(∈,∈∨q(λ,μ))-模糊子半群
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摘要
首先,给出了广义反模糊子半群和(∈,∈∨q(λ,μ))-模糊子半群的定义,并讨论了它们的等价刻画.其次,给出了(∈,∈∨q(λ,μ))-模糊子半群的基本性质:(∈,∈∨q(λ,μ))-模糊子半群的并仍是(∈,∈∨q(λ,μ))-模糊子半群,并获得了(∈,∈∨q(λ,μ))-模糊子半群在反直积运算下封闭的结论.最后,基于反扩张原理获得了(∈,∈∨q(λ,μ))-模糊子半群同态像与同态原像的相关性质。
In this paper,we firstly introduce the definition of generalized anti-fuzzy subsemigroups and the concept of(∈,∈∨ q(λ,μ))-fuzzy subsemigroups. We also discuss some equivalent characterizations.Then,the new property of them is given:the union of(∈,∈∨ q(λ,μ))-fuzzy subsemigroups is still(∈,∈∨q(λ,μ))-fuzzy subsemigroups.We also prove the anti-direct product of(∈,∈∨q(λ,μ))-fuzzy subsemigroups is also(∈,∈∨q(λ,μ))-fuzzy subsemigroups.Finally,based on the anti-extension principle,some relative results of its homomorphic image and homomorphic preimage are obtained.
In this paper,we firstly introduce the definition of generalized anti-fuzzy subsemigroups and the concept of(∈,∈∨ q(λ,μ))-fuzzy subsemigroups. We also discuss some equivalent characterizations.Then,the new property of them is given:the union of(∈,∈∨ q(λ,μ))-fuzzy subsemigroups is still(∈,∈∨q(λ,μ))-fuzzy subsemigroups.We also prove the anti-direct product of(∈,∈∨q(λ,μ))-fuzzy subsemigroups is also(∈,∈∨q(λ,μ))-fuzzy subsemigroups.Finally,based on the anti-extension principle,some relative results of its homomorphic image and homomorphic preimage are obtained.
引文
[1]Rosenfeld A.Fuzzy groups[J].Journal of Mathematical Analysis and Applications,1971,35:512~517.
[2]Pu Baoming,Liu Yingming.Fuzzy topology of fuzzy neighborhood structure and Moore-Smith type convergence point[J].Journal of Sichuan University(Natural Science Edition),1977,l:31~50(in Chinese).(蒲保明,刘应明.不分明拓扑学Ⅰ—不分明点的邻近构造与Moore—Smith式收敛[J].四川大学学报(自然科学版),1977,l:31~50).
[3]Bhakat S K,Das P.On the definition of a fuzzy subgroup[J].Fuzzy Sets and Systems,1992,51(2):235~241.
[4]Bhakat S K,Das P.(∈,∈∨q)-fuzzy subgroup[J].Fuzzy Sets and Systems,1996,80(3):359~368.
[5]Liao Zuhua,Gu Hui.(∈,∈∨q(λ,μ))-fuzzy normal subgroup[J].Fuzzy Systems and Mathematics,2006,20(5):47~53.
[6]Chen Min,Liao Zuhua.(∈,∈∨q(λ,μ))-fuzzy subsemigroups and(∈,∈∨q(λ,μ))-fuzzy ideals of semigroups[C]//Proc of the Sixth International Conference on Information and Management Sciences.Lhasa,Tibet:IMS,2007:575~579.
[7]Gu Hui,Liao Zuhua.(∈,∈∨q(λ,μ))-fuzzy convex sets[J].Fuzzy Systems and Mathematics,2007,21(1):92~96.
[8]Liao Zuhua,Chen Min.(∈,∈∨q(λ,μ))-fuzzy subnear-rings and ideals[J].Fuzzy Sets and Systems,2009,23(6):12~16(in Chinese).(廖祖华,陈敏.(∈,∈∨q(λ,μ))-模糊子近环和思想[J].模糊系统与数学,2009,23(6):12~16).
[9]Yi Lihua,Liao Zuhua.,(∈,∈∨q(λ,μ))-fuzzy Drazin subsemigroup[C]//Proc of Information Technology And Environmental System Sciences.ITESS,2008,6:997~1002.
[10]Liao Zuhua,Chen Min.(∈,∈∨q(λ,μ))-fuzzy subsemigroups and(∈,∈∨q(λ,μ))-Fuzzy completely regular subsemigroups[J].Journal of Jiangnan University(Natural Science Edition),2009,8(2):242~244(in Chinese)(廖祖华,陈敏.(∈,∈∨q(λ,μ))-模糊子半群和(∈,∈∨q(λ,μ))-模糊完全正则子半群[J].江南大学学报(自然科学版),2009,8(2):242~244).
[11]Liao Zuhua,Yi Lihua,Hu Miaohan.(∈,∈∨q(λ,μ))-fuzzy k-ideals of semigroups[J].Journal of Mathematics,2012,32(2):191~205.
[12]Fan Yingying,Liao Zuhua,Zeng Junqiao,et al.(∈,∈∨q(λ,μ))-fuzzyΓ-subsemigroups[J].Fuzzy Sets and Systems,2013,27(5):69~73(in Chinese)(范莹莹,廖祖华,曾俊俏等.(∈,∈∨q(λ,μ))-模糊Γ-子半群[J].模糊系统与数学,2013,27(5):69~73).
[13]Zeng Junqiao,Liao Zuhua,Fan Yingying,et al.(∈,∈∨q(λ,μ))-fuzzyΓ-subrings and homomorphism[J].Fuzzy Sets and Systems,2014,28(1):15~22(in Chinese).(曾俊俏,廖祖华,范莹莹等.(∈,∈∨q(λ,μ))-模糊Γ-子环及同态[J].模糊系统与数学,2014,28(1):15~22).
[14]Biswas R.Fuzzy subgroups and anti—fuzzy subgroups[J].Fuzzy Sets and Systems,1990,35(1):121~124.
[15]Zhang Cheng,Yuan Xuehai.(∈′,∈′∨q′)-fuzzy subgroup[J].Journal of Liaoning Normal University(natural science),1997,20,186~189.
[16]Yuan Xuehai,Fu Yunpeng,Liu Xin,et al.(β-,α-)-fuzzy subgroups[J].Journal of Liaoning Normal University(Nature Science Edition),2003,26(1):1~4.
[17]Guo Ziyan,Liao Zuhua,Guo Jianfu.(∈-,∈-∨q-(λ,μ))-fuzzy subgroup[C]//Proc of the Sixth International Conference on Information and Management Sciences.Lhasa,Tibet:IMS,2007:570~574.
[18]Yang Jihui,Liu Xin,Han Ying.(∈-,∈-∨q-)-Normal fuzzy subgroup[J].Journal of Liaoning Normal University,2003,26(4):350~352(in Chinese).(杨吉会,刘新,韩滢.(∈-,∈-∨q-)-正规模糊子群[J].辽宁师范大学学报(自然科学版),2003,26(4):350~352).
[19]Hao Cuiyun,Liao Zuhua,Ji Min,et al.Some properties of(∈-,∈-∨q-(λ,μ))-fuzzy subgroups[J].Fuzzy Sets and Systems,2012,26(6):26~30(in Chinese).(郝翠芸,廖祖华,嵇敏等.(∈-,∈-∨q-(λ,μ))-模糊子群的若干性质[J].模糊系统与数学,2012,26(6):26~30).
[20]Hao Cuiyun,Liao Zuhua.Some properties of(∈-,∈-∨q-(λ,μ))-fuzzy normal subgroups[J].Fuzzy Sets and Systems,2013,33(5):857~864(in Chinese).(郝翠芸,廖祖华.(∈-,∈-∨q-(λ,μ))-模糊正规子群的若干性质[J].数学杂志.2013,33(5):857~864).
[21]Zhu Chan,Liao Zuhua.(∈-,∈-∨q-(λ,μ))-fuzzy Complemented Semirings[J].Fuzzy Sets and Systems(accepted)(in Chinese).(朱婵,廖祖华等.(∈-,∈-∨q-(λ,μ))-模糊可补半环[J].模糊系统与数学(已录用)).
[22]Chen T Y.Lower cut set decomposition theorem and representation theorem[J].BUSEFAL,1995,63:46~48.
[23]Zhang Cheng,Chen Tuyun.Fuzzy sets and order set embedding[J].Journal of Liaoning Normal University,1995(4):275~279(in Chinese).(张成,陈图云.模糊集与顺序集合套[J].辽宁师范大学学报,1995(4):275~279).
[24]Qu Wanling,Geng Suyun,Zhang Li’ang.Discrete mathematics[M].Beijing:Higher Education Press,2008(in Chinese).(屈婉玲,耿素云,张立昂.离散数学[M].北京:高等教育出版社,2008).
[25]Liao Zuhua.Anti-fuzzy subgroupoids and anti-fuzzy completely regular subsemigroups[J].Journal of Lanzhou University(Nature Sciences),1996(32):120~123(in Chinese).(廖祖华.反F-子广群与反F-完全正则子半群[J].兰州大学学报(自然科学版)(96年模糊理论与应用国际会议暨八届年会),1996(32):120~123).
[26]Petrich M,Norman R.Completely regular semigroups[M].New York:John Wiley&Sons Inc,1999.
[2]Pu Baoming,Liu Yingming.Fuzzy topology of fuzzy neighborhood structure and Moore-Smith type convergence point[J].Journal of Sichuan University(Natural Science Edition),1977,l:31~50(in Chinese).(蒲保明,刘应明.不分明拓扑学Ⅰ—不分明点的邻近构造与Moore—Smith式收敛[J].四川大学学报(自然科学版),1977,l:31~50).
[3]Bhakat S K,Das P.On the definition of a fuzzy subgroup[J].Fuzzy Sets and Systems,1992,51(2):235~241.
[4]Bhakat S K,Das P.(∈,∈∨q)-fuzzy subgroup[J].Fuzzy Sets and Systems,1996,80(3):359~368.
[5]Liao Zuhua,Gu Hui.(∈,∈∨q(λ,μ))-fuzzy normal subgroup[J].Fuzzy Systems and Mathematics,2006,20(5):47~53.
[6]Chen Min,Liao Zuhua.(∈,∈∨q(λ,μ))-fuzzy subsemigroups and(∈,∈∨q(λ,μ))-fuzzy ideals of semigroups[C]//Proc of the Sixth International Conference on Information and Management Sciences.Lhasa,Tibet:IMS,2007:575~579.
[7]Gu Hui,Liao Zuhua.(∈,∈∨q(λ,μ))-fuzzy convex sets[J].Fuzzy Systems and Mathematics,2007,21(1):92~96.
[8]Liao Zuhua,Chen Min.(∈,∈∨q(λ,μ))-fuzzy subnear-rings and ideals[J].Fuzzy Sets and Systems,2009,23(6):12~16(in Chinese).(廖祖华,陈敏.(∈,∈∨q(λ,μ))-模糊子近环和思想[J].模糊系统与数学,2009,23(6):12~16).
[9]Yi Lihua,Liao Zuhua.,(∈,∈∨q(λ,μ))-fuzzy Drazin subsemigroup[C]//Proc of Information Technology And Environmental System Sciences.ITESS,2008,6:997~1002.
[10]Liao Zuhua,Chen Min.(∈,∈∨q(λ,μ))-fuzzy subsemigroups and(∈,∈∨q(λ,μ))-Fuzzy completely regular subsemigroups[J].Journal of Jiangnan University(Natural Science Edition),2009,8(2):242~244(in Chinese)(廖祖华,陈敏.(∈,∈∨q(λ,μ))-模糊子半群和(∈,∈∨q(λ,μ))-模糊完全正则子半群[J].江南大学学报(自然科学版),2009,8(2):242~244).
[11]Liao Zuhua,Yi Lihua,Hu Miaohan.(∈,∈∨q(λ,μ))-fuzzy k-ideals of semigroups[J].Journal of Mathematics,2012,32(2):191~205.
[12]Fan Yingying,Liao Zuhua,Zeng Junqiao,et al.(∈,∈∨q(λ,μ))-fuzzyΓ-subsemigroups[J].Fuzzy Sets and Systems,2013,27(5):69~73(in Chinese)(范莹莹,廖祖华,曾俊俏等.(∈,∈∨q(λ,μ))-模糊Γ-子半群[J].模糊系统与数学,2013,27(5):69~73).
[13]Zeng Junqiao,Liao Zuhua,Fan Yingying,et al.(∈,∈∨q(λ,μ))-fuzzyΓ-subrings and homomorphism[J].Fuzzy Sets and Systems,2014,28(1):15~22(in Chinese).(曾俊俏,廖祖华,范莹莹等.(∈,∈∨q(λ,μ))-模糊Γ-子环及同态[J].模糊系统与数学,2014,28(1):15~22).
[14]Biswas R.Fuzzy subgroups and anti—fuzzy subgroups[J].Fuzzy Sets and Systems,1990,35(1):121~124.
[15]Zhang Cheng,Yuan Xuehai.(∈′,∈′∨q′)-fuzzy subgroup[J].Journal of Liaoning Normal University(natural science),1997,20,186~189.
[16]Yuan Xuehai,Fu Yunpeng,Liu Xin,et al.(β-,α-)-fuzzy subgroups[J].Journal of Liaoning Normal University(Nature Science Edition),2003,26(1):1~4.
[17]Guo Ziyan,Liao Zuhua,Guo Jianfu.(∈-,∈-∨q-(λ,μ))-fuzzy subgroup[C]//Proc of the Sixth International Conference on Information and Management Sciences.Lhasa,Tibet:IMS,2007:570~574.
[18]Yang Jihui,Liu Xin,Han Ying.(∈-,∈-∨q-)-Normal fuzzy subgroup[J].Journal of Liaoning Normal University,2003,26(4):350~352(in Chinese).(杨吉会,刘新,韩滢.(∈-,∈-∨q-)-正规模糊子群[J].辽宁师范大学学报(自然科学版),2003,26(4):350~352).
[19]Hao Cuiyun,Liao Zuhua,Ji Min,et al.Some properties of(∈-,∈-∨q-(λ,μ))-fuzzy subgroups[J].Fuzzy Sets and Systems,2012,26(6):26~30(in Chinese).(郝翠芸,廖祖华,嵇敏等.(∈-,∈-∨q-(λ,μ))-模糊子群的若干性质[J].模糊系统与数学,2012,26(6):26~30).
[20]Hao Cuiyun,Liao Zuhua.Some properties of(∈-,∈-∨q-(λ,μ))-fuzzy normal subgroups[J].Fuzzy Sets and Systems,2013,33(5):857~864(in Chinese).(郝翠芸,廖祖华.(∈-,∈-∨q-(λ,μ))-模糊正规子群的若干性质[J].数学杂志.2013,33(5):857~864).
[21]Zhu Chan,Liao Zuhua.(∈-,∈-∨q-(λ,μ))-fuzzy Complemented Semirings[J].Fuzzy Sets and Systems(accepted)(in Chinese).(朱婵,廖祖华等.(∈-,∈-∨q-(λ,μ))-模糊可补半环[J].模糊系统与数学(已录用)).
[22]Chen T Y.Lower cut set decomposition theorem and representation theorem[J].BUSEFAL,1995,63:46~48.
[23]Zhang Cheng,Chen Tuyun.Fuzzy sets and order set embedding[J].Journal of Liaoning Normal University,1995(4):275~279(in Chinese).(张成,陈图云.模糊集与顺序集合套[J].辽宁师范大学学报,1995(4):275~279).
[24]Qu Wanling,Geng Suyun,Zhang Li’ang.Discrete mathematics[M].Beijing:Higher Education Press,2008(in Chinese).(屈婉玲,耿素云,张立昂.离散数学[M].北京:高等教育出版社,2008).
[25]Liao Zuhua.Anti-fuzzy subgroupoids and anti-fuzzy completely regular subsemigroups[J].Journal of Lanzhou University(Nature Sciences),1996(32):120~123(in Chinese).(廖祖华.反F-子广群与反F-完全正则子半群[J].兰州大学学报(自然科学版)(96年模糊理论与应用国际会议暨八届年会),1996(32):120~123).
[26]Petrich M,Norman R.Completely regular semigroups[M].New York:John Wiley&Sons Inc,1999.