模糊等价关系下的模糊粗糙群
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摘要
将粗糙结构与经典代数结构，拓扑结构，序结构不断整合必将涌现出新的富有生机的数学分支，目前粗糙结构与经典代数结构结合起来的研究已有文章出现。Kuroki N研究了半群中的粗理想。首次提出了粗子半群和粗理想的概念，证明了同余关系下，半群的粗糙集是半群，左(右，双)理想的粗糙集是左(右，双)理想，并首次提出了粗子群和正规粗子群的概念，证明了在经典群中一固定的经典的正规子群所决定的同余关系下，子群的粗糙集是子群，正规子群的粗糙集是正规子群。粗糙结构与模糊代数结构结合起来的研究也已有文章出现，如张京玲研究了，在一经典群中一固定的经典正规子群所决定的同余关系下，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。
本文在以上的研究的基础之上，首次研究了在模糊等价关系下，粗糙集理论与模糊代数结构的关系，得出了在一经典群中一固定的模糊正规子群决定一个模糊等价关系，两个模糊正规子群决定的两个模糊等价关系的合成等于这两个模糊正规子群的积所决定的模糊等价关系，在一经典群中一固定的模糊正规子群所决定的模糊等价关系下，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，和模糊粗糙集的一些有意义的结论。
在此基础之上本文又首次提出了T模糊粗子群的概念。T模糊正规子群决定一个T模糊等价关系，两个T模糊正规子群决定的两个T模糊等价关系的合成等于这两个T模糊正规子群的积所决定的T模糊等价关系，并证明了在经典群中一固定的T模糊正规子群所决定的T模糊等价关系下，与模糊粗糙集相似的结论。
Connected Rough structure with Algebra structure, Topology structure and order structure, many new prosperous mathematics branches will appear currently, there have been some articles on connecting Rough structure with Algebra structure. Rough ideals in a semigroups, Rough ideals been first introduced by KuroKi N. Under the condition of the congruence relation, Rough sets of a subsemigroup was proved to be its subsemigroup , while that of a left (right, besides ) ideals was also proved to be its a left (right, besides ) ideals, next, Rough sets in a group first introduced. Under the condition of the congruence relation determined by a given normal subgroup in a group, Rough sets of a subgroup was proved to be its subgroup, while that of a normal one was also proved to be its normal subgroup. Connected Rough structure with Fuzzy Algebra structure, there also have been some articles on connecting Rough structure with Fuzzy Algebra structure by zhang Jinglin. Under the condition of the congruence relation determined by a given Fuzzy normal group in a group, Rough sets of a Fuzzy subgroup was proved to be its Fuzzy subgroup, while that of a Fuzzy normal group was also proved to be its a Fuzzy normal group.
In this paper, under the condition of the Fuzzy congruence relation determined by a given Fuzzy normal subgroup in a group, Connecting Rough structure with Algebra structure was discussed. It is proved that Under the condition of the fuzzy congruence relation determined by a given Fuzzy normal subgroup in a group, Rough sets of a Fuzzy subgroup was a Fuzzy subgroup, while that of a Fuzzy normal group was also proved to be its a Fuzzy normal group. Next T fuzzy congruence relation, T fuzzy Rough sets properties been first introduced in this paper and get some similar results.
本文在以上的研究的基础之上，首次研究了在模糊等价关系下，粗糙集理论与模糊代数结构的关系，得出了在一经典群中一固定的模糊正规子群决定一个模糊等价关系，两个模糊正规子群决定的两个模糊等价关系的合成等于这两个模糊正规子群的积所决定的模糊等价关系，在一经典群中一固定的模糊正规子群所决定的模糊等价关系下，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，和模糊粗糙集的一些有意义的结论。
在此基础之上本文又首次提出了T模糊粗子群的概念。T模糊正规子群决定一个T模糊等价关系，两个T模糊正规子群决定的两个T模糊等价关系的合成等于这两个T模糊正规子群的积所决定的T模糊等价关系，并证明了在经典群中一固定的T模糊正规子群所决定的T模糊等价关系下，与模糊粗糙集相似的结论。
Connected Rough structure with Algebra structure, Topology structure and order structure, many new prosperous mathematics branches will appear currently, there have been some articles on connecting Rough structure with Algebra structure. Rough ideals in a semigroups, Rough ideals been first introduced by KuroKi N. Under the condition of the congruence relation, Rough sets of a subsemigroup was proved to be its subsemigroup , while that of a left (right, besides ) ideals was also proved to be its a left (right, besides ) ideals, next, Rough sets in a group first introduced. Under the condition of the congruence relation determined by a given normal subgroup in a group, Rough sets of a subgroup was proved to be its subgroup, while that of a normal one was also proved to be its normal subgroup. Connected Rough structure with Fuzzy Algebra structure, there also have been some articles on connecting Rough structure with Fuzzy Algebra structure by zhang Jinglin. Under the condition of the congruence relation determined by a given Fuzzy normal group in a group, Rough sets of a Fuzzy subgroup was proved to be its Fuzzy subgroup, while that of a Fuzzy normal group was also proved to be its a Fuzzy normal group.
In this paper, under the condition of the Fuzzy congruence relation determined by a given Fuzzy normal subgroup in a group, Connecting Rough structure with Algebra structure was discussed. It is proved that Under the condition of the fuzzy congruence relation determined by a given Fuzzy normal subgroup in a group, Rough sets of a Fuzzy subgroup was a Fuzzy subgroup, while that of a Fuzzy normal group was also proved to be its a Fuzzy normal group. Next T fuzzy congruence relation, T fuzzy Rough sets properties been first introduced in this paper and get some similar results.
引文
[1] PAWLAKZ RoughSet,[J] Comput Inform Sci, 1982, 11:341-356
[2] MORST N N, YAKOUT M M Axiomatics for fuzzy rough sets[J], Fuzzy Set and Systems, 1988, 100:327-342
[3] Chen De-Gang Properties of the T-fuzzy factor groups[J] Fuzzy Set and Systems, 1998, 99:187-189
[4] The Lower and Upper ApproxIMAtionS In a Fuzzy Groups[J] Intelligent Systems[J] 203-220
[5] LI Yong-Hua, XU Cheng-Xian Fuzzy Groups Congruences On a Semigroup[J] 模糊系统与数学 2000, Vol, 16, No4 57-62
[6] Chmielewski M R, Grzymala-BussE J W.Global Discretization of continuous attributes as preprocessing for machine learing,J.international of approximate Reasoning,1996,15:319-331
[7] 张文修 粗糙集理论与方法[M] 科学出版社2001版
[8] 罗承中 模糊集引论[M] 北师大出版社
[9] Jacobson N, 基础代数,M.北京: 高等教育出版社,1987
[10] Pomerol J C,. Artificial intelligence and human decision making[J]. European Journal of Operational Research,1997,99:3-25
[11] Chanas S, Kuchta D. Further remarks on the relation between rough and fuzzy sets[J] Fuzzy Sets and Systems, 1992,47:391-394
[12] Yao Y Y, Relational interpretations of neighborhood operators and rough set approximation[J] Information Sciences,1998,111:239-259
[13] Yao Y Y, Lingras P, Interpretations of belief functions in the theory of rough set[J] Informa- tion Sciences,1998,104:81-106
[14] Yao Y Y, Lin T Y, Gerneralization of rough sets using modal logic[J] Intelligent Automation and Soft Computing, 1996,2:103-120
[15] Nanda S,Majumdar S, Fuzzy rough sets[J] Fuzzy Sets and Systems,
1992,45:157-160
[16] Beaubouef T, Petry F, Arora G, Information -theoretic measures of uncertainty for rough sets and rough relational database[J] Information Sciences, 1998, 109:185-195
[17J Yao Y Y, Constructive and algebraic methods of the theory of rough sets[J] Information Sciences, 1998, 109:21-47
[18] Kuroki N Rough ideals in semigroups[J] Information Sciences, 1997,100:139-163
[19] Lin T Y. A rough logic formalism for fuzzy controllers: A hard and soft computing view[J] International Journal of Approximate Reasoning 1996, 15:395-411
[20] Banerjee M,Pal S K, Roughness of a fuzzy set[J] Information Sciences, 1996, 93:235-246
[21] Dubois D,Prade H Rough fuzzy sets and fuzzy rough sets[J] Int.J.Genneral Systems, 1990, 17:191-209
[22] Krolikowski R, Czyzewski A, Noise Reduction in Telecommunication Channela Using Rough Sets and Neural Networks[J] 7th International Workshop on New Directions in Rough Sets, Data Minong, and Granular- Soft Computing, Yamaguchi, Japan, 1999, 100-108
[23] 石峰 娄臻亮 张永清 一种改进的粗糙集属性约简启发式算法[J]上海交通大学学报 2002 Vol．36 No．4：478-481
[24] 石峰 娄臻亮 张永清 陆金桂 基于模糊-粗糙集模型的一种归纳学习方法[J] 上海交通大学学报 2002 Vol．36 No．7：920-924
[25] 施恩伟 粗糙集中不可分辨关系的某些性质[J] 科学通报 (英文辑)1990 35：338-341
[26] 张梅 李怀祖 张文修 Fuzzy 信息系统的 Rough集理论[J] 模糊系统与数学 2002 Vol．16 N0．3：44-49
[27] 张文修 吴伟志 梁吉业 李德玉 粗糙集理论与方法[M] 科学出版社2001年
[28] 李兵 吴孟达 粗糙集研究中的模糊集方法[J] 模糊系统与数学 2002 Vol．16 No．2：69-73
[29] 米据生，张文修 基于粗糙集的间接学习[J] 计算机科学 2002 Vol．29 No．6：96-97
[30] 陈遵德 Rough Set 神经网络智能系统及其应用[J] 模式识别与人工智能1999，12(1)：1-5
[31] 王珏 苗夺谦 周育健 关于Rough Set理论与应用的综述[J] 模式识别与人工智能 1996 Vol．9 No．4：337-344
[32] 张文修 吴伟志 粗糙集理论介绍和研究综述[J] 模糊系统与数学2000 Vol．14 No．4：1-12
[33] 汪培庄 模糊集合论与应用[M] 上海科学技术出版社 1983年
[34] 王国胤 Rough 集理论与知识获取 西安交通大学出版社 2001年
[35] 刘文奇 吴从炘 相似关系粗集理论与相似关系信息系统[J] 模糊系统与数学 2002 Vol．16 No．3：50-58
[36] 罗承忠 模糊集引论(上册)[M] 北师大出版社
[37] 王亚英 张春慨 邵惠鹤 变论域知识约简算法[J] 上海交通大学学报2002 Vol．36 No．4：566-569
[38] 李玉榕 乔斌 蒋静坪 粗糙集理论中不确定性的粗糙信息熵表示[J] 计算机科学 2002 Vol．29 No．5：101-103
[39] 陈德刚 张文修 Palawk粗糙集模型的随机集表示及合成[J] 计算机科学 2002 Vol．29 No．10：18-19
[40] 张文修 吴伟志 基于随机集的粗糙集模型[J] 西安交通大学学报2000 34(12)：15-19
[41] 程畎 莫智文 模糊粗糙集及粗糙模糊集的模糊度[J] 模糊系统与数学2001 Vol．15 No．3：15-18
[42] 王基一 许黎明 概率粗糙集模型[J] 计算机科学 2002 Vol．29 No．8：76-78
[2] MORST N N, YAKOUT M M Axiomatics for fuzzy rough sets[J], Fuzzy Set and Systems, 1988, 100:327-342
[3] Chen De-Gang Properties of the T-fuzzy factor groups[J] Fuzzy Set and Systems, 1998, 99:187-189
[4] The Lower and Upper ApproxIMAtionS In a Fuzzy Groups[J] Intelligent Systems[J] 203-220
[5] LI Yong-Hua, XU Cheng-Xian Fuzzy Groups Congruences On a Semigroup[J] 模糊系统与数学 2000, Vol, 16, No4 57-62
[6] Chmielewski M R, Grzymala-BussE J W.Global Discretization of continuous attributes as preprocessing for machine learing,J.international of approximate Reasoning,1996,15:319-331
[7] 张文修 粗糙集理论与方法[M] 科学出版社2001版
[8] 罗承中 模糊集引论[M] 北师大出版社
[9] Jacobson N, 基础代数,M.北京: 高等教育出版社,1987
[10] Pomerol J C,. Artificial intelligence and human decision making[J]. European Journal of Operational Research,1997,99:3-25
[11] Chanas S, Kuchta D. Further remarks on the relation between rough and fuzzy sets[J] Fuzzy Sets and Systems, 1992,47:391-394
[12] Yao Y Y, Relational interpretations of neighborhood operators and rough set approximation[J] Information Sciences,1998,111:239-259
[13] Yao Y Y, Lingras P, Interpretations of belief functions in the theory of rough set[J] Informa- tion Sciences,1998,104:81-106
[14] Yao Y Y, Lin T Y, Gerneralization of rough sets using modal logic[J] Intelligent Automation and Soft Computing, 1996,2:103-120
[15] Nanda S,Majumdar S, Fuzzy rough sets[J] Fuzzy Sets and Systems,
1992,45:157-160
[16] Beaubouef T, Petry F, Arora G, Information -theoretic measures of uncertainty for rough sets and rough relational database[J] Information Sciences, 1998, 109:185-195
[17J Yao Y Y, Constructive and algebraic methods of the theory of rough sets[J] Information Sciences, 1998, 109:21-47
[18] Kuroki N Rough ideals in semigroups[J] Information Sciences, 1997,100:139-163
[19] Lin T Y. A rough logic formalism for fuzzy controllers: A hard and soft computing view[J] International Journal of Approximate Reasoning 1996, 15:395-411
[20] Banerjee M,Pal S K, Roughness of a fuzzy set[J] Information Sciences, 1996, 93:235-246
[21] Dubois D,Prade H Rough fuzzy sets and fuzzy rough sets[J] Int.J.Genneral Systems, 1990, 17:191-209
[22] Krolikowski R, Czyzewski A, Noise Reduction in Telecommunication Channela Using Rough Sets and Neural Networks[J] 7th International Workshop on New Directions in Rough Sets, Data Minong, and Granular- Soft Computing, Yamaguchi, Japan, 1999, 100-108
[23] 石峰 娄臻亮 张永清 一种改进的粗糙集属性约简启发式算法[J]上海交通大学学报 2002 Vol．36 No．4：478-481
[24] 石峰 娄臻亮 张永清 陆金桂 基于模糊-粗糙集模型的一种归纳学习方法[J] 上海交通大学学报 2002 Vol．36 No．7：920-924
[25] 施恩伟 粗糙集中不可分辨关系的某些性质[J] 科学通报 (英文辑)1990 35：338-341
[26] 张梅 李怀祖 张文修 Fuzzy 信息系统的 Rough集理论[J] 模糊系统与数学 2002 Vol．16 N0．3：44-49
[27] 张文修 吴伟志 梁吉业 李德玉 粗糙集理论与方法[M] 科学出版社2001年
[28] 李兵 吴孟达 粗糙集研究中的模糊集方法[J] 模糊系统与数学 2002 Vol．16 No．2：69-73
[29] 米据生，张文修 基于粗糙集的间接学习[J] 计算机科学 2002 Vol．29 No．6：96-97
[30] 陈遵德 Rough Set 神经网络智能系统及其应用[J] 模式识别与人工智能1999，12(1)：1-5
[31] 王珏 苗夺谦 周育健 关于Rough Set理论与应用的综述[J] 模式识别与人工智能 1996 Vol．9 No．4：337-344
[32] 张文修 吴伟志 粗糙集理论介绍和研究综述[J] 模糊系统与数学2000 Vol．14 No．4：1-12
[33] 汪培庄 模糊集合论与应用[M] 上海科学技术出版社 1983年
[34] 王国胤 Rough 集理论与知识获取 西安交通大学出版社 2001年
[35] 刘文奇 吴从炘 相似关系粗集理论与相似关系信息系统[J] 模糊系统与数学 2002 Vol．16 No．3：50-58
[36] 罗承忠 模糊集引论(上册)[M] 北师大出版社
[37] 王亚英 张春慨 邵惠鹤 变论域知识约简算法[J] 上海交通大学学报2002 Vol．36 No．4：566-569
[38] 李玉榕 乔斌 蒋静坪 粗糙集理论中不确定性的粗糙信息熵表示[J] 计算机科学 2002 Vol．29 No．5：101-103
[39] 陈德刚 张文修 Palawk粗糙集模型的随机集表示及合成[J] 计算机科学 2002 Vol．29 No．10：18-19
[40] 张文修 吴伟志 基于随机集的粗糙集模型[J] 西安交通大学学报2000 34(12)：15-19
[41] 程畎 莫智文 模糊粗糙集及粗糙模糊集的模糊度[J] 模糊系统与数学2001 Vol．15 No．3：15-18
[42] 王基一 许黎明 概率粗糙集模型[J] 计算机科学 2002 Vol．29 No．8：76-78
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