半群与Fuzzy半群及其粗糙集的代数性质
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摘要
半群的代数理论是在数学内部和外部双重条件下，从20世纪50年代到60年代发展起来的一个崭新的代数分支。1990年，Biswas R提出了反Fuzzy子群的定义；1995年沈正维提出了一个群的反Fuzzy子群和正规反Fuzzy子群的定义。在此基础上，本文首先以一个半群作为基本集，给出了半群的反Fuzzy半群及其补的定义，并给出了其充要条件；在同态映射的条件下，讨论了反Fuzzy半群与正规反Fuzzy半群的同态问题。然后，给出了反Fuzzy(左、右、双、内禀)理想的定义与其充要条件；在同态映射下，讨论了其同态问题。
1965年，美国计算机与控制论专家Zadeh LA提出了Fuzzy集理论；1982年，波兰数学家pawlak Z提出了一种数据分析理论——粗糙集理论。二者是处理Fuzzy性和不确定性的数学工具，将二者结合起来研究，近年来越来越受到国际学术界的关注。本文以一个Fuzzy半群作为基本集，将粗糙集理论应用于Fuzzy半群中来，主要讨论了Fuzzy半群中两个非空Fuzzy子集积的粗理想的性质，进一步补充和完善了Fuzzy半群中粗糙集的数学结构。
1994年，Biswas R与Nanda S提出了粗糙群和粗糙子群的定义；2005年谢祥云提出了粗糙半群的粗糙同态映射。在此基础上，本文进一步将粗糙集的思想引入到一个Fuzzy代数系统，导出了Fuzzy粗糙半群与Fuzzy粗糙商半群的定义，并讨论了Fuzzy粗糙同态与Fuzzy粗糙同构的问题。
The semigroup algebra theory,which has a huge development under interior and exterior double Mathematical conditions form 1950's to 1960's,is a new branch of algebra theory.
In 1990,R.Biswas gave definition of anti-fuzzy subgroup.In 1995,Shen gave definitions of anti-fuzzy subgroup and normal anti-fiizzy subgroup of group.Basing on this,the definition of anti-fiizzy semigroup and its complements based on semigroup are introduced, and the sufficient and necessary of the exsistence of the anti-fuzzy semigroup is given in this paper. The homomorphism of anti-fuzzy semigroups and the hononorphism of normal anti-fuzzy semigroups are discussed. At the same time, the correlative definitions anti-fuzzy ideals of semigroup and the sufficient and necessary of the exsistence of anti-fuzzy ideals are introducted,and the homomorphism of anti-fuzzy ideals are discussed.
In 1965,L.A.Zadeh, a famous American electron engineer and cybernetics expert, put forward the theory of Fuzzy Sets. In 1982,Z.Pawlak, a famous Poland mathematician, put forward Rough Sets for the data analysis. Fuzzy sets and rough sets are means handling on fuzzy and indetermination. Today, fuzzy set theory, and rough set theory are combined in mathematics, and the combination is been adverted more and more by international academy. The properties of fuzzy rough ideals of two non-empty fuzzy subsets products in fuzzy semigroups are discussed in this paper,which based on a fuzzy semigroup. Thus the rough theory in fuzzy semigroups is completed and perfected.
In 1994,the definitions of rough groups and rough subgroups were introduced by R.Biswas and S.Nanda. In 2005,Xie introduced a rough homomorphism mapping of rough semigroups.In this paper,by using the rough theory which is applied to the fuzzy algebra systems, the definitions of fuzzy rough semigroups and fuzzy rough quotient semigroups are educed. Moreover, the fuzzy rough homomorphism and the fuzzy rough isomorphism are discussed.
1965年，美国计算机与控制论专家Zadeh LA提出了Fuzzy集理论；1982年，波兰数学家pawlak Z提出了一种数据分析理论——粗糙集理论。二者是处理Fuzzy性和不确定性的数学工具，将二者结合起来研究，近年来越来越受到国际学术界的关注。本文以一个Fuzzy半群作为基本集，将粗糙集理论应用于Fuzzy半群中来，主要讨论了Fuzzy半群中两个非空Fuzzy子集积的粗理想的性质，进一步补充和完善了Fuzzy半群中粗糙集的数学结构。
1994年，Biswas R与Nanda S提出了粗糙群和粗糙子群的定义；2005年谢祥云提出了粗糙半群的粗糙同态映射。在此基础上，本文进一步将粗糙集的思想引入到一个Fuzzy代数系统，导出了Fuzzy粗糙半群与Fuzzy粗糙商半群的定义，并讨论了Fuzzy粗糙同态与Fuzzy粗糙同构的问题。
The semigroup algebra theory,which has a huge development under interior and exterior double Mathematical conditions form 1950's to 1960's,is a new branch of algebra theory.
In 1990,R.Biswas gave definition of anti-fuzzy subgroup.In 1995,Shen gave definitions of anti-fuzzy subgroup and normal anti-fiizzy subgroup of group.Basing on this,the definition of anti-fiizzy semigroup and its complements based on semigroup are introduced, and the sufficient and necessary of the exsistence of the anti-fuzzy semigroup is given in this paper. The homomorphism of anti-fuzzy semigroups and the hononorphism of normal anti-fuzzy semigroups are discussed. At the same time, the correlative definitions anti-fuzzy ideals of semigroup and the sufficient and necessary of the exsistence of anti-fuzzy ideals are introducted,and the homomorphism of anti-fuzzy ideals are discussed.
In 1965,L.A.Zadeh, a famous American electron engineer and cybernetics expert, put forward the theory of Fuzzy Sets. In 1982,Z.Pawlak, a famous Poland mathematician, put forward Rough Sets for the data analysis. Fuzzy sets and rough sets are means handling on fuzzy and indetermination. Today, fuzzy set theory, and rough set theory are combined in mathematics, and the combination is been adverted more and more by international academy. The properties of fuzzy rough ideals of two non-empty fuzzy subsets products in fuzzy semigroups are discussed in this paper,which based on a fuzzy semigroup. Thus the rough theory in fuzzy semigroups is completed and perfected.
In 1994,the definitions of rough groups and rough subgroups were introduced by R.Biswas and S.Nanda. In 2005,Xie introduced a rough homomorphism mapping of rough semigroups.In this paper,by using the rough theory which is applied to the fuzzy algebra systems, the definitions of fuzzy rough semigroups and fuzzy rough quotient semigroups are educed. Moreover, the fuzzy rough homomorphism and the fuzzy rough isomorphism are discussed.
引文
[1] Biswas R. Fuzzy Subgroups and Anti-Fuzzy Subgrouops[J].Fuzzy Sets and Systems, 1990, 35(1):121-124
[2] 董百志．反Fuzzy子群的直积[J]．绍兴师专学报，1992，5：29-34
[3] 沈正维．一个群的反模糊子群[J]．辽宁大学学报，1995，18(2)：99-101
[4] 袁学海，李洪兴，罗承忠．几种新的截集及其应用[J]．模糊系统与数学，1997，11(1)：37-43
[5] 谭宜家．Fuzzy半群中的Fuzzy素理想[J]．模糊系统与数学，2001，15(1)：50-54
[6] 张文修等．模糊数学引论[M]．西安：西安交通大学出版社，1991
[7] 张文修，吴伟志，梁吉业等．粗糙集理论与方法[M]．北京：科学出版社，2001
[8] 刘文奇．Pawlak代数及其性质[J]．模糊系统与数学，1999，13(2)：3-10
[9] 刘文奇．F格上的逼近算子[J]．数学学报，2003，46(6)：1163-1170
[10] 谭宜家．Fuzzy半群中的Fuzzy理想[J]．模糊系统与数学，1995，9(2)：59-64
[11] 于纯海．Fuzzy集上的Fuzzy关系及其分类[J]．模糊数学，1987，7(3-4)：45-50
[12] 刘文奇，罗承忠．集合环[J]．模糊系统与数学，1997，11(2)：34-39
[13] 刘文奇，吴从忻．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，2002，16(3)：50-58
[14] 张仕念，刘文奇．可测空间的算子扩张及其上的等价关系[J]．纯粹数学与应用数学，2002，18(1)：83-85
[15] 张仕念，刘文奇．可测空间与Pawlak代数[J]．模糊系统与数学，2001，15(4)：40-43
[16] 蒲保明，刘应明．不分明拓扑学Ⅰ[J]．四川大学学报(自然科学版)，1997(1)：31-50
[17] 谢祥云，吴明芬．半群的模糊理论[M]．北京：科学出版社，2005
[18] Pawlak Z. Rough sets[J].Int J.Inform.Comp.Sci.1982,11:341-356
[19] Kuroki N. Rough Ideals in Semigroup[J].Inform.Sci.1997,100:139-163
[20] Kuroki N.Wang P P. The Lower and Upper Approximation in A Fuzzy Group[J]. Inform.sci.1996,90:203-220
[21] 罗承忠．模糊集引论[M]．北京：北京师范大学出版社，1989
[22] 刘文奇．Fuzzy集的表现理论及其应用[M]．昆明：云南科技出版社，1999
[23] Zhao Xiaoyan, Zhang Zhenliang. Rough Set and Roughness of Fuzzy Set on The Fuzzy Approximation Spaces [J]. Southeast Asian Bulletin of Mathematics,2005, 29:617-625
[24] Biswas R,Nanda S. Rough Groups and Rough Subgroups[J].Bulletin of The Polish Academy of Sciences Mathmatics,1994,42:251-254
[25] 韩素清．粗糙群的同态与同构[J]．山西大学学报，2001，24(4)：303-305
[26] 刘文奇．完全Boole代数的配对格[J]．昆明理工大学学报，1997，22(4)：102-105
[27] 刘文奇．从Boole代数到一般软代数的对称扩张[J]．兰州大学学报，1996，32：124-128
[28] 刘文奇．YN(X)上的拓扑结构[J]．昆明理工大学学报，1997，22(6)：12-15
[29] 刘文奇．软代数的表示定理[J]．模糊系统与数学，1999，13(1)：4-9
[30] 罗承忠．模糊集与集合套[J]．模糊数学，1983，4：113-126
[31] 罗承忠．可列集合套[J]．模糊系统与数学，1991，1：1-6
[32] 张振良．集合套的落影函数[J]．模糊系统与数学，1990，2：38-44
[33] 李洪兴，罗承忠，汪培庄．如何定义模糊集映射[J]．北京师范大学学报，1993，29(1)：1-9
[34] 彭家寅，李洪兴．FuzzyHX环的推广[J]．数学进展，2006，35(1)：67-74
[35] 史福贵．L-Fuzzy集与素元集合套[J]．数学研究与评论，1996，16(1)：398-402
[36] 史福贵．L-Fuzzy Mapping of L-Fuzzy Sets[J]．模糊系统与数学，2000，14(3)：16-24
[37] 李洪兴，汪培庄．模糊数学[M]．北京：国防工业出版社，1994
[38] 张金玲．模糊粗糙子群和模糊粗糙子环：[硕士毕业论文]．昆明：昆明理工大学，2002
[39] 李洪兴，汪培庄．幂群[J]．应用数学，1988，1(1)：1-4
[40] 史开泉．函数S-粗集[J]．山东大学学报，2005，(1)：1-10
[41] 史开泉，崔玉泉．S-粗集与粗决策[M]．北京：科学出版社，2006
[42] 李洪兴．正则HX群的同态与同构[J]．模糊系统与数学，1990，4(1)1-7
[43] 张禾瑞．近世代数基础[M]．北京：高等教育出版社，1978
[44] 戚振开．点态化的Fuzzy群[J]．模糊数学，1981(2)：29-36
[45] 于佳丽，舒兰．半群中粗理想的性质[J]．电子科技大学学报，2002，31(5)：539-541
[46] 于佳丽，舒兰．粗糙商半群的性质[J]．模糊系统与数学，2003，17(4)：25-27
[47] 蒋家尚．Fuzzy半群的Fuzzy理想[J]．华东船舶工业学院学报，2001，15(2)：28-30
[48] 邦德列雅金(曹锡华译)．连续群[M]．北京：科学出版社，1978
[49] 贺仲雄．模糊数学及其应用[M]．天津：天津科技出版社，1981
[50] 刘应明，任平．模糊数学[M]．上海：上海教育出版社，1987
[51] 马骥良，于纯海．模糊代数选论[M]．学苑出版社，1989
[52] 汪诚义．模糊数学引论[M]．北京：北京工业学院出版社．1988
[53] (日) 浅尼喜代治等(赵汝译)．模糊系统理论入门[M]．北京：北京师范大学出版社，1982
[54] (日)水本雅晴(欧阳绵译)．最新模糊集理论[J]．应用数学与计算数学，1980(3)
[55] 盛德成．抽象代数[M]．北京：科学出版社，2001
[56] Zhang Zhcnliang.Classifications of HX groups and their chains of normal subgroups[J].Italian Journal of Pure and Applied Mathematics,1999:125-134
[57] Zhang Zhenliang.The Properies and HX groups[J].Italian Journal of Pure and Applied Mathematics,1997(2):97-106
[58] Zhang Zhenliang.Structures and Classifications of HX Rings[J].Italian Journal of Pure and Applies mathematics,2002(12):9-20
[59] Howio J M. An Introduction to Semigroup Theory[M].New York:Academic,1976
[60] Kuroki N. On Fuzzy Semigroup[J].Inform Sci.1991,53:203-236
[61] Zadeh LA. Fuzzy Sets[J].Inform and Control.1965,8:338-353
[62] X Y Xie. Fuzzy Ideals in Semigroups[J].puzzy Math,1999,7:357-365
[63] Pawlak Z. Rough Sets and Fuzzy Sets[J].Fuzzy Sets and Systems,1985,17:99-102
[64] Kuroki N. Fuzzy Congruences and Fuzzy Normal Subgroups[J].Inform Sci,1992, 60:247-259
[65] Chakrabarty K,Biswas R and Nanda S. Fuzziness In Rough Sets[J].Fuzzy Sets and Systems,2000,110:247-251
[66] Mohua Banerjee. Roughness of A Fuzzy Sets[J].Inform Comput Sci,1996, 93:235-246
[67] Dib K A and Galhum N. Fuzzy Ideals and Fuzzy Bi-ideals in Fuzzy Scmigroups[J]. Fuzzy Sets and Systems,1997,92:103-111
[68] Mordeson J N. Rough Set Theory Applied (Fuzzy) Ideal Theory[J]. Fuzzy Sets and Systems,2001,12:315-324
[69] Murali V. Fuzzy Equivalent Relations[J]. Fuzzy Sets and Systems,1989,30:155-163
[70] MuraliV.Fuzzy Congruence Relations[J]. Fuzzy Sets and Systcms,1991, 41:359-369
[71] Nemitz W C. Fuzzy Relations and Fuzzy Functions[J]. Fuzzy Sets and Systems,1986,19:177-191
[72] Tan Y J. Fuzzy Congruences on A Regular Semigroup[J]. Fuzzy Sets and Systems, 2001,117:447-453
[73] Xie X Y. Fuzzy Iideals Extensions in Semigroups[J].Kyungpook Math J,2002, 42: 39-49
[74] Xie X Y. Fuzzy Ideals Extensions of Semigroups[J].Soochow J Math,2001,27: 125-138
[75] Yao Y Y. A Comparative Study of Fuzzy Sets and Rough Sets[J].Inform Sci,1998,109: 227-241
[76] Yao Y Y. Two Views of The Theory of Rough Sets in Finite Universe[J].Ineter Approx Resson,1996,15:291-317
[77] Zadeh L A. Similimity Relations and Fuzzy Ordering[J].Inform Sci,1971, 3: 117-200
[78] Abu Osman M T. On The Product of Fuzzy Subgroups[J]. Fuzzy Sets and Systems.1987,24: 79-86
[79] Eroglu M S. The Homomorphic Image of A Fuzzy Subgroups Is Always A Fuzzy Subgroups[J]. Fuzzy Sets and Systems.1989,3:225-226
[80] Makamba B B. Direct Products and Isomorphism of Fuzzy Subgroups[J].Infrom Science. 1992, 65:33-43
[81] Ajmal N and Prajpat A S. Fuzzy Cosets and Fuzzy Normal Subgroups[J].Inform Science.1992,64:17-25
[82] Anthony J A and Sherwood H. Fuzzy Groups Redefined[J].J.Math.Anal.Appl.1979, 69:124-130
[83] Bhattacharya P. Fuzzy Subgroups:Some Characterization Ⅱ[J].Inform Science. 1986,38:293-297
[84] Bhattacharya P. Fuzzy Subgroups:Some Characterization[J]. J.Math.Anal.Appl. 1987,128:241-252
[85] Bhattacharya P and Mukherjee N P. Fuzzy Relation and Fuzzy Groups[J]. Inform Science.1985,36:267-282
[86] Bhattacharya P and Mukherjee N P. Some Group-thoretic Analogues Ⅱ[J]. Inform Science. 1987,41:77-91
[87] Das P S. Fuzzy Groups and Level Subgroups[J]. J.Math.Anal.Appl.1981,84: 264-269
[88] Ibrahim A S and Khatab S A. On Fuzzy Quotient Subgroups[J].Proc Pakistan Acad Sci. 1990, 27(1):45-54
[89] Ray S. The Free Product of Fuzzy Subgroups[J].Fuzzy Sets and Systems. 1992, 50:225-235
[90] Ray S. Isomorphic Fuzzy Subgroups[J].Fuzzy Sets and Systems. 1992, 50:201-207
[91] Xiao Qimei,Zhang Zhenliang. Rough Prime Ideals and Rough Fuzzy Prime Ideals in Semigroups[J].Information Sciences,2006(176):725-733
[2] 董百志．反Fuzzy子群的直积[J]．绍兴师专学报，1992，5：29-34
[3] 沈正维．一个群的反模糊子群[J]．辽宁大学学报，1995，18(2)：99-101
[4] 袁学海，李洪兴，罗承忠．几种新的截集及其应用[J]．模糊系统与数学，1997，11(1)：37-43
[5] 谭宜家．Fuzzy半群中的Fuzzy素理想[J]．模糊系统与数学，2001，15(1)：50-54
[6] 张文修等．模糊数学引论[M]．西安：西安交通大学出版社，1991
[7] 张文修，吴伟志，梁吉业等．粗糙集理论与方法[M]．北京：科学出版社，2001
[8] 刘文奇．Pawlak代数及其性质[J]．模糊系统与数学，1999，13(2)：3-10
[9] 刘文奇．F格上的逼近算子[J]．数学学报，2003，46(6)：1163-1170
[10] 谭宜家．Fuzzy半群中的Fuzzy理想[J]．模糊系统与数学，1995，9(2)：59-64
[11] 于纯海．Fuzzy集上的Fuzzy关系及其分类[J]．模糊数学，1987，7(3-4)：45-50
[12] 刘文奇，罗承忠．集合环[J]．模糊系统与数学，1997，11(2)：34-39
[13] 刘文奇，吴从忻．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，2002，16(3)：50-58
[14] 张仕念，刘文奇．可测空间的算子扩张及其上的等价关系[J]．纯粹数学与应用数学，2002，18(1)：83-85
[15] 张仕念，刘文奇．可测空间与Pawlak代数[J]．模糊系统与数学，2001，15(4)：40-43
[16] 蒲保明，刘应明．不分明拓扑学Ⅰ[J]．四川大学学报(自然科学版)，1997(1)：31-50
[17] 谢祥云，吴明芬．半群的模糊理论[M]．北京：科学出版社，2005
[18] Pawlak Z. Rough sets[J].Int J.Inform.Comp.Sci.1982,11:341-356
[19] Kuroki N. Rough Ideals in Semigroup[J].Inform.Sci.1997,100:139-163
[20] Kuroki N.Wang P P. The Lower and Upper Approximation in A Fuzzy Group[J]. Inform.sci.1996,90:203-220
[21] 罗承忠．模糊集引论[M]．北京：北京师范大学出版社，1989
[22] 刘文奇．Fuzzy集的表现理论及其应用[M]．昆明：云南科技出版社，1999
[23] Zhao Xiaoyan, Zhang Zhenliang. Rough Set and Roughness of Fuzzy Set on The Fuzzy Approximation Spaces [J]. Southeast Asian Bulletin of Mathematics,2005, 29:617-625
[24] Biswas R,Nanda S. Rough Groups and Rough Subgroups[J].Bulletin of The Polish Academy of Sciences Mathmatics,1994,42:251-254
[25] 韩素清．粗糙群的同态与同构[J]．山西大学学报，2001，24(4)：303-305
[26] 刘文奇．完全Boole代数的配对格[J]．昆明理工大学学报，1997，22(4)：102-105
[27] 刘文奇．从Boole代数到一般软代数的对称扩张[J]．兰州大学学报，1996，32：124-128
[28] 刘文奇．YN(X)上的拓扑结构[J]．昆明理工大学学报，1997，22(6)：12-15
[29] 刘文奇．软代数的表示定理[J]．模糊系统与数学，1999，13(1)：4-9
[30] 罗承忠．模糊集与集合套[J]．模糊数学，1983，4：113-126
[31] 罗承忠．可列集合套[J]．模糊系统与数学，1991，1：1-6
[32] 张振良．集合套的落影函数[J]．模糊系统与数学，1990，2：38-44
[33] 李洪兴，罗承忠，汪培庄．如何定义模糊集映射[J]．北京师范大学学报，1993，29(1)：1-9
[34] 彭家寅，李洪兴．FuzzyHX环的推广[J]．数学进展，2006，35(1)：67-74
[35] 史福贵．L-Fuzzy集与素元集合套[J]．数学研究与评论，1996，16(1)：398-402
[36] 史福贵．L-Fuzzy Mapping of L-Fuzzy Sets[J]．模糊系统与数学，2000，14(3)：16-24
[37] 李洪兴，汪培庄．模糊数学[M]．北京：国防工业出版社，1994
[38] 张金玲．模糊粗糙子群和模糊粗糙子环：[硕士毕业论文]．昆明：昆明理工大学，2002
[39] 李洪兴，汪培庄．幂群[J]．应用数学，1988，1(1)：1-4
[40] 史开泉．函数S-粗集[J]．山东大学学报，2005，(1)：1-10
[41] 史开泉，崔玉泉．S-粗集与粗决策[M]．北京：科学出版社，2006
[42] 李洪兴．正则HX群的同态与同构[J]．模糊系统与数学，1990，4(1)1-7
[43] 张禾瑞．近世代数基础[M]．北京：高等教育出版社，1978
[44] 戚振开．点态化的Fuzzy群[J]．模糊数学，1981(2)：29-36
[45] 于佳丽，舒兰．半群中粗理想的性质[J]．电子科技大学学报，2002，31(5)：539-541
[46] 于佳丽，舒兰．粗糙商半群的性质[J]．模糊系统与数学，2003，17(4)：25-27
[47] 蒋家尚．Fuzzy半群的Fuzzy理想[J]．华东船舶工业学院学报，2001，15(2)：28-30
[48] 邦德列雅金(曹锡华译)．连续群[M]．北京：科学出版社，1978
[49] 贺仲雄．模糊数学及其应用[M]．天津：天津科技出版社，1981
[50] 刘应明，任平．模糊数学[M]．上海：上海教育出版社，1987
[51] 马骥良，于纯海．模糊代数选论[M]．学苑出版社，1989
[52] 汪诚义．模糊数学引论[M]．北京：北京工业学院出版社．1988
[53] (日) 浅尼喜代治等(赵汝译)．模糊系统理论入门[M]．北京：北京师范大学出版社，1982
[54] (日)水本雅晴(欧阳绵译)．最新模糊集理论[J]．应用数学与计算数学，1980(3)
[55] 盛德成．抽象代数[M]．北京：科学出版社，2001
[56] Zhang Zhcnliang.Classifications of HX groups and their chains of normal subgroups[J].Italian Journal of Pure and Applied Mathematics,1999:125-134
[57] Zhang Zhenliang.The Properies and HX groups[J].Italian Journal of Pure and Applied Mathematics,1997(2):97-106
[58] Zhang Zhenliang.Structures and Classifications of HX Rings[J].Italian Journal of Pure and Applies mathematics,2002(12):9-20
[59] Howio J M. An Introduction to Semigroup Theory[M].New York:Academic,1976
[60] Kuroki N. On Fuzzy Semigroup[J].Inform Sci.1991,53:203-236
[61] Zadeh LA. Fuzzy Sets[J].Inform and Control.1965,8:338-353
[62] X Y Xie. Fuzzy Ideals in Semigroups[J].puzzy Math,1999,7:357-365
[63] Pawlak Z. Rough Sets and Fuzzy Sets[J].Fuzzy Sets and Systems,1985,17:99-102
[64] Kuroki N. Fuzzy Congruences and Fuzzy Normal Subgroups[J].Inform Sci,1992, 60:247-259
[65] Chakrabarty K,Biswas R and Nanda S. Fuzziness In Rough Sets[J].Fuzzy Sets and Systems,2000,110:247-251
[66] Mohua Banerjee. Roughness of A Fuzzy Sets[J].Inform Comput Sci,1996, 93:235-246
[67] Dib K A and Galhum N. Fuzzy Ideals and Fuzzy Bi-ideals in Fuzzy Scmigroups[J]. Fuzzy Sets and Systems,1997,92:103-111
[68] Mordeson J N. Rough Set Theory Applied (Fuzzy) Ideal Theory[J]. Fuzzy Sets and Systems,2001,12:315-324
[69] Murali V. Fuzzy Equivalent Relations[J]. Fuzzy Sets and Systems,1989,30:155-163
[70] MuraliV.Fuzzy Congruence Relations[J]. Fuzzy Sets and Systcms,1991, 41:359-369
[71] Nemitz W C. Fuzzy Relations and Fuzzy Functions[J]. Fuzzy Sets and Systems,1986,19:177-191
[72] Tan Y J. Fuzzy Congruences on A Regular Semigroup[J]. Fuzzy Sets and Systems, 2001,117:447-453
[73] Xie X Y. Fuzzy Iideals Extensions in Semigroups[J].Kyungpook Math J,2002, 42: 39-49
[74] Xie X Y. Fuzzy Ideals Extensions of Semigroups[J].Soochow J Math,2001,27: 125-138
[75] Yao Y Y. A Comparative Study of Fuzzy Sets and Rough Sets[J].Inform Sci,1998,109: 227-241
[76] Yao Y Y. Two Views of The Theory of Rough Sets in Finite Universe[J].Ineter Approx Resson,1996,15:291-317
[77] Zadeh L A. Similimity Relations and Fuzzy Ordering[J].Inform Sci,1971, 3: 117-200
[78] Abu Osman M T. On The Product of Fuzzy Subgroups[J]. Fuzzy Sets and Systems.1987,24: 79-86
[79] Eroglu M S. The Homomorphic Image of A Fuzzy Subgroups Is Always A Fuzzy Subgroups[J]. Fuzzy Sets and Systems.1989,3:225-226
[80] Makamba B B. Direct Products and Isomorphism of Fuzzy Subgroups[J].Infrom Science. 1992, 65:33-43
[81] Ajmal N and Prajpat A S. Fuzzy Cosets and Fuzzy Normal Subgroups[J].Inform Science.1992,64:17-25
[82] Anthony J A and Sherwood H. Fuzzy Groups Redefined[J].J.Math.Anal.Appl.1979, 69:124-130
[83] Bhattacharya P. Fuzzy Subgroups:Some Characterization Ⅱ[J].Inform Science. 1986,38:293-297
[84] Bhattacharya P. Fuzzy Subgroups:Some Characterization[J]. J.Math.Anal.Appl. 1987,128:241-252
[85] Bhattacharya P and Mukherjee N P. Fuzzy Relation and Fuzzy Groups[J]. Inform Science.1985,36:267-282
[86] Bhattacharya P and Mukherjee N P. Some Group-thoretic Analogues Ⅱ[J]. Inform Science. 1987,41:77-91
[87] Das P S. Fuzzy Groups and Level Subgroups[J]. J.Math.Anal.Appl.1981,84: 264-269
[88] Ibrahim A S and Khatab S A. On Fuzzy Quotient Subgroups[J].Proc Pakistan Acad Sci. 1990, 27(1):45-54
[89] Ray S. The Free Product of Fuzzy Subgroups[J].Fuzzy Sets and Systems. 1992, 50:225-235
[90] Ray S. Isomorphic Fuzzy Subgroups[J].Fuzzy Sets and Systems. 1992, 50:201-207
[91] Xiao Qimei,Zhang Zhenliang. Rough Prime Ideals and Rough Fuzzy Prime Ideals in Semigroups[J].Information Sciences,2006(176):725-733