否定非对合剩余格上基于正规模糊理想的一致拓扑空间
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摘要
拓扑结构是逻辑代数研究领域的重要研究内容之一,为了揭示否定非对合剩余格上的拓扑结构,基于正规模糊理想诱导的同余关系在否定非对合剩余格上构造一致拓扑空间并讨论其拓扑性质.证明了:(1)一致拓扑空间是第一可数,零维,非连通,局部紧的完全正则空间;(2)一致拓扑空间是T_1空间当且仅当是T_2空间;(3)否定非对合剩余格中格运算和伴随运算关于一致拓扑都是连续的,从而构成拓扑否定非对合剩余格.同时,获得了一致拓扑空间是紧空间和离散空间的充分必要条件.最后,讨论了拓扑否定非对合剩余格中代数同构与拓扑同胚间的关系.对从拓扑层面进一步揭示否定非对合剩余格的内部特征具有一定的促进作用.
Topological structure is one of important research contents in the field of logical algebra. In order to describe the topological structure of negative non-involutive residuated lattices, based on the congruences induced by normal fuzzy ideals, uniform topological spaces are established and some of their properties are discussed. The following conclusions are proved:(1) every uniform topological space is first-countable, zero-dimensional, disconnected,locally compact and completely regular.(2) a uniform topological space is a T_1 space iff it is a T_2 space.(3) the lattice and adjoint operations in a negative non-involutive residuated lattice are continuous under the uniform topology, which make the negative non-involutive residuated lattice to be topological negative non-involutive residuated lattice.Meanwhile, some necessary and sufficient conditions for the uniform topological spaces to be compact and discrete are obtained. Finally, the relationships between algebraic isomorphism and topological homeomorphism in topological negative non-involutive residuated lattice are discussed. The results of this paper have a positive role to reveal internal features of negative non-involutive residuated lattices on a topological level.
Topological structure is one of important research contents in the field of logical algebra. In order to describe the topological structure of negative non-involutive residuated lattices, based on the congruences induced by normal fuzzy ideals, uniform topological spaces are established and some of their properties are discussed. The following conclusions are proved:(1) every uniform topological space is first-countable, zero-dimensional, disconnected,locally compact and completely regular.(2) a uniform topological space is a T_1 space iff it is a T_2 space.(3) the lattice and adjoint operations in a negative non-involutive residuated lattice are continuous under the uniform topology, which make the negative non-involutive residuated lattice to be topological negative non-involutive residuated lattice.Meanwhile, some necessary and sufficient conditions for the uniform topological spaces to be compact and discrete are obtained. Finally, the relationships between algebraic isomorphism and topological homeomorphism in topological negative non-involutive residuated lattice are discussed. The results of this paper have a positive role to reveal internal features of negative non-involutive residuated lattices on a topological level.
引文
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[11]刘春辉.剩余格的模糊滤子理论[J].高校应用数学学报,2016, 31(2):233-247.
[12]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.
[13] Liu Yi, QinYao, Qin Xiaoyan, et al. Ideals and fuzzy ideals on residuated lattices[J]. Internat J Mach Learn Cyber, 2017, 8(1):239-253.
[14]刘春辉.关于否定非对合剩余格的LI-理想理论[J].模糊系统与数学,2018, 32(1):66-76.
[15]罗清君·R_0-代数中素滤子的拓扑性质[J].数学学报,2008, 51(4):795-802.
[16] Liu Chunhui, Xu Luoshan. Prime MP-filter spaces of fuzzy implication algebras[J]. Chin Quart J Math, 2012, 27(2):246-253.
[17] Mahmood B. Spectrum Topology of a residuated lattices[J]. Fuzzy Inform Engineer, 2013,5(2):159-172.
[18] Masoud H, Esfandiar E, Arsham B S. A topology induced by uniformity on BL-algebras[J].Math Logic Quart, 2007, 53(2):162-169.
[19]刘春辉. FI代数上基于模糊滤子的一致拓扑空间[J].浙江大学学报(理学版),2018, 45(5):521-528.
[20] Kelley J L. General Topology[M]. New York:Springer, 2001.
[21]熊金城.点集拓扑讲义[M].北京:高等教育出版社,2011.
[22]李庆国,汤灿琴,李纪波.一般拓扑学[M].长沙:湖南大学出版社,2006.
[23]程吉树,陈水利.点集拓扑学[M].北京:科学出版社,2008.
[2] Ward M, Dilworth R P. Residuated Lattices[J]. Trans Amer Math Soc, 1939, 45:335-354.
[3] Pavelka J. On fuzzy logic I, II, III[J]. Zeitschr F Math Logic and Grundlagend Math, 1979,25:45-52; 119-134; 447-464.
[4] Wang Guojun,Zhou Hongjun. Introduction to Mathematical Logic and Resolution Principle(Second Edition)[M]. Beijing:Science Press, 2009.
[5] Hájek P. Metamathematics of Fuzzy Logic[M]. Dordrecht:Kluwer Acade. Publishers, 1998.
[6] Xu Yang, Ruan D, Qin Keyun, et al. Lattice-Valued Logics[M]. Berlin:Springer, 2003.
[7] Michiro K, Wieslaw A D. Filter theory of BL-algebras[J]. Soft Computing, 2008, 12(5):419-423.
[8] Dumitru B, Dana P. Some types of filters on residuated lattices[J]. Soft Computing, 2014,18:825-837.
[9] Liu Yonglin, Liu Sanyang, Xu Yang, et al. ILI-ideals and prime LI-ideals in lattice implication algebras[J]. Inform Science, 2003, 155(1-2):157-175.
[10]刘春辉.Fuzzy蕴涵代数的滤子理论[J].山东大学学报(理学版),2013, 48(9):73-77.
[11]刘春辉.剩余格的模糊滤子理论[J].高校应用数学学报,2016, 31(2):233-247.
[12]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.
[13] Liu Yi, QinYao, Qin Xiaoyan, et al. Ideals and fuzzy ideals on residuated lattices[J]. Internat J Mach Learn Cyber, 2017, 8(1):239-253.
[14]刘春辉.关于否定非对合剩余格的LI-理想理论[J].模糊系统与数学,2018, 32(1):66-76.
[15]罗清君·R_0-代数中素滤子的拓扑性质[J].数学学报,2008, 51(4):795-802.
[16] Liu Chunhui, Xu Luoshan. Prime MP-filter spaces of fuzzy implication algebras[J]. Chin Quart J Math, 2012, 27(2):246-253.
[17] Mahmood B. Spectrum Topology of a residuated lattices[J]. Fuzzy Inform Engineer, 2013,5(2):159-172.
[18] Masoud H, Esfandiar E, Arsham B S. A topology induced by uniformity on BL-algebras[J].Math Logic Quart, 2007, 53(2):162-169.
[19]刘春辉. FI代数上基于模糊滤子的一致拓扑空间[J].浙江大学学报(理学版),2018, 45(5):521-528.
[20] Kelley J L. General Topology[M]. New York:Springer, 2001.
[21]熊金城.点集拓扑讲义[M].北京:高等教育出版社,2011.
[22]李庆国,汤灿琴,李纪波.一般拓扑学[M].长沙:湖南大学出版社,2006.
[23]程吉树,陈水利.点集拓扑学[M].北京:科学出版社,2008.