Fuzzy幂环
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摘要
随着社会和科学技术更加进步、发展，人类进入了“自然——社会——思维”的认知阶段的更复杂的社会。人们要求数学也能处理更为复杂的不确定现象，特别是人文科学、社会科学和思维科学中的不确定现象，也要求计算机像人脑一样， 能自行识别和处理客观世界中的不确定问题。
美国著名的电子工程学家和控制论专家扎德(L.A.Zadeh)是位很有见识的科学家，他正视并为解决这类问题于1965年，发表了《模糊集合》一文，大胆地对现代数学的基石—集合论进行修改和扩充，提出了用模糊集合(Fuzzy set)作为表现模糊事物的数学模型。宣告了新兴学科—模糊数学的诞生。
模糊数学将沿着两条途径发展：一方面是研究模糊性的内在规律，也就是探讨模糊语言和模糊逻辑。在这个方向上，模糊数学与人工智能、知识工程、专家系统等分支的有机结合，以增进电脑活性，更好地模拟人的思维。另一方面是把模糊集合当作一个能概括更加多样化数学概念框架，建立处理模糊现象的确切性的数学理论，以拓广数学基础，使经典数学的若干方面在更广阔更深刻的意义下向前推进。
模糊数学理论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，拓扑结构的提升，可测结构的提升等等。文考虑了代数结构的提升问题，首次提出了幂环的概念。文分别研究了正规幂环和一致幂环。文研究了各种幂环的性质，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。文提出Fuzzy幂群的概念讨论了Fuzzy幂群的结构及其同态问题。文分别讨论了Fuzzy幂群的性质，结构及分类。
作者由代数学中群，环的结构得出启发，在本文中首次提出Fuzzy幂环的概念，讨论了Fuzzy幂环的及其性质，完整地研究了各种Fuzzy幂环的结构，对Fuzzy幂环进行了分类，并构造了各类的子环列，进一步研究了Fuzzy幂环的同态，同构。
With a further progress and development of society and science & technology, we have come into a more complicated society with a recognizant stage characterized by "nature - society - thought" . Mathematics is necessitated to dealing with more complicated and indefinite phenomenon, especially in field of humanism science, social science and thinking science. Similarly, people need computer to automatically recognize and manage indefinite phenomenon in impersonal world, like human cerebra.
L. A. Zadeh, a famous American electronic engineering and cybernetics scientist with profound insight, played emphasis on this kind of problem and published <> for resolving it in 1965. He amended and extended set theory - the basis of modern mathematics with a bold hand and brought forward the thinking of using Fuzzy set as a mathematical model to representing fuzzy business. It announced the birth of a new subject - Fuzzy mathematics.
Fuzzy mathematics will grow along two directions in the future: first, it will research the inner rule of fuzzy character, i. e the fuzzy language and fuzzy logic. In this direction, computer would be more active and perform more excellent in simulating human thinking through the combination between fuzzy mathematics and some branches such as artificial intelligence, knowledge engineering and expert system. Second, we will regard fuzzy set as a frame that can generalize more kind of mathematical concept and establish definite mathematical theory which can deal with fuzzy phenomenon to extend mathematical basis. Consequently, some aspects of classic mathematics will be advanced more widely and more profoundly.
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 ordered, topological measurable structure, etc, have been considered . The concept of HX ring was raised firstly in the paper[3] Paper[6][7]studied normal HX ring and uniform HX ring . The properties, structures, classifications, homomorphism and isomorphism of all kinds of HX ring have been considered in the paper [1] . With the development of fuzzy set theory, all kinds of the structure are upgraded not only from their universes to their power sets but also from their universes to their fuzzy power sets. In the paper[5], the concept of fuzzy power group was raised and in paper[8][9] the structure and homomorphism of fuzzy power group was considered.
Enlightened by the structures of group and ring in algebra, in this paper , Author firstly raise the concept of fuzzy power ring . First, the properties of the fuzzy power ring will be considered. Second, the structure of each fuzzy power ring will be described systematically, and attempts will bo made to classify the fuzzy power ring and the chains of sub rings will bo constructed. Finally it's homoinorphism, isonmrphism and direct product will be considert-d.
美国著名的电子工程学家和控制论专家扎德(L.A.Zadeh)是位很有见识的科学家，他正视并为解决这类问题于1965年，发表了《模糊集合》一文，大胆地对现代数学的基石—集合论进行修改和扩充，提出了用模糊集合(Fuzzy set)作为表现模糊事物的数学模型。宣告了新兴学科—模糊数学的诞生。
模糊数学将沿着两条途径发展：一方面是研究模糊性的内在规律，也就是探讨模糊语言和模糊逻辑。在这个方向上，模糊数学与人工智能、知识工程、专家系统等分支的有机结合，以增进电脑活性，更好地模拟人的思维。另一方面是把模糊集合当作一个能概括更加多样化数学概念框架，建立处理模糊现象的确切性的数学理论，以拓广数学基础，使经典数学的若干方面在更广阔更深刻的意义下向前推进。
模糊数学理论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，拓扑结构的提升，可测结构的提升等等。文考虑了代数结构的提升问题，首次提出了幂环的概念。文分别研究了正规幂环和一致幂环。文研究了各种幂环的性质，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。文提出Fuzzy幂群的概念讨论了Fuzzy幂群的结构及其同态问题。文分别讨论了Fuzzy幂群的性质，结构及分类。
作者由代数学中群，环的结构得出启发，在本文中首次提出Fuzzy幂环的概念，讨论了Fuzzy幂环的及其性质，完整地研究了各种Fuzzy幂环的结构，对Fuzzy幂环进行了分类，并构造了各类的子环列，进一步研究了Fuzzy幂环的同态，同构。
With a further progress and development of society and science & technology, we have come into a more complicated society with a recognizant stage characterized by "nature - society - thought" . Mathematics is necessitated to dealing with more complicated and indefinite phenomenon, especially in field of humanism science, social science and thinking science. Similarly, people need computer to automatically recognize and manage indefinite phenomenon in impersonal world, like human cerebra.
L. A. Zadeh, a famous American electronic engineering and cybernetics scientist with profound insight, played emphasis on this kind of problem and published <
Fuzzy mathematics will grow along two directions in the future: first, it will research the inner rule of fuzzy character, i. e the fuzzy language and fuzzy logic. In this direction, computer would be more active and perform more excellent in simulating human thinking through the combination between fuzzy mathematics and some branches such as artificial intelligence, knowledge engineering and expert system. Second, we will regard fuzzy set as a frame that can generalize more kind of mathematical concept and establish definite mathematical theory which can deal with fuzzy phenomenon to extend mathematical basis. Consequently, some aspects of classic mathematics will be advanced more widely and more profoundly.
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 ordered, topological measurable structure, etc, have been considered . The concept of HX ring was raised firstly in the paper[3] Paper[6][7]studied normal HX ring and uniform HX ring . The properties, structures, classifications, homomorphism and isomorphism of all kinds of HX ring have been considered in the paper [1] . With the development of fuzzy set theory, all kinds of the structure are upgraded not only from their universes to their power sets but also from their universes to their fuzzy power sets. In the paper[5], the concept of fuzzy power group was raised and in paper[8][9] the structure and homomorphism of fuzzy power group was considered.
Enlightened by the structures of group and ring in algebra, in this paper , Author firstly raise the concept of fuzzy power ring . First, the properties of the fuzzy power ring will be considered. Second, the structure of each fuzzy power ring will be described systematically, and attempts will bo made to classify the fuzzy power ring and the chains of sub rings will bo constructed. Finally it's homoinorphism, isonmrphism and direct product will be considert-d.
引文
[1] Zhang zhenliang. Structures and classifications of HX ring[J]. Italian Journal of pure and Applied Mathematics, 2001(11).131—142.
[2] 钟育彬．Fuzzy幂群的基数定理[J]．模糊系统与数学，1998(1)，61，69．
[3] 李洪兴．HX环[J]．数学季刊，1991(1)，17-21．
[4] 姚炳学，李洪兴．幂环[J]．模糊系统与数学，2000(2)，15-19．
[5] 罗承忠，米洪海．Fuzzy幂群[J]．模糊系统与数学，1994(1)，1-7．
[6] 张振良．正规幂环和一致幂环[J]．纯粹数学与应用数学，2001(1)，6-13．
[7] 李洪兴．正则HX环的同态与同构[J]．模糊系统与数学，1990(1)：1—7．
[8] 杨培亮，张振良．F幂群的分类[J]．模糊系统与数学，2002(4)
[9] 杨培亮，张振良．F幂群的性质与结构[J]．纯粹数学与应用数学，2002(3)．
[10] 罗承忠．模糊集引论[M]．北京师范大学出版社，1989．
[11] T.W.Hungerford，代数学，冯克勤译，湖南教育出版社，1985．
[12] 熊全淹．近世代数．武汉大学出版社，1995．
[13] 姚生，抽象代数学，复旦大学出版社，1997．
[14] 李洪兴，汪培庄编著．模糊数学．国防工业出版社，1994．
[15] 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．
[16] 张文修著．模糊数学基础．西安交通大学出版社，1984．
[17] 孟道骥．Fuzzy群．模糊数学，1982(4)．
[18] 林楠．幂群与拟商群．辽宁师范大学出版社，1998(2)．
[19] Zhang Zhenliang. Equivalent Conditions under Which the Normal Hypergrops are quotient groups. BUSEFAL(法刊), 1987(31).
[20] 李洪兴，汪培庄．幂群．应用数学．1998(2)．
[21] 李洪兴．正则HX群的同态与同构．模糊系统与数学，1990(1)．
[22] 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)．
[23] 张振良．幂群的结构与分类．纯粹数学与应用数学，1998(1)．
[24] Zhang Zhenliang. The properties of HX Groups. Italian Journal of Pure and Applied Mathematics(意刊), 1997 (2).97—106.
[25] Zhang Zhen liang. Classification of HX Groups and Their Chair of normal subgroup. Italian Journal of Pure and Applied Mathematils(意刊), 1999 (5)125-134.
[26] 钟育彬．幂群的结构与相互关系．数学季刊，1990(4)．
[27] 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)．
[28] Li Hongxing, Duan Qinzhi, Wang Pei zhuang, Hypergroups, BUSEFAL(法刊), 1985 (23)
[29] 罗承忠．模糊集引论．北京：北京师范大学出版社，1989．
[30] 段钦治．超代数结构的几个问题[J]．纺织高校基础科学学报，1996(2)：43—45．
[31] Zhong Yubin. The Structure of Hypergroup on cyclical Group[J]. Chinese Quarterly Journal of Mathematics, 1994(1):26-31.
[32] 钟育彬．一类环上的HX环的结构[J]．模糊系统与数学，1995(4)：73—77．
[33] 李洪兴，段钦治．基于幂群的超拓扑群[J]．北京师范大学出版社，1996(1)：7—9．
[34] LI Hongxing. HX ring[J]. BUSEFAL, 1988,34:3-8.
[35] LI Hongxing. HX group[J]. BUSEFAL, 33,1988.
[36] Carl Faith, Algebra 2, Ring Theory, Springer-Verlag, Berling Heidelberg New York, 1976.
[37] 张诚一，党平安．Fuzzy幂群的基数及表示[J]．模糊系统与数学，2001(4)．
[38] M．赫尔，《群论》，裘光明译，科学出版社，1981．
[39] 钟育彬．幂集的同构关系．数学季刊，1990(4)．
[40] 李洪兴，罗承忠，汪培庄，袁学海．Fuzzy的基数．模糊系统与数学．1999(3)．
[41] Dubois D,Prade H. Fuzzy sets and systems. Academic Press Inc.(London)LTD.1980.
[42] 马骥良，于纯海，Fuzzy环(1)，东北师大学报(自然科学版)，1 (1982)23—28．
[2] 钟育彬．Fuzzy幂群的基数定理[J]．模糊系统与数学，1998(1)，61，69．
[3] 李洪兴．HX环[J]．数学季刊，1991(1)，17-21．
[4] 姚炳学，李洪兴．幂环[J]．模糊系统与数学，2000(2)，15-19．
[5] 罗承忠，米洪海．Fuzzy幂群[J]．模糊系统与数学，1994(1)，1-7．
[6] 张振良．正规幂环和一致幂环[J]．纯粹数学与应用数学，2001(1)，6-13．
[7] 李洪兴．正则HX环的同态与同构[J]．模糊系统与数学，1990(1)：1—7．
[8] 杨培亮，张振良．F幂群的分类[J]．模糊系统与数学，2002(4)
[9] 杨培亮，张振良．F幂群的性质与结构[J]．纯粹数学与应用数学，2002(3)．
[10] 罗承忠．模糊集引论[M]．北京师范大学出版社，1989．
[11] T.W.Hungerford，代数学，冯克勤译，湖南教育出版社，1985．
[12] 熊全淹．近世代数．武汉大学出版社，1995．
[13] 姚生，抽象代数学，复旦大学出版社，1997．
[14] 李洪兴，汪培庄编著．模糊数学．国防工业出版社，1994．
[15] 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．
[16] 张文修著．模糊数学基础．西安交通大学出版社，1984．
[17] 孟道骥．Fuzzy群．模糊数学，1982(4)．
[18] 林楠．幂群与拟商群．辽宁师范大学出版社，1998(2)．
[19] Zhang Zhenliang. Equivalent Conditions under Which the Normal Hypergrops are quotient groups. BUSEFAL(法刊), 1987(31).
[20] 李洪兴，汪培庄．幂群．应用数学．1998(2)．
[21] 李洪兴．正则HX群的同态与同构．模糊系统与数学，1990(1)．
[22] 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)．
[23] 张振良．幂群的结构与分类．纯粹数学与应用数学，1998(1)．
[24] Zhang Zhenliang. The properties of HX Groups. Italian Journal of Pure and Applied Mathematics(意刊), 1997 (2).97—106.
[25] Zhang Zhen liang. Classification of HX Groups and Their Chair of normal subgroup. Italian Journal of Pure and Applied Mathematils(意刊), 1999 (5)125-134.
[26] 钟育彬．幂群的结构与相互关系．数学季刊，1990(4)．
[27] 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)．
[28] Li Hongxing, Duan Qinzhi, Wang Pei zhuang, Hypergroups, BUSEFAL(法刊), 1985 (23)
[29] 罗承忠．模糊集引论．北京：北京师范大学出版社，1989．
[30] 段钦治．超代数结构的几个问题[J]．纺织高校基础科学学报，1996(2)：43—45．
[31] Zhong Yubin. The Structure of Hypergroup on cyclical Group[J]. Chinese Quarterly Journal of Mathematics, 1994(1):26-31.
[32] 钟育彬．一类环上的HX环的结构[J]．模糊系统与数学，1995(4)：73—77．
[33] 李洪兴，段钦治．基于幂群的超拓扑群[J]．北京师范大学出版社，1996(1)：7—9．
[34] LI Hongxing. HX ring[J]. BUSEFAL, 1988,34:3-8.
[35] LI Hongxing. HX group[J]. BUSEFAL, 33,1988.
[36] Carl Faith, Algebra 2, Ring Theory, Springer-Verlag, Berling Heidelberg New York, 1976.
[37] 张诚一，党平安．Fuzzy幂群的基数及表示[J]．模糊系统与数学，2001(4)．
[38] M．赫尔，《群论》，裘光明译，科学出版社，1981．
[39] 钟育彬．幂集的同构关系．数学季刊，1990(4)．
[40] 李洪兴，罗承忠，汪培庄，袁学海．Fuzzy的基数．模糊系统与数学．1999(3)．
[41] Dubois D,Prade H. Fuzzy sets and systems. Academic Press Inc.(London)LTD.1980.
[42] 马骥良，于纯海，Fuzzy环(1)，东北师大学报(自然科学版)，1 (1982)23—28．