Heyting代数的扩张模糊滤子
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摘要
运用代数学与模糊集的方法和原理对Heyting代数的模糊滤子理论作进一步深入研究。引入了Heyting代数(H,≤,→)的模糊滤子f关于H上模糊子集μ的扩张模糊滤子和不变模糊滤子概念,获得了扩张模糊滤子和不变模糊滤子的若干性质。建立了扩张模糊滤子和生成模糊滤子间的关系,并利用这一关系给出了扩张模糊滤子在格结构研究中的应用,证明了一个Heyting代数(H,≤,→)的全体模糊滤子之集FFil(H)的3个特殊子集关于模糊集合包含序都构成完备Heyting代数。
The theory of fuzzy filters is studied in Heyting algebras by using the methods and principles of algebra and fuzzy sets.The notions of expand fuzzy filter and invariant fuzzy filter of a fuzzy filter f associated to a fuzzy subset μ in a Heyting algebra( H,≤,→) are introduced. Some properties of expand and invariant fuzzy filters are obtained. The relation between expand fuzzy filters and generated fuzzy filters is built,and the application of expand fuzzy filters in study of lattice structures is given by using this relation. We proved that three subsets of the set FFil( H) of containing all fuzzy filters in a Heyting algebra( H,≤,→),under fuzzy set-inclusion order,are form complete Heyting algebras.
The theory of fuzzy filters is studied in Heyting algebras by using the methods and principles of algebra and fuzzy sets.The notions of expand fuzzy filter and invariant fuzzy filter of a fuzzy filter f associated to a fuzzy subset μ in a Heyting algebra( H,≤,→) are introduced. Some properties of expand and invariant fuzzy filters are obtained. The relation between expand fuzzy filters and generated fuzzy filters is built,and the application of expand fuzzy filters in study of lattice structures is given by using this relation. We proved that three subsets of the set FFil( H) of containing all fuzzy filters in a Heyting algebra( H,≤,→),under fuzzy set-inclusion order,are form complete Heyting algebras.
引文
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[13]刘春辉.Heyting代数的扩张滤子[J].数学的实践与认识,2017,47(22):255-261.LIU Chunhui.Expand filters in Heyting algebras[J].Journal of M athematics in Practice and Theory,2017,47(22):255-261.
[14]刘春辉.Heyting代数的不变滤子[J].数学的实践与认识,2018,48(12):240-246.LIU Chunhui.Invariant filters in Heyting algebras[J].Journal of M athematics in Practice and Theory,2018,48(12):240-246.
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[2]贺伟.Heyting代数的谱空间[J].数学进展,1998,27(2):44-47.HE Wei.Spectrums of Heyting algebras[J].Advances in M athematics,1998,27(2):44-47.
[3]徐晓泉,熊华平,杨金波.近性Heyting代数[J].数学年刊:A辑,2000,21(2):165-174.XU Xiaoquan,XIONG Huaping,YANG Jinbo.Proximity Heyting algenras[J].Chinese Annals of M athematics:A,2000,21(2):165-174.
[4]李志伟,郑崇友.Heyting代数与fuzzy蕴涵代数[J].数学杂志,2002,22(2):237-240.LI Zhiw ei,ZHENG Chongyou.Heyting algebras and fuzzy implication algebras[J].Journal of M athematics,2002,22(2):237-240.
[5]王习娟,贺伟.关于topos中的内蕴Heyting代数对象[J].数学杂志,2011,31(6):979-998.WANG Xijuan,HE Wei.On the Heyting algebra objects in topos[J].Journal of M athematics,2011,31(6):979-998.
[6]吴洪博,石慧君.Heyting系统及其H-Locale化形式[J].数学学报,2012,55(6):1119-1130.WU Hongbo,SHI Huijun.Heyting system and its H-Localification[J].Acta M athematica Sinica,2012,55(6):1119-1130.
[7]吴洪博,石慧君.Heyting系统及其H-空间化表示形式[J].电子学报,2012,50(5):995-999.WU Hongbo,SHI Huijun.Heyting system and its representation by H-spatilization[J].Acta Electronica Sinica,2012,50(5):995-999.
[8]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.ZHENG Chongyou,FAN Lei,CUI Hongbin.Frame and continuous lattices[M].Beijing:Capital Normal University Press,2000.
[9]姚卫.Heyting代数中的滤子、同构定理及其范畴Heyt[D].西安:陕西师范大学,2005.YAO Wei.Filters and isomorphims theorems in Heyting algebra and its category Heyt[D].Xian:Shaanxi Normal University,2005.
[10]王伟.非可换逻辑代数的滤子及模糊化理论[D].西安:西北大学,2010.WANG Wei.Filters and its fuzzification theory of non-commutative logical algebras[D].Xian:Northw est University,2010.
[11]ZADEH L A.Fuzzy sets[J].Information and Control,1965,8:338-353.
[12]刘春辉.Heyting代数的模糊滤子格[J].山东大学学报(理学版),2013,48(12):57-60.LIU Chunhui.Lattice of fuzzy filters in a Heyting algebra[J].Journal of Shandong University(Natural Science),2013,48(12):57-60.
[13]刘春辉.Heyting代数的扩张滤子[J].数学的实践与认识,2017,47(22):255-261.LIU Chunhui.Expand filters in Heyting algebras[J].Journal of M athematics in Practice and Theory,2017,47(22):255-261.
[14]刘春辉.Heyting代数的不变滤子[J].数学的实践与认识,2018,48(12):240-246.LIU Chunhui.Invariant filters in Heyting algebras[J].Journal of M athematics in Practice and Theory,2018,48(12):240-246.
[15]DAVEY B A,PRIESTLEY H A.Introduction to lattices and order[M].Cambridge:Cambridge University Press,2002.