粗糙S-系和模糊S-系
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摘要
我们首先引入了粗糙S-系和模糊S-系的概念,然后讨论了粗糙S-系的性质,比较详细地介绍了在S-子系诱导的同余下的粗糙S-系的性质,以及两个粗糙S-子系得直积的上、下近似集的性质;我们重点讨论了模糊S-系的交和并的上、下近似集的性质。
在第二章中我们介绍了S-系上模糊同余的概念,讨论了S-系上模糊同余和模糊等价关系间的联系;重点讨论了由S-系上的任意模糊二元关系生成的模糊同余的结构和性质;在介绍了模糊同余类后,我们证明了S-系上的模糊同态定理以及模糊Rees同余的性质。
在第三章中,我们给出了模糊素S-系和模糊半素S-系的定义,并研究了它们的上、下近似集的性质,然后讨论了模糊素S-系和模糊半素S-系的扩张性质。最后,我们证明了:如果模糊S-系f是模糊不可约的,那么对于f模糊素性和模糊半素性是等价的;另外,利用模糊素S-系给出半群中的素理想的一个等价刻画。
In this paper,we introduce the notions of approximate S-acts and fuzzy S-acts,and discuss some of their properties.Specially,we investigate the characters of approximate S-act under a congruence induced by a S-subact.The properties which are of the upper and lower approximation sets of the products of S-acts,the union and meet of two fuzzy S-subacts,are also discussed.
In chapter two,we give the definitions of fuzzy congruences on a S-act,and we discuss the relations between fuzzy congruences and fuzzy equivalences;we obtain the properties of fuzzy Rees congruences;and fuzzy morphisms theorem of S-acts.
In chapter three,we introduce the concepts of fuzzy prime(semiprime)S-acts,and investigate the properties of their upper and lower approximate sets.Then we discuss the extension properties of fuzzy prime(semiprime)S-acts.Finally,we prove that a fuzzy irreducible semiprime S-act must be a fuzzy prime S-act;and give an equivalent description of prime ideals of semigroups by using fuzzy prime S-acts.
在第二章中我们介绍了S-系上模糊同余的概念,讨论了S-系上模糊同余和模糊等价关系间的联系;重点讨论了由S-系上的任意模糊二元关系生成的模糊同余的结构和性质;在介绍了模糊同余类后,我们证明了S-系上的模糊同态定理以及模糊Rees同余的性质。
在第三章中,我们给出了模糊素S-系和模糊半素S-系的定义,并研究了它们的上、下近似集的性质,然后讨论了模糊素S-系和模糊半素S-系的扩张性质。最后,我们证明了:如果模糊S-系f是模糊不可约的,那么对于f模糊素性和模糊半素性是等价的;另外,利用模糊素S-系给出半群中的素理想的一个等价刻画。
In this paper,we introduce the notions of approximate S-acts and fuzzy S-acts,and discuss some of their properties.Specially,we investigate the characters of approximate S-act under a congruence induced by a S-subact.The properties which are of the upper and lower approximation sets of the products of S-acts,the union and meet of two fuzzy S-subacts,are also discussed.
In chapter two,we give the definitions of fuzzy congruences on a S-act,and we discuss the relations between fuzzy congruences and fuzzy equivalences;we obtain the properties of fuzzy Rees congruences;and fuzzy morphisms theorem of S-acts.
In chapter three,we introduce the concepts of fuzzy prime(semiprime)S-acts,and investigate the properties of their upper and lower approximate sets.Then we discuss the extension properties of fuzzy prime(semiprime)S-acts.Finally,we prove that a fuzzy irreducible semiprime S-act must be a fuzzy prime S-act;and give an equivalent description of prime ideals of semigroups by using fuzzy prime S-acts.
引文
[1]Rosenfeld A,Fuzzy groups[J],J.Math.Anal.Appl.,1971,35:512-517.
[2]Liu WJ,Fuzzy invariant subgroups and fuzzy idals[J],FSs,1982,8:133-139.
[3]Kuroki N,Fuzzy.Bi-ideals in semigroups[J],comment.Math.Univ.St.Paul.,1980,80:17-21.
[4]Kuroki N and Wang P P.The low and upper approximations in a fuzzy.group[J].Inform.Sci.1996,90:203-220.
[5]Pawlak Z,Rough sets[J],Inter.J Computer and Inform.Sci,1982,11:341-356.
[6]Kuroki N,Wang PP,The loer and upper approximations ina fuzzy group[J],Inform.Sci.,1996,90:203-220.
[7]B.Davvaz,M.Mahdavipour,Roughness in modules[J],Inform.Sci.,2006,2:1-16.
[8]陈倩华等,弱P·反演半群上的模糊P-核正规系,(待发)
[9]刘仲奎.半群的S-系理论[M].北京:科学出版社,1992,1-2.
[10]谢祥云.半群的模糊理论[M].北京:科学出版社,2005
[11]张文修,吴伟志,梁吉业,李德玉.粗糙集理论与方法[M].北京:科学出版社,2001
[12]Kuroki N.Rough ideals in semigroups[J].Inform.Sci.1997,100:139-163.
[13]Davvaz B.Roughness in rings[J].Inform.Sci.2004,164:147-163.
[14]Jun B Y.Roughness of Gamma-subsemigroups/Ideals in Gamma-semigroups[J].Bull.Korea Math.Soc.2003,40(3):531-536.
[15]Kuroki N and Wang P P.The low and upper approximations in a fuzzy group[J].Inform.Sci.1996,90:203-220.
[16]Xiao Q M and Zhang Z L.Rough prime ideals and rough fuzzy prime ideals in semigroups[J].Inform.Sci.2006,176:725-733.
[17]Javed Ahsan,Lin Zhongkui,Prime and semiprime acts over monoids with zero,i.Mathematical J.Ibaraki University,2001,33,9-15.
[18]R.Biswas and S.Nanda S,Rough groups and rough subgroups[J],Bull.Polish Acad.Math.1994,42:251-254
[19]Z.Bonikowaski,Algebraic structures of rough sets,in W.P.Ziarko(Ed),Rough sets fuzzysets and knowledge discovery,Springer.Berlin,1995,pp.242-247.
[20]Z.Pawlak,Rogh sets,Int.J.Inf.Comp.Sci.1982,11:342-356.
[21]N.Kuroki,Fuzzy congruences and fuzzy norma subgroups,Inform.Sci.Math.1992,60:247-259.
[22]L.A.Zadeh,Fuzzy Sets,Inform.And control,8(1965)338-353
[23]Qi-Mei xiao,Zhen-Lliang Zhang,Rough prime ideals and rough fuzzy prime ideals in semigroups[J],Inform.Sci,176(2006)725-733
[24]Xiang -Yun Xie,Jian Tang,Feng Yan,A characterization of prime fuzzy ideals of orded semigroups,Fuzzy Sys.Math(to appear)
[25]郭增晓,粗糙模糊子群在满同态下的性质,河北师范大学学报(自然版),29(2005),560-562
[26]张金玲,张振良,模糊粗糙子群,模糊数学与系统,18(2004),46-48.
[27]高井贵等,群中模糊同余关系下的模糊粗糙群,模糊数学与系统,19(2005),62-66
[2]Liu WJ,Fuzzy invariant subgroups and fuzzy idals[J],FSs,1982,8:133-139.
[3]Kuroki N,Fuzzy.Bi-ideals in semigroups[J],comment.Math.Univ.St.Paul.,1980,80:17-21.
[4]Kuroki N and Wang P P.The low and upper approximations in a fuzzy.group[J].Inform.Sci.1996,90:203-220.
[5]Pawlak Z,Rough sets[J],Inter.J Computer and Inform.Sci,1982,11:341-356.
[6]Kuroki N,Wang PP,The loer and upper approximations ina fuzzy group[J],Inform.Sci.,1996,90:203-220.
[7]B.Davvaz,M.Mahdavipour,Roughness in modules[J],Inform.Sci.,2006,2:1-16.
[8]陈倩华等,弱P·反演半群上的模糊P-核正规系,(待发)
[9]刘仲奎.半群的S-系理论[M].北京:科学出版社,1992,1-2.
[10]谢祥云.半群的模糊理论[M].北京:科学出版社,2005
[11]张文修,吴伟志,梁吉业,李德玉.粗糙集理论与方法[M].北京:科学出版社,2001
[12]Kuroki N.Rough ideals in semigroups[J].Inform.Sci.1997,100:139-163.
[13]Davvaz B.Roughness in rings[J].Inform.Sci.2004,164:147-163.
[14]Jun B Y.Roughness of Gamma-subsemigroups/Ideals in Gamma-semigroups[J].Bull.Korea Math.Soc.2003,40(3):531-536.
[15]Kuroki N and Wang P P.The low and upper approximations in a fuzzy group[J].Inform.Sci.1996,90:203-220.
[16]Xiao Q M and Zhang Z L.Rough prime ideals and rough fuzzy prime ideals in semigroups[J].Inform.Sci.2006,176:725-733.
[17]Javed Ahsan,Lin Zhongkui,Prime and semiprime acts over monoids with zero,i.Mathematical J.Ibaraki University,2001,33,9-15.
[18]R.Biswas and S.Nanda S,Rough groups and rough subgroups[J],Bull.Polish Acad.Math.1994,42:251-254
[19]Z.Bonikowaski,Algebraic structures of rough sets,in W.P.Ziarko(Ed),Rough sets fuzzysets and knowledge discovery,Springer.Berlin,1995,pp.242-247.
[20]Z.Pawlak,Rogh sets,Int.J.Inf.Comp.Sci.1982,11:342-356.
[21]N.Kuroki,Fuzzy congruences and fuzzy norma subgroups,Inform.Sci.Math.1992,60:247-259.
[22]L.A.Zadeh,Fuzzy Sets,Inform.And control,8(1965)338-353
[23]Qi-Mei xiao,Zhen-Lliang Zhang,Rough prime ideals and rough fuzzy prime ideals in semigroups[J],Inform.Sci,176(2006)725-733
[24]Xiang -Yun Xie,Jian Tang,Feng Yan,A characterization of prime fuzzy ideals of orded semigroups,Fuzzy Sys.Math(to appear)
[25]郭增晓,粗糙模糊子群在满同态下的性质,河北师范大学学报(自然版),29(2005),560-562
[26]张金玲,张振良,模糊粗糙子群,模糊数学与系统,18(2004),46-48.
[27]高井贵等,群中模糊同余关系下的模糊粗糙群,模糊数学与系统,19(2005),62-66
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