格蕴涵代数的Ω-模糊子代数
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摘要
给定一个集合Ω,将Ω-模糊集与格蕴涵代数相结合,引入了格蕴涵代数Ω-模糊子代数的概念,并研究了其相关性质;给出了格蕴涵代数Ω-模糊子代数的若干等价刻画,讨论了格蕴涵代数Ω-模糊子代数与格蕴涵代数子代数之间的相互关系;研究了格蕴涵代数Ω-模糊子代数在Ω-模糊集下交、并等运算下的基本性质;讨论了格蕴涵代数Ω-模糊子代数的同态像与同态原像的基本性质;最后研究了格蕴涵代数Ω-模糊子代数的直积与投影。
Let Ω be a set. By combining the Ω-fuzzy sets and the lattice implication algebras, this paper introduces the concepts of Ω-fuzzy subalgebra in the lattice implication algebra, and investigates some related properties. Beside, this paper obtains some equivalent descriptions of the Ω-fuzzy subalgebra in the lattice implication algebra,and gives the relations between Ω-fuzzy subalgebra and the subalgebra in the lattice implication algebra. Then this paper studies some basic properties of the Ω-fuzzy subalgebra under the intersection, union and other operations ofΩ-fuzzy sets, and discusses some basic properties of homomorphic image and homomorphic preimage of Ω-fuzzy subalgebra. Finally, this paper studies the Ω-fuzzy subalgebra direct product in the lattice implication algebra.
Let Ω be a set. By combining the Ω-fuzzy sets and the lattice implication algebras, this paper introduces the concepts of Ω-fuzzy subalgebra in the lattice implication algebra, and investigates some related properties. Beside, this paper obtains some equivalent descriptions of the Ω-fuzzy subalgebra in the lattice implication algebra,and gives the relations between Ω-fuzzy subalgebra and the subalgebra in the lattice implication algebra. Then this paper studies some basic properties of the Ω-fuzzy subalgebra under the intersection, union and other operations ofΩ-fuzzy sets, and discusses some basic properties of homomorphic image and homomorphic preimage of Ω-fuzzy subalgebra. Finally, this paper studies the Ω-fuzzy subalgebra direct product in the lattice implication algebra.
引文
[1]Zadeh L A.Fuzzy sets[J].Information and Control,1965,8(3):338-353.
[2]Rosenfeld A.Fuzzy groups[J].Journal of Mathematical Analysis and Applications,1971,35(3):512-517.
[3]Zadeh L A.The concept of a linguistic variable and its application to approximate reasoning[J].Information Science,1975,8(3):199-249.
[4]Pu Paoming,Liu Yingming.Fuzzy topology(I)neighborhood structure of a fuzzy point and Moore-Smith convergence[J].Journal of Mathematical Analysis and Applications,1980,76(2):571-599.
[5]Pu Paoming,Liu Yingming.Fuzzy topology(II)product and quotient spaces[J].Journal of Mathematical Analysis and Applications,1980,77(1):20-37.
[6]Bhakat S K,Das P.On the definition of a fuzzy subgroup[J].Fuzzy Sets Systems,1992,51(2):235-241.
[7]Bhakat S K,Das P.Fuzzy subgroups[J].Fuzzy Sets Systems,1996,80(3):359-368.
[8]Bhakat S K,Das P.Fuzzy subrings and ideals redefined[J].Fuzzy Sets Systems,1996,81(3):383-393.
[9]Liao Zuhua,Gu Hui.(∈,∈∨q(λ,μ))-fuzzy normal subgroup fuzzy normal subgroup[J].Fuzzy Systems and Mathematics,2006,20(5):38-43.
[10]Zhang Jianzhong,Fu Xiaobo,Liao Zuhua.(∈,∈∨q(λ,μ))-fuzzy ideal of N(2,2,0)algebra[J].Journal of Frontiers of Computer Science and Technology,2013,7(11):1048-1056.
[11]Fan Xiaowei,Liao Zuhua,Fan Yingying,et al.(∈,∈∨q(λ,μ))-fuzzyГ-completely prime ideals ofГ-semigroups[J].Fuzzy Systems and Mathematics,2014,28(2):52-61.
[12]Young B J,Kyung H K.OnΩ-fuzzy ideals of BCK/BCI-algebras[J].The Journal of Fuzzy Mathematics,2001,9(1):173-180.
[13]Peng Jiayin.Ω-fuzzy p-ideals of BCI-algebras[J].Journal of Neijiang Normal University,2008,23(12):8-15.
[14]Peng Jiayin.Ω-fuzzy H-ideals of BCI-algebras[J].Journal of Neijiang Normal University,2009,24(2):5-10.
[15]Peng Jiayin.Ω-fuzzy dot ideals of BCH-algebras[J].Fuzzy Systems and Mathematics,2009,23(6):5-11.
[16]Peng Jiayin.Ω-fuzzy positive implicative ideals in BCK-algebras[J].Mathematics in Practice and Theory,2010,40(2):157-163.
[17]Zhu Xiaoying,Liao Zuhua,Luo Xiaotang,et al.Ω-fuzzy subsemigroup of semigroups[J].Journal of Jiangnan University:Natural Science Edition,2013,12(3):343-346.
[18]Luo Xiaotang,Liao Zuhua,Zhu Xiaoying,et al.GeneralizedΩ-fuzzy bi-ideals of semigroups[J].Journal of Jiangnan University:Natural Science Edition,2013,12(4):480-484.
[19]Luo Xiaotang,Liao Zuhua,Zhu Xiaoying,et al.Ω-fuzzy interior ideals of semigroups[J].Journal of Jiangnan University:Natural Science Edition,2014,12(4):400-494.
[20]Zhu Chan,Liao Zuhua,Luo Xiaotang,et al.Ω-fuzzy complemented subsemiring[J].Fuzzy Systems and Mathematics,2014,28(5):11-18.
[21]Liu Weifeng.Ω-fuzzy subalgebra of boolean algebra[J].Journal of Hubei University:Natural Science,2013,35(2):144-148.
[22]Tang Huajing,Zhang Hui.Ω-fuzzy soft rings[J].Computer Engineering and Applications,2015,51(9):122-124.
[23]Wu Wangming.Fuzzy implication algebras[J].Fuzzy Systems and Mathematics,1990,4(1):56-64.
[24]Xu Yang.Lattice implication algebras[J].Journal of Southwest Jiaotong University,1993,28(1):20-27.
[25]Xu Yang,Ruan Da,Qin Keyun.Lattice-valued logic[M].Berlin:Springer,2003.
[26]Liu Yi,Xu Yang,Qin Ya.Interval-valued(α,β)-fuzzy lattice implication subalgebras[J].Computer Science,2011,38(4):263-266.
[27]Qin Xuecheng,Liu Chunhui.Interval-valued fuzzy subalgebras of lattice implication algebras[J].Pure and Applied Mathematics,2011,27(6):801-807.
[28]Liu Chunhui.On IV-(∈,∈∨q)-fuzzy lattice implication subalgebras[J].Computer Engineering and Applications,2012,48(26):39-43.
[29]Fu Xiaobo,Liao Zuhua,Zheng Gaoping.(∈,∈∨q(λ,μ))-fuzzy subalgebra of lattice implication algebras[J].Fuzzy Systems and Mathematics,2014,28(6):44-52.
[10]张建忠,傅小波,廖祖华.N(2,2,0)代数的(∈,∈∨q(λ,μ))-理想[J].计算机科学与探索,2013,7(11):1048-1056.
[11]范晓威,廖祖华,范莹莹,等.Γ-半群的(∈,∈∨q(λ,μ))-模糊Γ-完全素理想[J].模糊系统与数学,2014,28(2):52-61.
[15]彭家寅.BCH-代数的Ω-模糊点理想[J].模糊系统与数学,2009,23(6):5-11.
[16]彭家寅.BCH-代数的Ω-模糊正定关联理想[J].数学的实践与认识,2010,40(2):157-163.
[17]朱晓英,廖祖华,罗晓棠,等.半群的Ω-模糊子半群[J].江南大学学报:自然科学版,2013,12(3):343-346.
[18]罗晓棠,廖祖华,朱晓英,等.半群的广义Ω-模糊双理想[J].江南大学学报:自然科学版,2013,12(4):480-484.
[19]罗晓棠,廖祖华,朱晓英,等.半群的Ω-模糊内理想[J].江南大学学报:自然科学版,2014,12(4):400-494.
[20]朱婵,廖祖华,罗晓棠,等.Ω-模糊可补子半环[J].模糊系统与数学,2014,28(5):11-18.
[21]汤华晶,张慧.Ω-模糊软环[J].计算机工程与应用,2015,51(9):122-124.
[22]刘卫锋.布尔代数的Ω-模糊子代数[J].湖北大学学报:自然科学版,2013,35(2):144-148.
[23]吴望名.Fuzzy蕴涵代数[J].模糊系统与数学,1990,4(1):56-64.
[24]徐扬.格蕴涵代数[J].西南交通大学学报,1993,28(1):20-27.
[26]刘熠,徐扬,秦亚.区间值(α,β)-模糊格蕴涵子代数[J].计算机科学,2011,38(4):263-266.
[27]秦学成,刘春辉.格蕴涵代数的区间值模糊子代数[J].纯粹数学与应用数学,2011,27(6):801-807.
[28]刘春辉.关于IV-(∈,∈∨q)模糊格蕴涵子代数[J].计算机工程与应用,2012,48(26):39-43.
[29]傅小波,廖祖华,郑高平,等.格蕴涵代数(∈,∈∨q(λ,μ))-模糊子代数[J].模糊系统与数学,2014,28(6):44-52.
[2]Rosenfeld A.Fuzzy groups[J].Journal of Mathematical Analysis and Applications,1971,35(3):512-517.
[3]Zadeh L A.The concept of a linguistic variable and its application to approximate reasoning[J].Information Science,1975,8(3):199-249.
[4]Pu Paoming,Liu Yingming.Fuzzy topology(I)neighborhood structure of a fuzzy point and Moore-Smith convergence[J].Journal of Mathematical Analysis and Applications,1980,76(2):571-599.
[5]Pu Paoming,Liu Yingming.Fuzzy topology(II)product and quotient spaces[J].Journal of Mathematical Analysis and Applications,1980,77(1):20-37.
[6]Bhakat S K,Das P.On the definition of a fuzzy subgroup[J].Fuzzy Sets Systems,1992,51(2):235-241.
[7]Bhakat S K,Das P.Fuzzy subgroups[J].Fuzzy Sets Systems,1996,80(3):359-368.
[8]Bhakat S K,Das P.Fuzzy subrings and ideals redefined[J].Fuzzy Sets Systems,1996,81(3):383-393.
[9]Liao Zuhua,Gu Hui.(∈,∈∨q(λ,μ))-fuzzy normal subgroup fuzzy normal subgroup[J].Fuzzy Systems and Mathematics,2006,20(5):38-43.
[10]Zhang Jianzhong,Fu Xiaobo,Liao Zuhua.(∈,∈∨q(λ,μ))-fuzzy ideal of N(2,2,0)algebra[J].Journal of Frontiers of Computer Science and Technology,2013,7(11):1048-1056.
[11]Fan Xiaowei,Liao Zuhua,Fan Yingying,et al.(∈,∈∨q(λ,μ))-fuzzyГ-completely prime ideals ofГ-semigroups[J].Fuzzy Systems and Mathematics,2014,28(2):52-61.
[12]Young B J,Kyung H K.OnΩ-fuzzy ideals of BCK/BCI-algebras[J].The Journal of Fuzzy Mathematics,2001,9(1):173-180.
[13]Peng Jiayin.Ω-fuzzy p-ideals of BCI-algebras[J].Journal of Neijiang Normal University,2008,23(12):8-15.
[14]Peng Jiayin.Ω-fuzzy H-ideals of BCI-algebras[J].Journal of Neijiang Normal University,2009,24(2):5-10.
[15]Peng Jiayin.Ω-fuzzy dot ideals of BCH-algebras[J].Fuzzy Systems and Mathematics,2009,23(6):5-11.
[16]Peng Jiayin.Ω-fuzzy positive implicative ideals in BCK-algebras[J].Mathematics in Practice and Theory,2010,40(2):157-163.
[17]Zhu Xiaoying,Liao Zuhua,Luo Xiaotang,et al.Ω-fuzzy subsemigroup of semigroups[J].Journal of Jiangnan University:Natural Science Edition,2013,12(3):343-346.
[18]Luo Xiaotang,Liao Zuhua,Zhu Xiaoying,et al.GeneralizedΩ-fuzzy bi-ideals of semigroups[J].Journal of Jiangnan University:Natural Science Edition,2013,12(4):480-484.
[19]Luo Xiaotang,Liao Zuhua,Zhu Xiaoying,et al.Ω-fuzzy interior ideals of semigroups[J].Journal of Jiangnan University:Natural Science Edition,2014,12(4):400-494.
[20]Zhu Chan,Liao Zuhua,Luo Xiaotang,et al.Ω-fuzzy complemented subsemiring[J].Fuzzy Systems and Mathematics,2014,28(5):11-18.
[21]Liu Weifeng.Ω-fuzzy subalgebra of boolean algebra[J].Journal of Hubei University:Natural Science,2013,35(2):144-148.
[22]Tang Huajing,Zhang Hui.Ω-fuzzy soft rings[J].Computer Engineering and Applications,2015,51(9):122-124.
[23]Wu Wangming.Fuzzy implication algebras[J].Fuzzy Systems and Mathematics,1990,4(1):56-64.
[24]Xu Yang.Lattice implication algebras[J].Journal of Southwest Jiaotong University,1993,28(1):20-27.
[25]Xu Yang,Ruan Da,Qin Keyun.Lattice-valued logic[M].Berlin:Springer,2003.
[26]Liu Yi,Xu Yang,Qin Ya.Interval-valued(α,β)-fuzzy lattice implication subalgebras[J].Computer Science,2011,38(4):263-266.
[27]Qin Xuecheng,Liu Chunhui.Interval-valued fuzzy subalgebras of lattice implication algebras[J].Pure and Applied Mathematics,2011,27(6):801-807.
[28]Liu Chunhui.On IV-(∈,∈∨q)-fuzzy lattice implication subalgebras[J].Computer Engineering and Applications,2012,48(26):39-43.
[29]Fu Xiaobo,Liao Zuhua,Zheng Gaoping.(∈,∈∨q(λ,μ))-fuzzy subalgebra of lattice implication algebras[J].Fuzzy Systems and Mathematics,2014,28(6):44-52.
[10]张建忠,傅小波,廖祖华.N(2,2,0)代数的(∈,∈∨q(λ,μ))-理想[J].计算机科学与探索,2013,7(11):1048-1056.
[11]范晓威,廖祖华,范莹莹,等.Γ-半群的(∈,∈∨q(λ,μ))-模糊Γ-完全素理想[J].模糊系统与数学,2014,28(2):52-61.
[15]彭家寅.BCH-代数的Ω-模糊点理想[J].模糊系统与数学,2009,23(6):5-11.
[16]彭家寅.BCH-代数的Ω-模糊正定关联理想[J].数学的实践与认识,2010,40(2):157-163.
[17]朱晓英,廖祖华,罗晓棠,等.半群的Ω-模糊子半群[J].江南大学学报:自然科学版,2013,12(3):343-346.
[18]罗晓棠,廖祖华,朱晓英,等.半群的广义Ω-模糊双理想[J].江南大学学报:自然科学版,2013,12(4):480-484.
[19]罗晓棠,廖祖华,朱晓英,等.半群的Ω-模糊内理想[J].江南大学学报:自然科学版,2014,12(4):400-494.
[20]朱婵,廖祖华,罗晓棠,等.Ω-模糊可补子半环[J].模糊系统与数学,2014,28(5):11-18.
[21]汤华晶,张慧.Ω-模糊软环[J].计算机工程与应用,2015,51(9):122-124.
[22]刘卫锋.布尔代数的Ω-模糊子代数[J].湖北大学学报:自然科学版,2013,35(2):144-148.
[23]吴望名.Fuzzy蕴涵代数[J].模糊系统与数学,1990,4(1):56-64.
[24]徐扬.格蕴涵代数[J].西南交通大学学报,1993,28(1):20-27.
[26]刘熠,徐扬,秦亚.区间值(α,β)-模糊格蕴涵子代数[J].计算机科学,2011,38(4):263-266.
[27]秦学成,刘春辉.格蕴涵代数的区间值模糊子代数[J].纯粹数学与应用数学,2011,27(6):801-807.
[28]刘春辉.关于IV-(∈,∈∨q)模糊格蕴涵子代数[J].计算机工程与应用,2012,48(26):39-43.
[29]傅小波,廖祖华,郑高平,等.格蕴涵代数(∈,∈∨q(λ,μ))-模糊子代数[J].模糊系统与数学,2014,28(6):44-52.