格蕴涵代数的犹豫模糊LI-理想
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摘要
将犹豫模糊集应用于格蕴涵代数,提出了格蕴涵代数犹豫模糊LI-理想的概念。给出了格蕴涵代数犹豫模糊LI-理想若干等价刻画,研究了格蕴涵代数犹豫模糊LI-理想的一些性质;给出了犹豫模糊特征函数的定义,讨论了犹豫模糊LI-理想和LI-理想的关系。最后给出了犹豫模糊集下像与原像的定义,研究了犹豫模糊LI-理想像与原像的性质。
By applying the hesitant fuzzy set theory to lattice implications algebras, the concepts of hesitant fuzzy LI-ideals of lattice implications algebras are introduced. Some equivalent characterizations of hesitant fuzzy LI-ideals of lattice implications algebras are also discussed; Several properties of the lattice implications algebras are investigated. The nation of the characteristic hesitant fuzzy set is proposed, and the relationship between hesitant fuzzy LI-ideals and LI-ideals is discussed. Finally, we introduce the concept of homomorphic image and homomorphic preimage of hesitant fuzzy set, and discuss the properties of homomorphic image and homomorphic preimage of hesitant fuzzy LI-ideals.
By applying the hesitant fuzzy set theory to lattice implications algebras, the concepts of hesitant fuzzy LI-ideals of lattice implications algebras are introduced. Some equivalent characterizations of hesitant fuzzy LI-ideals of lattice implications algebras are also discussed; Several properties of the lattice implications algebras are investigated. The nation of the characteristic hesitant fuzzy set is proposed, and the relationship between hesitant fuzzy LI-ideals and LI-ideals is discussed. Finally, we introduce the concept of homomorphic image and homomorphic preimage of hesitant fuzzy set, and discuss the properties of homomorphic image and homomorphic preimage of hesitant fuzzy LI-ideals.
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,1971,35:512～517.
[3] Chen N,Xu Z S ,Xia M M.Interval-valued hesitant preference relations and their applications to group decision making[J].Knowledge-Based Systems,2013,37:528～540.
[4] Atanassow K T.Intuitionistic fuzzy sets[J].Fuzzy Sets and Systems,1986,20(1):87～96.
[5] Young B J,Kyung H K.On Ω-fuzzy ideals of BCK/BCI-algebras[J].The Journal of Fuzzy Mathematics,2001,9(1):4173～180.
[6] Torra V.Hesitant fuzzy sets[J].International Journal of Intelligent Systems,2010,25(6):529～539.
[7] Torra V,Narukawa Y.On hesitant fuzzy sets and decision[C]//The 18th IEEE International Conference on Fuzzy Systems,Jeju Island,Korea,2009:1378～1382.
[8] Xia M M,Xu Z H.Hesitant fuzzy information aggregation in decision making[J].International Journal of Approximate Reasoning,2011,52(3):395～407.
[9] Xu Z H,Xia M M.Distance and similarity measures for hesitant fuzzy sets[J].Information Sciences,2011,181(11):2128～2138.
[10] Xu Z H,Xia M M.On distance and correlation measures of hesitant fuzzy information[J].International Journal of Intelligent Systems,2011,26(5):410～425.
[11] Rodrguez R M,Martnez L,Herrera F.Hesitant fuzzy linguistic term sets for decision making[J].IEEE Transactions on Fuzzy Systems,2012,20(1):109～119.
[12] Zhang H D,et al.Dual hesitant fuzzy rough set and its application[J].Soft Computing,2017,21(12):3287～3305.
[13] Xia M M,Xu Z S.Some studies on properties of hesitant fuzzy sets[J].International Journal of Machine Learning and Cybernetics,2017,8(2):489～495.
[14] Xia M M,Xu Z S,Zhu B.Some hesitant fuzzy aggregation operators with their application in group decision making[J].Group Decision and Negotiation,2013,22(2):259～279.
[15] Zhu B,Xu Z S,Xia M M.Hesitant fuzzy geometric Bonferroni means[J].Information Sciences,2012,205(1):72～85.
[16] Chen N,Xu Z S ,Xia M M.Interval-valued hesitant preference relations and their applications to group decision making[J].Knowledge-Based Systems,2013,37:528～540.
[17] Zhao H,Xu Z S,Liu S D.Dual hesitant fuzzy information aggregation with Einstein t-conorm and t-norm[J].Journal of Systems Science and Systems Engineering,2017,26(2):240～264.
[18] Zhu B,Xu Z S.Extended hesitant fuzzy sets[J].Technological and Economic Development of Economy,2016,22(1):100～121.
[19] 吴望名.Fuzzy蕴涵代数[J].模糊系统与数学,1990,4(1):56～64.
[20] 徐扬.格蕴涵代数[J].西南交通大学学报,1993,(1):20～27.
[21] Xu Y,Ruan D,Qin K Y.Lattice-valued logic[M].Berlin:Springer,2003.
[22] Jun Y B,Roh E H,Xu Y.LI-ideals in lattice implication algebras[J].Bulletin of the Korean Mathematical Society,1998,5(1):13～24.
[23] Young B,Xu Y,Qin K Y.Positive implicative and associative ideals of lattice implication algebras[J].Bulletin of the Korean Mathematical Society,1998,1(35):3～61.
[24] Zhao G F,Xu Y,Song Z M.Implicative and fuzzy implicative ideals of lattice implication algebras[J].Chinese Quarterly Journal of Mathematics,2001,16(4):105～110.
[25] Qin K Y,Xu Y,Jun Y B.Ultra LI-ideals in lattice implication algebras[J].Czechoslovak Mathematical Journal,2002,52(127):463～468.
[26] Liu Y L,Liu S Y,Xu Y.ILI-ideals and prime LI- ideals in lattice implication algebra[J].Information Sciences,2003,155(1/2):157～175.
[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.格蕴涵代数的(∈,∈∨q(λ,μ))-模糊滤子[J] .计算机科学与探索,2014,8(3):376～384.
[30] 傅小波,廖祖华.格蕴涵代数的(∈,∈∨q(λ,μ))-模糊素滤子[J] .计算机科学与探索,2015,9(2):227～233.
[31] 傅小波,廖祖华.格蕴涵代数的Ω-模糊子代数[J].计算机科学与探索,2017,11(7):1183～1190.
[32] Jun Y B,Lee K J,Song S Z.Hesitant fuzzy bi-ideals in semigroups[J].Journal of the Korean Mathematical Society,2015,30(3):43～154.
[33] Jun Y B,Ahn S S.Hesitant fuzzy set theory applied to BCK/BCI-algebras[J].Journal of Computational Analysis and Applitations,2016,20(4):635～645.
[34] Jun Y B,Song S Z.Hesitant fuzzy set theory applied to filters in MTL-algebras[J].Honam Mathematical Journal,2014,36(4):813～830.
[35] Muhiudin G,et al.Hesitant fuzzy filters in lattice algebras[J].Journal of Computational Analysis and Applications,2017,22(6):1105～1113.
[36] Jun Y B,Ahn S S.On hesitant fuzzy filters in BE-algebras[J].Journal of Computational Analysis and Applications,2017,22(2):346～358.
[37] Deepa V,Sunil J J.Hesitant fuzzy subrings and ideals[J].Far East Journal of Mathematical Sciences,2017,101(12):2685～2702.
[2] Rosenfeld A.Fuzzy groups [J].Journal of Mathematical Analysis and Applications,1971,35:512～517.
[3] Chen N,Xu Z S ,Xia M M.Interval-valued hesitant preference relations and their applications to group decision making[J].Knowledge-Based Systems,2013,37:528～540.
[4] Atanassow K T.Intuitionistic fuzzy sets[J].Fuzzy Sets and Systems,1986,20(1):87～96.
[5] Young B J,Kyung H K.On Ω-fuzzy ideals of BCK/BCI-algebras[J].The Journal of Fuzzy Mathematics,2001,9(1):4173～180.
[6] Torra V.Hesitant fuzzy sets[J].International Journal of Intelligent Systems,2010,25(6):529～539.
[7] Torra V,Narukawa Y.On hesitant fuzzy sets and decision[C]//The 18th IEEE International Conference on Fuzzy Systems,Jeju Island,Korea,2009:1378～1382.
[8] Xia M M,Xu Z H.Hesitant fuzzy information aggregation in decision making[J].International Journal of Approximate Reasoning,2011,52(3):395～407.
[9] Xu Z H,Xia M M.Distance and similarity measures for hesitant fuzzy sets[J].Information Sciences,2011,181(11):2128～2138.
[10] Xu Z H,Xia M M.On distance and correlation measures of hesitant fuzzy information[J].International Journal of Intelligent Systems,2011,26(5):410～425.
[11] Rodrguez R M,Martnez L,Herrera F.Hesitant fuzzy linguistic term sets for decision making[J].IEEE Transactions on Fuzzy Systems,2012,20(1):109～119.
[12] Zhang H D,et al.Dual hesitant fuzzy rough set and its application[J].Soft Computing,2017,21(12):3287～3305.
[13] Xia M M,Xu Z S.Some studies on properties of hesitant fuzzy sets[J].International Journal of Machine Learning and Cybernetics,2017,8(2):489～495.
[14] Xia M M,Xu Z S,Zhu B.Some hesitant fuzzy aggregation operators with their application in group decision making[J].Group Decision and Negotiation,2013,22(2):259～279.
[15] Zhu B,Xu Z S,Xia M M.Hesitant fuzzy geometric Bonferroni means[J].Information Sciences,2012,205(1):72～85.
[16] Chen N,Xu Z S ,Xia M M.Interval-valued hesitant preference relations and their applications to group decision making[J].Knowledge-Based Systems,2013,37:528～540.
[17] Zhao H,Xu Z S,Liu S D.Dual hesitant fuzzy information aggregation with Einstein t-conorm and t-norm[J].Journal of Systems Science and Systems Engineering,2017,26(2):240～264.
[18] Zhu B,Xu Z S.Extended hesitant fuzzy sets[J].Technological and Economic Development of Economy,2016,22(1):100～121.
[19] 吴望名.Fuzzy蕴涵代数[J].模糊系统与数学,1990,4(1):56～64.
[20] 徐扬.格蕴涵代数[J].西南交通大学学报,1993,(1):20～27.
[21] Xu Y,Ruan D,Qin K Y.Lattice-valued logic[M].Berlin:Springer,2003.
[22] Jun Y B,Roh E H,Xu Y.LI-ideals in lattice implication algebras[J].Bulletin of the Korean Mathematical Society,1998,5(1):13～24.
[23] Young B,Xu Y,Qin K Y.Positive implicative and associative ideals of lattice implication algebras[J].Bulletin of the Korean Mathematical Society,1998,1(35):3～61.
[24] Zhao G F,Xu Y,Song Z M.Implicative and fuzzy implicative ideals of lattice implication algebras[J].Chinese Quarterly Journal of Mathematics,2001,16(4):105～110.
[25] Qin K Y,Xu Y,Jun Y B.Ultra LI-ideals in lattice implication algebras[J].Czechoslovak Mathematical Journal,2002,52(127):463～468.
[26] Liu Y L,Liu S Y,Xu Y.ILI-ideals and prime LI- ideals in lattice implication algebra[J].Information Sciences,2003,155(1/2):157～175.
[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.格蕴涵代数的(∈,∈∨q(λ,μ))-模糊滤子[J] .计算机科学与探索,2014,8(3):376～384.
[30] 傅小波,廖祖华.格蕴涵代数的(∈,∈∨q(λ,μ))-模糊素滤子[J] .计算机科学与探索,2015,9(2):227～233.
[31] 傅小波,廖祖华.格蕴涵代数的Ω-模糊子代数[J].计算机科学与探索,2017,11(7):1183～1190.
[32] Jun Y B,Lee K J,Song S Z.Hesitant fuzzy bi-ideals in semigroups[J].Journal of the Korean Mathematical Society,2015,30(3):43～154.
[33] Jun Y B,Ahn S S.Hesitant fuzzy set theory applied to BCK/BCI-algebras[J].Journal of Computational Analysis and Applitations,2016,20(4):635～645.
[34] Jun Y B,Song S Z.Hesitant fuzzy set theory applied to filters in MTL-algebras[J].Honam Mathematical Journal,2014,36(4):813～830.
[35] Muhiudin G,et al.Hesitant fuzzy filters in lattice algebras[J].Journal of Computational Analysis and Applications,2017,22(6):1105～1113.
[36] Jun Y B,Ahn S S.On hesitant fuzzy filters in BE-algebras[J].Journal of Computational Analysis and Applications,2017,22(2):346～358.
[37] Deepa V,Sunil J J.Hesitant fuzzy subrings and ideals[J].Far East Journal of Mathematical Sciences,2017,101(12):2685～2702.