究,构造了一个非平凡的ＨＸ环,并建立了ＨＸ环上的同态与同构定理,为进上步研究ＨＸ环奠定了基础。     

Pro-homology and isomorphisms of pro-groups

We extend the scope of some useful theorems of E. Dror and J. Stallings of which the following result of Stallings is easy to state: A homomorphism θ: A→B of nilpotent groups is an isomorphism if and only if θ induces an isomorphism and an epimorphism on the first and the second homology groups with integer coefficients, respectively. We extend and enlarge the algebraic framework to the category of pro-groups in a natural way so that an anaogue of these results of Dror and Stallings can be appropriately stated and proved. This is done in a natural manner so that our extended results cań be useful in pro-homotopy theory or shape theory which is analogous to the usefulness of the results of Dror and Stallings in the context of homotopy theory. As an application, we obtain a version of a Whitehead Theorem in pro-homotopy and shape theory which only involves homology pro-groups; a complete proof of this result will be given elsewhere.     

L-Fuzzy代数的L-Fuzzy代数同态(英文)

本文研究了L-fuzzy代数的L-fuzzy代数同态的定义及其性质.利用L-fuzzy集的水平截集,得到TL-fuzzy代数的L-fuzzy代数同态的若干等价刻画.     

On Subgroups and Quotient Groups of L-fuzzy Topological Groups

In paper [1], it gave the concept and basic properties of L-fuzzy topological groups. This paper will enter the concept of quotient groups of L-fuzzy topological groups and will prove some properties of subgroups and quotient groups. The persent work is the continuation of [1] and hence all the conventions in [1] still hold good in the present paper.     

Conceptual metaphors for isomorphism and homomorphism: Instructors’ descriptions for themselves and when teaching

Group isomorphism and homomorphism are topics central to abstract algebra, yet research on instructors’ views of these concepts is limited. Based on interviews from two instructors as well as classroom video from eight class periods, this paper examines the language used to discuss isomorphism and homomorphism. Language used by instructors in interviews and classroom settings are identified and classified into four main categories: formal definition, mapping, sameness, and combinations of sameness and mapping language. How the two instructors drew on language classified into those four categories in the interview and instruction settings are examined for isomorphism and homomorphism. Similarities and differences between the interview and instruction contexts reveal the wide variety of ways of understanding isomorphism and homomorphism as well as a research need to examine mathematicians’ content knowledge in more than one context.