否定非对合剩余格的LI-理想理论
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摘要
综合运用泛代数和格序理论的方法和原理研究否定非对合剩余格的理想问题.首先,在否定非对合剩余格L中引入LI-理想以及由L的非空子集生成的LI-理想的概念并考察它们的相关性质.其次,在L的全体LI-理想之集Id(L)上定义了格运算和,证明了(Id(L),?,,)构成一个分配的连续格,从而构成一个Frame.然后,在L中引入素LI-理想概念并讨论其性质,建立了预线性否定非对合剩余格的素LI-理想定理.最后,借助于素LI-理想之特性获得了预线性否定非对合剩余格的LI-理想格(Id(L),?,,)中素元的若干等价刻画.
In this paper, the problem of ideals in negative non-involutive residuated lattices is studied by using the principle and method of universal algebras and lattice-order theory. Firstly, the notions of LI-ideals and LI-ideal generated by a non-empty subset are introduced in negative noninvolutive residuated L, and some of their properties are investigated. Secondly, lattice operations and are defined on the set Id(L) of all LI-ideals in L. It is proved that(Id(L), ?,,) forms a distributive continuous lattice, and particularly forms a frame. Then, the notion of prime LI-ideals is introduced in L and its properties are discussed. The prime LI-ideals theorem is established in prelinear negative non-involutive residuated lattices. Finally, some equivalent characterizations of prime elements of LI-ideal lattice(Id(L), ?,,) are obtained by means of prime LI-ideals in pre-linear negative non-involutive residuated lattice L.
In this paper, the problem of ideals in negative non-involutive residuated lattices is studied by using the principle and method of universal algebras and lattice-order theory. Firstly, the notions of LI-ideals and LI-ideal generated by a non-empty subset are introduced in negative noninvolutive residuated L, and some of their properties are investigated. Secondly, lattice operations and are defined on the set Id(L) of all LI-ideals in L. It is proved that(Id(L), ?,,) forms a distributive continuous lattice, and particularly forms a frame. Then, the notion of prime LI-ideals is introduced in L and its properties are discussed. The prime LI-ideals theorem is established in prelinear negative non-involutive residuated lattices. Finally, some equivalent characterizations of prime elements of LI-ideal lattice(Id(L), ?,,) are obtained by means of prime LI-ideals in pre-linear negative non-involutive residuated lattice L.
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,2005.
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[7]Wang Guojun,Zhou Hongjun.Introduction to Mathematical Logic and Resolution Principle(Second Edition),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.
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[17]Liu Yonglin,Liu Sanyang,Xu Yang,Qin Keyun.ILI-ideals and prime LI-ideals in lattice implication algebras[J].Information Sciences,2003,155:157-175.
[18]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.
[19]Michiro K,Wieslaw A D.Filter theory of BL-algebras[J].Soft Comput.,2008,12:419-423.
[20]Luo Qingjun.The filter lattices on R0-algebras[J].Journal of Mathematical Research&Exposition,2009,29(1):169-176.
[21]Zhu Yiquan,Xu Yang.On filter theory of residuated lattices[J].Information Sciences,2010,180:3614-3632.
[22]LIU Chunhui,XU Luoshan.On-ideals and lattice of-ideals in regular residuated lattices[J].Advance in Intelligent and Soft Computing,2010,82:425-434.
[23]Dumitru B,Dana P.Some types of filters on residuated lattices[J].Soft Comput.,2014,18:825-837.
[24]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.
[25]Gierz G,et al.Continuous Lattices and Domains[M].London:Cambridge University Press,2003.
[26]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]H′ajek P.Metamathematics of Fuzzy Logic[M].Dordrecht:Kluwer Academic Publishers,1998.
[5]Xu Yang,Ruan D,Qin Keyun,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 Guojun,Zhou Hongjun.Introduction to Mathematical Logic and Resolution Principle(Second Edition),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]梁聪,张小红.预线性与对合非结合剩余格[J].模糊系统与数学,2012,26(3):17-23.
[13]朱怡权,曹喜望.关于PFI代数与剩余格[J].数学进展,2006,35(2):223-231.
[14]任兰,张小红.幂等剩余格[J].模糊系统与数学,2011,25(3):21-29.
[15]方进明.剩余格与模糊集[M].北京:科学出版社,2012.
[16]Jun Y B.On LI-ideals and prime LI-ideals of lattice implication algebras[J].Bulletin of the Korean Mathematical Society,1999,36(2):369-380.
[17]Liu Yonglin,Liu Sanyang,Xu Yang,Qin Keyun.ILI-ideals and prime LI-ideals in lattice implication algebras[J].Information Sciences,2003,155:157-175.
[18]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.
[19]Michiro K,Wieslaw A D.Filter theory of BL-algebras[J].Soft Comput.,2008,12:419-423.
[20]Luo Qingjun.The filter lattices on R0-algebras[J].Journal of Mathematical Research&Exposition,2009,29(1):169-176.
[21]Zhu Yiquan,Xu Yang.On filter theory of residuated lattices[J].Information Sciences,2010,180:3614-3632.
[22]LIU Chunhui,XU Luoshan.On-ideals and lattice of-ideals in regular residuated lattices[J].Advance in Intelligent and Soft Computing,2010,82:425-434.
[23]Dumitru B,Dana P.Some types of filters on residuated lattices[J].Soft Comput.,2014,18:825-837.
[24]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.
[25]Gierz G,et al.Continuous Lattices and Domains[M].London:Cambridge University Press,2003.
[26]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.