关于否定非对合剩余格的模糊LI-理想
|
摘要
运用模糊集及分析学的方法和技巧对否定非对合剩余格的模糊LI-理想问题作深入研究。证明了一个给定的否定非对合剩余格L的全体模糊LI-理想之集FLI(L)关于模糊集合包含序?构成完备Heyting代数。并给出了完备Heyting代数(FLI(L),?)中蕴涵算子的表示定理。
In this paper,we further study the problem of fuzzy LI-ideals in negative non-involutive residuated lattices by using the principle and method of fuzzy sets and analysis.It is proved that the set of all fuzzy LI-ideals FLI(L)in a given negative non-involutive residuated lattice L,under fuzzy set-inclusion order?,forms a complete Heyting algebra.Finally,a representation theorem of implication in the complete Heyting algebra(FLI(L),?)is given.
In this paper,we further study the problem of fuzzy LI-ideals in negative non-involutive residuated lattices by using the principle and method of fuzzy sets and analysis.It is proved that the set of all fuzzy LI-ideals FLI(L)in a given negative non-involutive residuated lattice L,under fuzzy set-inclusion order?,forms a complete Heyting algebra.Finally,a representation theorem of implication in the complete Heyting algebra(FLI(L),?)is given.
引文
[1]王国俊.非经典数理逻辑与近似推理[M].北京:科学出版社,2003.
[2]Borns D W,Mack J M.An algebraic introduction on mathematical logic[M].Berlin:Springer,1975.
[3]Bolc L,Borowik P.Many-valued logics[M].Berlin:Springer-Verlag,1992.
[4]Hájek P.Metamathematics of fuzzy logic[M].Dordrecht:Kluwer Academic Publishers,1998.
[5]Xu Y,et al.Lattice-valued logics[M].Berlin:Springer-Verlag,2003.
[6]张小红.模糊逻辑及其代数分析[M].北京:科学出版社,2008.
[7]Wang G J,Zhou H J.Introduction to mathematical logic and resolution principle(second edition)[M].Beijing:Science Press,2009.
[8]裴道武.基于三角模的模糊逻辑理论及其应用[M].北京:科学出版社,2013.
[9]Ward M,Dilworth R P.Residuated lattices[J].Trans.Amer.Math.Soc.,1939,45:335~354.
[10]Pavelka J.On fuzzy logic I,II,III[J].Zeitschr.F Math Logic and Grundlagend Math,1979,25:45~52;119~134;447~464.
[11]裴道武.剩余格与正则剩余格的特征[J].数学学报,2002,45(2):271~278.
[12]方进明.剩余格与模糊集[M].北京:科学出版社,2012.
[13]Liu Y L,Liu S Y,Xu Y,Qin K Y.ILI-ideals and prime LI-ideals in lattice implication algebras[J].Information Sciences,2003,155:157~175.
[14]Zhang X H,Jun Y B,Doh M I.Fuzzy filters and fuzzy ideals of BL-algebras[J].Fuzzy Systems and Mathematics,2006,20(3):8~20.
[15]Michiro K,Wieslaw A D.Filter theory of BL-algebras[J].Soft Comput.,2008,12:419~423.
[16]Zhang X H,Zhou H J,Mao X Y.IMTL(MV)-filters and fuzzy IMTL(MV)-filters of residuated lattices[J].J.Intelligent&Fuzzy Systems,2014,26(2):589~596.
[17]Dumitru B,Dana P.Some types of filters on residuated lattices[J].Soft Comput.,2014,18:825~837.
[18]刘春辉.格蕴涵代数的素模糊LI-理想及其谱空间[J].高校应用数学学报,2014,29(1):115~126.
[19]刘春辉.否定非对合剩余格的LI-理想理论[J].高校应用数学学报,2015,30(4):445~456.
[20]Liu Y,Qin Y,Qin X Y,Xu Y.Ideals and fuzzy ideals on residuated lattices[J].International Journal of Machine Learning and Cybernetics,2014:1~15.
[21]Zhang J L,Zhou H J.Fuzzy filters on the residuated lattices[J].New Mathematics and Natural Computation,2006,2(1):11~28.
[22]郑崇友,樊磊,崔宏斌.Frame与连续格[M].北京:首都师范大学出版社,2000.
[2]Borns D W,Mack J M.An algebraic introduction on mathematical logic[M].Berlin:Springer,1975.
[3]Bolc L,Borowik P.Many-valued logics[M].Berlin:Springer-Verlag,1992.
[4]Hájek P.Metamathematics of fuzzy logic[M].Dordrecht:Kluwer Academic Publishers,1998.
[5]Xu Y,et al.Lattice-valued logics[M].Berlin:Springer-Verlag,2003.
[6]张小红.模糊逻辑及其代数分析[M].北京:科学出版社,2008.
[7]Wang G J,Zhou H J.Introduction to mathematical logic and resolution principle(second edition)[M].Beijing:Science Press,2009.
[8]裴道武.基于三角模的模糊逻辑理论及其应用[M].北京:科学出版社,2013.
[9]Ward M,Dilworth R P.Residuated lattices[J].Trans.Amer.Math.Soc.,1939,45:335~354.
[10]Pavelka J.On fuzzy logic I,II,III[J].Zeitschr.F Math Logic and Grundlagend Math,1979,25:45~52;119~134;447~464.
[11]裴道武.剩余格与正则剩余格的特征[J].数学学报,2002,45(2):271~278.
[12]方进明.剩余格与模糊集[M].北京:科学出版社,2012.
[13]Liu Y L,Liu S Y,Xu Y,Qin K Y.ILI-ideals and prime LI-ideals in lattice implication algebras[J].Information Sciences,2003,155:157~175.
[14]Zhang X H,Jun Y B,Doh M I.Fuzzy filters and fuzzy ideals of BL-algebras[J].Fuzzy Systems and Mathematics,2006,20(3):8~20.
[15]Michiro K,Wieslaw A D.Filter theory of BL-algebras[J].Soft Comput.,2008,12:419~423.
[16]Zhang X H,Zhou H J,Mao X Y.IMTL(MV)-filters and fuzzy IMTL(MV)-filters of residuated lattices[J].J.Intelligent&Fuzzy Systems,2014,26(2):589~596.
[17]Dumitru B,Dana P.Some types of filters on residuated lattices[J].Soft Comput.,2014,18:825~837.
[18]刘春辉.格蕴涵代数的素模糊LI-理想及其谱空间[J].高校应用数学学报,2014,29(1):115~126.
[19]刘春辉.否定非对合剩余格的LI-理想理论[J].高校应用数学学报,2015,30(4):445~456.
[20]Liu Y,Qin Y,Qin X Y,Xu Y.Ideals and fuzzy ideals on residuated lattices[J].International Journal of Machine Learning and Cybernetics,2014:1~15.
[21]Zhang J L,Zhou H J.Fuzzy filters on the residuated lattices[J].New Mathematics and Natural Computation,2006,2(1):11~28.
[22]郑崇友,樊磊,崔宏斌.Frame与连续格[M].北京:首都师范大学出版社,2000.