基于正则FI代数的MT理想及其应用
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摘要
本文基于经典代数的角度对正则FI代数进行了再研究.首先,在正则FI代数中通过蕴涵算子提出了MT理想的概念,讨论了正则FI代数中MT理想与同余关系的联系;其次,在正则FI代数中引入素MT理想的概念,并以素MT理想为工具给出了正则FI代数的条件嵌入定理;最后,通过以蕴涵算子表示的隐式余三角模对MT理想的特征进行了描述,并通过特征定理给出了正则FI代数中MT理想的生成方法.
In the present paper,the regular FI algebras have been investigated again based on the idea of classical algebras.Firstly,in regular FI algebras the concept of the MT ideal is proposed by the implication operator,and the connections between MT ideals and congruence relations are discussed;Secondly,in regular FI algebras the prime MT ideals is defined,and the theorem of conditional embedding of regular FI regulars is gained by the prime MT ideals.Thirdly,the characteristic of MT ideals is described with the help of co-triangle-norm on regular FI algebras that is shown by the form of implication operators,and the generating method of MT ideals is given by the theorem of characteristic of MT ideals in regular FI algebras.
In the present paper,the regular FI algebras have been investigated again based on the idea of classical algebras.Firstly,in regular FI algebras the concept of the MT ideal is proposed by the implication operator,and the connections between MT ideals and congruence relations are discussed;Secondly,in regular FI algebras the prime MT ideals is defined,and the theorem of conditional embedding of regular FI regulars is gained by the prime MT ideals.Thirdly,the characteristic of MT ideals is described with the help of co-triangle-norm on regular FI algebras that is shown by the form of implication operators,and the generating method of MT ideals is given by the theorem of characteristic of MT ideals in regular FI algebras.
引文
[1]吴望名.Fuzzy蕴涵代数[J].模糊系统与数学,1990,4(1):56-63.Wu Wang-ming.Fuzzy implication algebras[J].Fuzzy Systemsand Mathematics,1990,4(1):56-63.(in Chinese)
[2]徐扬.格蕴涵代数[J].西南交通大学学报,1993,28(1):20-27.Xu Yang.Lattice implication algebras[J].Journal of SouthwestJiaotong University,1993,28(1):20-27.(in Chinese)
[3]XU Yang,YUAN Da,QIN Keyun,LIU Jun.Lattice-ValuedLogic[M].Berlin:Spinger-verlag,2003.
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[5]吴洪博.基础R0-代数与基础L*系统[J].数学进展,2003,32(5):565-576.WuHong-bo.Basis R0-algebra and basis L*systems[J].Ad-vances in Mathematics,2003,32(5):565-576.(in Chinese)
[6]P Hájek.Metamathematics of Fuzzy Logic[M].Dordrech:Kluwer Academic Publishers,1998.
[7]李壁镜,王国俊.正则蕴涵算子所对应的逻辑伪度量空间[J].电子学报,2010,38(3):497-502.Li B-i jing,Wang Guo-jun.Logic pseddo-metric spaces ofregu-lar implication operators[J].Acta Electronica Sinica,2010,38(3):497-502.(in Chinese)
[8]胡明娣,王国俊.对称逻辑公式在经典逻辑度量空间中的分布[J].电子学报,2011,39(2):419-423.Hu Ming-di,Wang Guo-jun.Distribution of the symmetricallogic formulas in the classical logic metric space[J].Acta Elec-tronica Sinica,2011,39(2):419-423.(in Chinese)
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[10]吴洪博,邵晓丽.完备剩余格中的全蕴涵推理方法[J].数学进展,2006,35(3):303-314.WuHong-bo,ShaoXiao-li.A method oftotal implication rea-soning on complete residuated lattice[J].Advances in Mathe-matics,2006,35(3):303-314.(in Chinese)
[11]LIU Huawen,WANG Guojun.Triple I method based onpointwise sustaining degrees[J].Computers and Mathematicswith Applications,2008,55(11):2680-2688.
[12]LIU Huawen,WANG Guojun.Unified forms offully implica-tion restriction methods for fuzzy reasoning[J].InformationSciences,2007,177(3):956-966.
[13]李洪兴,彭佳寅,王加银.基于三I算法的模糊系统及其响应性能[J].系统科学与数学,2005,25(5):578-590.Li Hong-xing,Peng Jia-yin,Wang Jia-yin.Fuzzy systemsbased on triple I algorithm and their response ability[J].Jour-nal of Systems Science and Mathematical Sciences,2005,25(5):578-590.(in Chinese)
[14]汪德刚,谷云东,李洪兴.模糊模态命题逻辑及其广义重言式[J].电子学报,2007,35(2):261-264.WangDe-gang,GuYun-dong,LiHong-xing.Generalizedtau-tology in fuzzy model propositional logic[J].Acta ElectronicaSinica,2007,35(2):261-264.(in Chinese)
[15]吴洪博,周建仁.计量逻辑中真度的均值表示及应用[J].电子学报,2012,40(9):1822-1828.WuHong-bo,Zhou Jian-ren.The form ofmean representationof truth degree with application in quantitative logic[J].ActaElectronica Sinica,2012,40(9):1822-1828.(in Chinese)
[16]王国俊,宋建设.命题逻辑中的程度化方法[J].电子学报,2006,34(2):252-257.Wang Guo-jun,Song jian-she.Graded method in propositionallogic[J].Acta Electronica Sinica,2006,34(2):252-257.(inChinese)
[17]吴洪博,周建仁,张琼.(3n+1)值逻辑系统R0L中公式的真度性质[J].电子学报,2011,39(10):2230-2234.Wu Hong-bo,Zhou Jian-ren,Zhang Qiong.The properties oftruth degrees of formulas in(3n+1)-valued logic system R0L[J].Acta Electronica Sinica,2011,39(10):2230-2234.(inChinese)
[18]程国胜.R0-代数中滤子与理想[J].模糊系统与数学,2001,15(1):58-61.Cheng Guo-sheng.The filters and the ideals in R0-algebras[J].Fuzzy Systems and Mathematics,2001,15(1):58-61.(in Chinese)
[19]郑崇友,樊磊,崔宏斌.FRAME与连续格(第二版)[M].北京:首都师范大学,2000.Zheng Chong-you,Fan Lei,Cui Hong-bin.Frame and Contin-uous Lattice(2nd edition)[M].Beijing:Capital Normal Un-iversity,2000.(in Chinese)
[20]李志伟,孙立民,郑崇友.正则Fuzzy蕴涵代数[J].模糊系统与数学,2002,16(2):22-26.Li Zh-i wei,Sun L-i min,Zheng Chong-you.Regular fuzzy im-plication algebra[J].Fuzzy Systems and Mathematics,2002,16(2):22-26.(in Chinese)
[2]徐扬.格蕴涵代数[J].西南交通大学学报,1993,28(1):20-27.Xu Yang.Lattice implication algebras[J].Journal of SouthwestJiaotong University,1993,28(1):20-27.(in Chinese)
[3]XU Yang,YUAN Da,QIN Keyun,LIU Jun.Lattice-ValuedLogic[M].Berlin:Spinger-verlag,2003.
[4]王国俊.非经典数理逻辑与近似推理(第二版)[M].北京:科学出版社,2003.Wang Guo-jun.Non-classical mathematical logic and approx-imate reasoning(2nd edition)[M].Beijing:Science Press,2003.(in Chinese)
[5]吴洪博.基础R0-代数与基础L*系统[J].数学进展,2003,32(5):565-576.WuHong-bo.Basis R0-algebra and basis L*systems[J].Ad-vances in Mathematics,2003,32(5):565-576.(in Chinese)
[6]P Hájek.Metamathematics of Fuzzy Logic[M].Dordrech:Kluwer Academic Publishers,1998.
[7]李壁镜,王国俊.正则蕴涵算子所对应的逻辑伪度量空间[J].电子学报,2010,38(3):497-502.Li B-i jing,Wang Guo-jun.Logic pseddo-metric spaces ofregu-lar implication operators[J].Acta Electronica Sinica,2010,38(3):497-502.(in Chinese)
[8]胡明娣,王国俊.对称逻辑公式在经典逻辑度量空间中的分布[J].电子学报,2011,39(2):419-423.Hu Ming-di,Wang Guo-jun.Distribution of the symmetricallogic formulas in the classical logic metric space[J].Acta Elec-tronica Sinica,2011,39(2):419-423.(in Chinese)
[9]王国俊.模糊推理的全蕴涵三I算法[J].中国科学(E辑),1999,29(1):43-53.WangGuo-jun.Fully implication triple I methods for fuzzy rea-soning system[J].Science in China(Series E),1999,29(1):43-53.(in Chinese)
[10]吴洪博,邵晓丽.完备剩余格中的全蕴涵推理方法[J].数学进展,2006,35(3):303-314.WuHong-bo,ShaoXiao-li.A method oftotal implication rea-soning on complete residuated lattice[J].Advances in Mathe-matics,2006,35(3):303-314.(in Chinese)
[11]LIU Huawen,WANG Guojun.Triple I method based onpointwise sustaining degrees[J].Computers and Mathematicswith Applications,2008,55(11):2680-2688.
[12]LIU Huawen,WANG Guojun.Unified forms offully implica-tion restriction methods for fuzzy reasoning[J].InformationSciences,2007,177(3):956-966.
[13]李洪兴,彭佳寅,王加银.基于三I算法的模糊系统及其响应性能[J].系统科学与数学,2005,25(5):578-590.Li Hong-xing,Peng Jia-yin,Wang Jia-yin.Fuzzy systemsbased on triple I algorithm and their response ability[J].Jour-nal of Systems Science and Mathematical Sciences,2005,25(5):578-590.(in Chinese)
[14]汪德刚,谷云东,李洪兴.模糊模态命题逻辑及其广义重言式[J].电子学报,2007,35(2):261-264.WangDe-gang,GuYun-dong,LiHong-xing.Generalizedtau-tology in fuzzy model propositional logic[J].Acta ElectronicaSinica,2007,35(2):261-264.(in Chinese)
[15]吴洪博,周建仁.计量逻辑中真度的均值表示及应用[J].电子学报,2012,40(9):1822-1828.WuHong-bo,Zhou Jian-ren.The form ofmean representationof truth degree with application in quantitative logic[J].ActaElectronica Sinica,2012,40(9):1822-1828.(in Chinese)
[16]王国俊,宋建设.命题逻辑中的程度化方法[J].电子学报,2006,34(2):252-257.Wang Guo-jun,Song jian-she.Graded method in propositionallogic[J].Acta Electronica Sinica,2006,34(2):252-257.(inChinese)
[17]吴洪博,周建仁,张琼.(3n+1)值逻辑系统R0L中公式的真度性质[J].电子学报,2011,39(10):2230-2234.Wu Hong-bo,Zhou Jian-ren,Zhang Qiong.The properties oftruth degrees of formulas in(3n+1)-valued logic system R0L[J].Acta Electronica Sinica,2011,39(10):2230-2234.(inChinese)
[18]程国胜.R0-代数中滤子与理想[J].模糊系统与数学,2001,15(1):58-61.Cheng Guo-sheng.The filters and the ideals in R0-algebras[J].Fuzzy Systems and Mathematics,2001,15(1):58-61.(in Chinese)
[19]郑崇友,樊磊,崔宏斌.FRAME与连续格(第二版)[M].北京:首都师范大学,2000.Zheng Chong-you,Fan Lei,Cui Hong-bin.Frame and Contin-uous Lattice(2nd edition)[M].Beijing:Capital Normal Un-iversity,2000.(in Chinese)
[20]李志伟,孙立民,郑崇友.正则Fuzzy蕴涵代数[J].模糊系统与数学,2002,16(2):22-26.Li Zh-i wei,Sun L-i min,Zheng Chong-you.Regular fuzzy im-plication algebra[J].Fuzzy Systems and Mathematics,2002,16(2):22-26.(in Chinese)