Fuzzy幂群
|
摘要
模糊数学的发展突出了集值映射的重要性，各种数学结构需要由论域向
其幂集上提升，如序结构的提升、可测结构的提升、拓扑结构的提升等等。
文考虑了代数结构的提升问题，首次提出了幂群的概念。文分别研究
了正规幂群和一致幂群。文分别研究了各种幂群的性质、结构、分类
和同态、同构关系。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。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。
本文在文的基础上进一步研究了Fuzzy幂群，讨论了Fuzzy幂群及其
性质，完整地研究了各种Fuzzy幂群的结构，对Fuzzy幂群进行了分类，并构
造了各类的子群列和正规子群列，进一步研究了Fuzzy幂群的同态，同构和
Fuzzy幂群的直积。
With developing fuzzy mathematics, the importance of set value mapping has
been highlighted, so that the upgrade of all kinds of the structures, such as or-
dered, topological measurable structure, etc, have been considered. The concept of
HX group was raised firstly in the [23 studied normal FIX group
and uniform I-{X group. The properties, structures, classifications, homomorphism
and isomorphism of all kinds of FIX group have been considered in the pa
t6J12J th
per . With e development of fuzzy set theory, all kinds of the structure
ar upgraded not only from their universes to their power sets but aTho from their
universes to their fuzzy power sets. In the paperJ, the concept of fuzzy power
group has been raised firstly and the structure and homomorphisms of fuzzy pow-
er group has been considered.
On the basis of paper], this paper gives some further description for fuzzy
power group. First, the properties of the fuzzy power group wiil be considered.
second, the structures of each fuzzy power group will be described systematically,
and attempts will be made to classify the fuzzy power groups and the chains of
subgroup and the chains of normal subgroup of each class will be constructed. Fi-
nally it homorphism, isomorphism and direct product will be considered.
其幂集上提升，如序结构的提升、可测结构的提升、拓扑结构的提升等等。
文考虑了代数结构的提升问题，首次提出了幂群的概念。文分别研究
了正规幂群和一致幂群。文分别研究了各种幂群的性质、结构、分类
和同态、同构关系。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。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。
本文在文的基础上进一步研究了Fuzzy幂群，讨论了Fuzzy幂群及其
性质，完整地研究了各种Fuzzy幂群的结构，对Fuzzy幂群进行了分类，并构
造了各类的子群列和正规子群列，进一步研究了Fuzzy幂群的同态，同构和
Fuzzy幂群的直积。
With developing fuzzy mathematics, the importance of set value mapping has
been highlighted, so that the upgrade of all kinds of the structures, such as or-
dered, topological measurable structure, etc, have been considered. The concept of
HX group was raised firstly in the [23 studied normal FIX group
and uniform I-{X group. The properties, structures, classifications, homomorphism
and isomorphism of all kinds of FIX group have been considered in the pa
t6J12J th
per . With e development of fuzzy set theory, all kinds of the structure
ar upgraded not only from their universes to their power sets but aTho from their
universes to their fuzzy power sets. In the paperJ, the concept of fuzzy power
group has been raised firstly and the structure and homomorphisms of fuzzy pow-
er group has been considered.
On the basis of paper], this paper gives some further description for fuzzy
power group. First, the properties of the fuzzy power group wiil be considered.
second, the structures of each fuzzy power group will be described systematically,
and attempts will be made to classify the fuzzy power groups and the chains of
subgroup and the chains of normal subgroup of each class will be constructed. Fi-
nally it homorphism, isomorphism and direct product will be considered.
引文
[1] 罗承忠，米洪海．Fuzzy幂群．模糊系统与数学，1994(1)
[2] 罗承忠．模糊集引论．北京：北京师范大学出版社，1989．
[3] Li Hong xing, Duan Qin zhi, Wang Pei zhuang, Hypergroups, BUSE-FAL(法刊), 1985(23) .
[4] 张振良，李洪兴，汪培庄，正规超群与商群的关系．数学季刊，1987 (3) ．
[5] 钟良彬．幂群的结构与相互关系．数学季刊，1990(4) ．
[6] Zhang Zhen ling. The properties of HX Groups. Italian Journal of Pure and Applied Mathematils (意刊), 1997(2) .
[7] Zhang Zhen liang. Classifications of HX Groups and Their Chair of normal subgroup. Italian Journal of pure and Applied Mathematils(意刊) ,1999．
[8] 张振良．幂群的结构与分类．纯粹数学与应用数学，1998(1) ．
[9] 张振良．幂群的结构与同态，同构关系．昆明理工大学学报，1998 (2) ．
[10] 李洪兴．正则HX群的同态与同构．模糊系统与数学，1990(1) ．
[11] 李洪兴，汪培庄．幂群．应用数学，1998(2) ．
[12] Zhang Zhen liang. Equivalent Conditions under which the Normal Hypergroups are quotient groups.BUSEFAL(法刊), 1986(31) .
[13] 钟良彬．Fuzzy幂群的基数定理．模糊系统与数学，1998(2) ．
[14] 林楠．幂群与拟商群．辽宁师范大学学报，1998(2) ．
[15] 孟道骥．Fuzzy群．模糊数学，1982(4) ．
[16] 冯克勤译．代数学．湖南教育出版社，1985．
[17] 张文修著．模糊数学基础．西安交通大学出版社，1984．
[18] 刘应明主编，刘旺金，何家儒著．模糊数学导论．四川教育出版 社，1992．
[19] 李洪兴，汪培庄编著．模糊数学．国防工业出版社，1994．
[20] 熊全淹．近世代数．武汉大学出版社，1995．
[21] 姚生．抽象代数学．复旦大学出版社，1997．
[2] 罗承忠．模糊集引论．北京：北京师范大学出版社，1989．
[3] Li Hong xing, Duan Qin zhi, Wang Pei zhuang, Hypergroups, BUSE-FAL(法刊), 1985(23) .
[4] 张振良，李洪兴，汪培庄，正规超群与商群的关系．数学季刊，1987 (3) ．
[5] 钟良彬．幂群的结构与相互关系．数学季刊，1990(4) ．
[6] Zhang Zhen ling. The properties of HX Groups. Italian Journal of Pure and Applied Mathematils (意刊), 1997(2) .
[7] Zhang Zhen liang. Classifications of HX Groups and Their Chair of normal subgroup. Italian Journal of pure and Applied Mathematils(意刊) ,1999．
[8] 张振良．幂群的结构与分类．纯粹数学与应用数学，1998(1) ．
[9] 张振良．幂群的结构与同态，同构关系．昆明理工大学学报，1998 (2) ．
[10] 李洪兴．正则HX群的同态与同构．模糊系统与数学，1990(1) ．
[11] 李洪兴，汪培庄．幂群．应用数学，1998(2) ．
[12] Zhang Zhen liang. Equivalent Conditions under which the Normal Hypergroups are quotient groups.BUSEFAL(法刊), 1986(31) .
[13] 钟良彬．Fuzzy幂群的基数定理．模糊系统与数学，1998(2) ．
[14] 林楠．幂群与拟商群．辽宁师范大学学报，1998(2) ．
[15] 孟道骥．Fuzzy群．模糊数学，1982(4) ．
[16] 冯克勤译．代数学．湖南教育出版社，1985．
[17] 张文修著．模糊数学基础．西安交通大学出版社，1984．
[18] 刘应明主编，刘旺金，何家儒著．模糊数学导论．四川教育出版 社，1992．
[19] 李洪兴，汪培庄编著．模糊数学．国防工业出版社，1994．
[20] 熊全淹．近世代数．武汉大学出版社，1995．
[21] 姚生．抽象代数学．复旦大学出版社，1997．