BR_0-代数中MT理想的扩展及素MT理想的存在性
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摘要
BR0-代数是一类重要的基础逻辑代数,其中著名的MV代数和R0-代数均是BR0-代数的特款,因而对BR0-代数研究结果具有普遍的实用性.首先,通过BR0-代数中极大并-理想的存在性证明了BR0-代数中素并-理想的存在性;其次,利用对偶范畴的思想方法和MP滤子的特征,在BR0-代数中提出了MT理想,极大MT理想,素MT理想等概念,讨论了它们的基本性质及相互关系,并通过素并-理想构造性的证明了素MT理想的存在性;最后,在非退化的BR0-代数中证明了任何一个真MT理想可以扩展为一个极大素MT理想.本文的工作是对BR0-代数研究内容和方法的有益补充.
BR0-algebras are important basic algebras,which take MV algebras and R0-algebras as their exceptions,so the study results of BR0-algebras have universal applicability. Firstly,the existence of prime union-ideals is proved through the existence of maximal union-ideals in BR0-algebras; Secondly,having utilized the ideals and methods of dual category and characteristic of MP filter,the concepts of MT ideal,maximal MT ideal,prime MT ideal are introduced to BR0-algebras,and the relationships among them are discussed,and the existence of prime MT ideals is proved by constructing of prime MT ideals through prime union-ideals in BR0-algebras; At last,it is proved that a proper MT ideal can be expended to a maximal prime MT ideal. The work of the paper is a beneficial supplement for the content and methods of studying series BR0-algebras.
BR0-algebras are important basic algebras,which take MV algebras and R0-algebras as their exceptions,so the study results of BR0-algebras have universal applicability. Firstly,the existence of prime union-ideals is proved through the existence of maximal union-ideals in BR0-algebras; Secondly,having utilized the ideals and methods of dual category and characteristic of MP filter,the concepts of MT ideal,maximal MT ideal,prime MT ideal are introduced to BR0-algebras,and the relationships among them are discussed,and the existence of prime MT ideals is proved by constructing of prime MT ideals through prime union-ideals in BR0-algebras; At last,it is proved that a proper MT ideal can be expended to a maximal prime MT ideal. The work of the paper is a beneficial supplement for the content and methods of studying series BR0-algebras.
引文
[1]吴望名.Fuzzy蕴涵代数[J].模糊系统与数学,1990,4(1):56-64.Wu Wangming.Fuzzy implication algebras[J].Fuzzy Systems and M athematics,1990,4(1):56-63.(in Chinese)
[2]吴洪博,汪宁.基于正则FI代数的MT理想及其应用[J].电子学报,2013,41(7):1389-1394.Wu Hongbo,Wang Ning.M T ideals of regular FI algebras with applications[J].Acta Electronica Sinica,2013,41(7):1389-1394.(in Chinese)
[3]徐扬.格蕴含代数[J].西南交通大学学报,1993,89(1):20-27.Xu Yang.Lattice implication algebras[J].Journal of South-west Jiaotong University,1993,28(1):20-27.(in Chinese)
[4]刘军,徐扬.格蕴含代数的滤子与结构[J].科学通报,1997,42(10):1049-1052.Liu Jun,Xu Yang.Filters and its structures in lattice implication algebra[J].Chinese Science Bulletin,1997,42(10):1049-1052.(in Chinese)
[5]Y Xu,D Ruan,K Y Qin,J Liu.Lattice-Valued Logic[M].Berlin Heridelberg:Springer-Verlag,2003.
[6]P Hajek.Metamathematics of Fuzzy Logic[M].Netherlands: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 Bijing,Wang Guojun.Logic pseudo-metric spaces of regular implication operators[J].Acta Electronica Sinica,2010,38(3):497-502.(in Chinese)
[8]汪德刚,谷云东,李洪兴.模糊模态模态逻辑及其广义重言式[J].电子学报,2007,35(2):261-264.Wang Degang,Gu Yundong,LI Hongxing.Fuzzy modal logic and its tautologies[J].Acta Electronica Sinica,2007,35(2):261-264.(in Chinese)
[9]王国俊.模糊命题演算的一种形式演绎系统L*[J].科学通报,1997,42(10):1041-1045.Wang Guo-jun.A formal deductive system L*for fuzzy propositional calculus[J].Chinese Science Bulletin,1997,42(10):1041-1045.(in Chinese)
[10]王国俊.数理逻辑引论与归结原理[M].北京:科学出版社,2000.
[11]G J Wang.On the logic foundation of fuzzy reasoning[J].Information Sciences,1999,177:47-88.
[12]F Esteva,L Godo.Monoidal t-norm based logic:towards a logic for left-continuous t-norms[J].Fuzzy Sets and Systems,2001,124:271-288.
[13]H B Wu.A kind of simplified form L*0for the system L*[J].Journal of Fuzzy Mathematics,2001,43(3):233-238.
[14]吴洪博.基础R0-代数与基础L*系统[J].数学进展,2003,32(5):565-576Wu Hongbo.Basic R0-algebra and basic L*system[J].Advances in M athematics,2003,32(5):565-576.(in Chinese)
[15]周建仁,吴洪博.R0-蕴涵算子所导出的逻辑函数的特征[J].数学学报(中文版),2014,57(2):235-248.Zhou Jianren,Wu Hong-bo.Characterizations of logic functions derived by R0-operator[J].Acta M athematics Sinica,2014,57(2):235-248.(in Chinese)
[16]周红军,折延宏.Lukasiewicz命题逻辑中命题的Choquet积分真度理论[J].电子学报,2013,23(3):557-563.Zhou Hongjun,She Yanhong.Theory of Choquet integral truth degrees of propositions in Lukasiewicz propositional logic[J].Acta Electronica Sinca,2013,23(3):557-563.(in Chinese)
[17]胡明娣,王国俊.经典逻辑度量空间中的模2次范整线性空间结构[J].电子学报,2011,39(4):899-905.Hu M ing-di,Wang Guo jun.Z(2)-normal linear structure on classical logic metric space[J].Acta Electronica Sinca,2011,39(4):899-905.(in Chinese)
[18]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):419-4239.Hu M ing-di,Wang Guo-jun.Distribution of the symmetrical logic formulas in the classical logic metric space[J].Acta Electronica Sinca,2011,39(2):419-423.(in Chinese)
[19]张东晓,李立峰.二值命题逻辑公式的语构程度化方法[J].电子学报,2008,36(2):320-325.Zhang Dongxiao,Li Lifeng.Syntactic graded method of 2-va;ued propositional logic formulas[J].Acta Electronica Sinca,2008,36(2):320-325.(in Chinese)
[20]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):2230-2234,2229.Wu Hongbo,Zhou Jianren,Zhang Qiong.The properties of truth degrees of formulas in(3n+1)-valued logic system R0L[J].Acta Electronica Sinca,2011,39(10):2230-2234,2229.(in Chinese)
[21]吴洪博,周建仁.计量逻辑中真度的均值表示及其应用[J].电子学报,2012,40(9):1822-1828.Wu Hongbo,Zhou Jianren.The form of mean representation of truth degree with application in quantitative logic[J].Acta Electronica Sinca,2012,40(9):1822-1828.(in Chinese)
[22]左卫兵.基于MV代数语义的格值逻辑的程度化方法[J].电子学报,2013,41(10):2036-2040.Zuo Weibing.Graded method of lattice-valued logic based on M V-algebra semantics[J].Acta Electronica Sinca,2013,41(10):2036-2040.(in Chinese)
[23]F Esteva,L Godo,C Noguera.On expansions of WNM tnorm based logics with truth-constants[J].Fuzzy Sets and Systems,2010,162:347-368.
[24]San-min Wang,Guo-jun Wang.A triangular norm-based propositional fuzzy logic[J].Fuzzy Sets and Systems,2003,136(1):55-70.
[25]刘敏,吴洪博.预线性剩余格与逻辑代数[J].工程数学学报,2008,25(2):199-203.Liu M in,Wu Hongbo.Prelinearity residuated-lattice and logic algebras[J].Chinese Journal of Engineering M athematics,2008,25(2):199-203.(in Chinese)
[26]周建仁,吴洪博.WBR0-代数的正则性及与其他逻辑代数的关系[J].山东大学学报(自然科学版),2012,47(2):86-92.Zhou Jianren,Wu Hongbo.The regularness of WBR0-algebra and relationship with logic algebras[J].Journal of Shandong University(Natural Edition),2012,47(2):86-92.(in Chinese)
[27]D W Pei.On equivalent forms of fuzzy logic systems NM and IM TL[J].Fuzzy Sets and Systems,2003,138:187-195.
[28]王国俊.MV-代数、BL-代数、R0-代数与多值逻辑[J].模糊系统与数学,2002,16(2):1-15.Wang Guojun.M V-algebras,BL-algebras,R0-algebras and many-valued logics[J].Fuzzy System and M athematics,2002,16(2):1-15.(in Chinese)
[29]郑崇友,樊磊,崔宏斌.Frame与连续格[M].第2版.北京:首都师范大学出版社,2000.
[2]吴洪博,汪宁.基于正则FI代数的MT理想及其应用[J].电子学报,2013,41(7):1389-1394.Wu Hongbo,Wang Ning.M T ideals of regular FI algebras with applications[J].Acta Electronica Sinica,2013,41(7):1389-1394.(in Chinese)
[3]徐扬.格蕴含代数[J].西南交通大学学报,1993,89(1):20-27.Xu Yang.Lattice implication algebras[J].Journal of South-west Jiaotong University,1993,28(1):20-27.(in Chinese)
[4]刘军,徐扬.格蕴含代数的滤子与结构[J].科学通报,1997,42(10):1049-1052.Liu Jun,Xu Yang.Filters and its structures in lattice implication algebra[J].Chinese Science Bulletin,1997,42(10):1049-1052.(in Chinese)
[5]Y Xu,D Ruan,K Y Qin,J Liu.Lattice-Valued Logic[M].Berlin Heridelberg:Springer-Verlag,2003.
[6]P Hajek.Metamathematics of Fuzzy Logic[M].Netherlands: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 Bijing,Wang Guojun.Logic pseudo-metric spaces of regular implication operators[J].Acta Electronica Sinica,2010,38(3):497-502.(in Chinese)
[8]汪德刚,谷云东,李洪兴.模糊模态模态逻辑及其广义重言式[J].电子学报,2007,35(2):261-264.Wang Degang,Gu Yundong,LI Hongxing.Fuzzy modal logic and its tautologies[J].Acta Electronica Sinica,2007,35(2):261-264.(in Chinese)
[9]王国俊.模糊命题演算的一种形式演绎系统L*[J].科学通报,1997,42(10):1041-1045.Wang Guo-jun.A formal deductive system L*for fuzzy propositional calculus[J].Chinese Science Bulletin,1997,42(10):1041-1045.(in Chinese)
[10]王国俊.数理逻辑引论与归结原理[M].北京:科学出版社,2000.
[11]G J Wang.On the logic foundation of fuzzy reasoning[J].Information Sciences,1999,177:47-88.
[12]F Esteva,L Godo.Monoidal t-norm based logic:towards a logic for left-continuous t-norms[J].Fuzzy Sets and Systems,2001,124:271-288.
[13]H B Wu.A kind of simplified form L*0for the system L*[J].Journal of Fuzzy Mathematics,2001,43(3):233-238.
[14]吴洪博.基础R0-代数与基础L*系统[J].数学进展,2003,32(5):565-576Wu Hongbo.Basic R0-algebra and basic L*system[J].Advances in M athematics,2003,32(5):565-576.(in Chinese)
[15]周建仁,吴洪博.R0-蕴涵算子所导出的逻辑函数的特征[J].数学学报(中文版),2014,57(2):235-248.Zhou Jianren,Wu Hong-bo.Characterizations of logic functions derived by R0-operator[J].Acta M athematics Sinica,2014,57(2):235-248.(in Chinese)
[16]周红军,折延宏.Lukasiewicz命题逻辑中命题的Choquet积分真度理论[J].电子学报,2013,23(3):557-563.Zhou Hongjun,She Yanhong.Theory of Choquet integral truth degrees of propositions in Lukasiewicz propositional logic[J].Acta Electronica Sinca,2013,23(3):557-563.(in Chinese)
[17]胡明娣,王国俊.经典逻辑度量空间中的模2次范整线性空间结构[J].电子学报,2011,39(4):899-905.Hu M ing-di,Wang Guo jun.Z(2)-normal linear structure on classical logic metric space[J].Acta Electronica Sinca,2011,39(4):899-905.(in Chinese)
[18]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):419-4239.Hu M ing-di,Wang Guo-jun.Distribution of the symmetrical logic formulas in the classical logic metric space[J].Acta Electronica Sinca,2011,39(2):419-423.(in Chinese)
[19]张东晓,李立峰.二值命题逻辑公式的语构程度化方法[J].电子学报,2008,36(2):320-325.Zhang Dongxiao,Li Lifeng.Syntactic graded method of 2-va;ued propositional logic formulas[J].Acta Electronica Sinca,2008,36(2):320-325.(in Chinese)
[20]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):2230-2234,2229.Wu Hongbo,Zhou Jianren,Zhang Qiong.The properties of truth degrees of formulas in(3n+1)-valued logic system R0L[J].Acta Electronica Sinca,2011,39(10):2230-2234,2229.(in Chinese)
[21]吴洪博,周建仁.计量逻辑中真度的均值表示及其应用[J].电子学报,2012,40(9):1822-1828.Wu Hongbo,Zhou Jianren.The form of mean representation of truth degree with application in quantitative logic[J].Acta Electronica Sinca,2012,40(9):1822-1828.(in Chinese)
[22]左卫兵.基于MV代数语义的格值逻辑的程度化方法[J].电子学报,2013,41(10):2036-2040.Zuo Weibing.Graded method of lattice-valued logic based on M V-algebra semantics[J].Acta Electronica Sinca,2013,41(10):2036-2040.(in Chinese)
[23]F Esteva,L Godo,C Noguera.On expansions of WNM tnorm based logics with truth-constants[J].Fuzzy Sets and Systems,2010,162:347-368.
[24]San-min Wang,Guo-jun Wang.A triangular norm-based propositional fuzzy logic[J].Fuzzy Sets and Systems,2003,136(1):55-70.
[25]刘敏,吴洪博.预线性剩余格与逻辑代数[J].工程数学学报,2008,25(2):199-203.Liu M in,Wu Hongbo.Prelinearity residuated-lattice and logic algebras[J].Chinese Journal of Engineering M athematics,2008,25(2):199-203.(in Chinese)
[26]周建仁,吴洪博.WBR0-代数的正则性及与其他逻辑代数的关系[J].山东大学学报(自然科学版),2012,47(2):86-92.Zhou Jianren,Wu Hongbo.The regularness of WBR0-algebra and relationship with logic algebras[J].Journal of Shandong University(Natural Edition),2012,47(2):86-92.(in Chinese)
[27]D W Pei.On equivalent forms of fuzzy logic systems NM and IM TL[J].Fuzzy Sets and Systems,2003,138:187-195.
[28]王国俊.MV-代数、BL-代数、R0-代数与多值逻辑[J].模糊系统与数学,2002,16(2):1-15.Wang Guojun.M V-algebras,BL-algebras,R0-algebras and many-valued logics[J].Fuzzy System and M athematics,2002,16(2):1-15.(in Chinese)
[29]郑崇友,樊磊,崔宏斌.Frame与连续格[M].第2版.北京:首都师范大学出版社,2000.