JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2014, Vol. 49 ›› Issue (12): 87-94.doi: 10.6040/j.issn.1671-9352.0.2014.325

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Fuzzy ultra ⊙-ideals in regular residuated lattices

LIU Chun-hui1,2   

  1. 1. Office of Teaching Affair, Chifeng University, Chifeng 024001, Inner Mongolia, China;
    2. Department of Mathematics and Statistics, Chifeng University, Chifeng 024001, Inner Mongolia, China
  • Received:2014-07-15 Revised:2014-10-21 Online:2014-12-20 Published:2014-12-20

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Abstract: Firstly, the notion of fuzzy ultra ⊙-ideal is introduced and its properties are investigated in regular residuated lattices, and some equivalent characterizations of fuzzy ultra ⊙-ideals are obtained. Secondly, the lattice operations ∨, ∧ and the order-reversing involution on the set FU(L) of all fuzzy ultra ⊙-ideals in a regular residuated lattice L are defined. It is proved that (FU(L),∨,∧,,0L,1L) formes a De Morgan algebra if L satisfies the condition (P). Finally, the adjoint pair () on FU(L) is defined. It is also proved that (FU(L),,0L,1L) formes a residuated lattice if L satisfies the condition (P).

Key words: fuzzy logic, fuzzy ⊙-ideal, fuzzy ultra ⊙-ideal, De Morgan algebra, (regular)residuated lattice

CLC Number: 

  • O141.1
[1] WANG Guojun, ZHOU Hongjun. Introduction to mathematical logic and resolution principle[M]. 2nd. Beijing: Science Press, 2009.
[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]. Z Math Logic Grund Math, 1979, 25:45-52; 119-134; 447-464.
[4] 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. MV-algebras, BL-algebras, R0-algebras and multiple valued logic[J]. Fuzzy Systems and Mathematics, 2002, 16(2):1-15.
[5] XU Yang, RUAN Da, QIN Keyun, et al. Lattice-valued logic[M]. Berlin: Springer, 2004.
[6] 刘春辉, 徐罗山. 关于剩余格的理想[J]. 山东大学学报:理学版, 2010, 45(4):66-71. LIU Chunhui, XU Luoshan. On ideals of residuated lattices[J]. Journal of Shandong University: Natural Science, 2010, 45(4):66-71.
[7] LIU Yonglin, LIU Anyang, QIN Keyun. ILI-ideals and prime LI-ideals in lattice implication algebras[J]. Information Sciences, 2003, 155:157-175.
[8] LIU Chunhui, XU Luoshan. On ⊙-ideals and lattice of ⊙-ideals in regular residuated lattices[J]. The 2nd International Conference on Quantitative Logic and Soft Computing, 2010: 425-434.
[9] ZADEH L A. Fuzzy sets[J]. Information and Control, 1965, 8:338-353.
[10] LIU Lianzhen, LI Kaitai. Fuzzy filters of BL-algebras[J]. Information Sciences, 2005, 173(1-3):141-154.
[11] 刘春辉. 格蕴涵代数的模糊LI-理想及其谱空间[J]. 高校应用数学学报:A辑, 2014, 29(1):115-126. LIU Chunhui. Prime fuzzy LI-ideals and its spectrum space of lattice implication algebras[J]. Applied Mathematics A Journal of Chinese Universities, 2014, 29(1):115-126.
[12] 秦学成, 刘春辉. 正则剩余格的fuzzy ⊙-理想[J]. 山东大学学报:理学版, 2010, 45(10):66-71. QIN Xuecheng, LIU Chunhui. Fuzzy ⊙-ideals in regular residuated lattices[J]. Journal of Shandong University: Natural Science, 2010, 45(10):66-71.
[13] 秦学成, 刘春辉. 正则剩余格的fuzzy ⊙-理想格[J]. 山东大学学报:理学版, 2011, 46(8):73-76. QIN Xuecheng, LIU Chunhui. The lattice of fuzzy ⊙-ideals in regular residuated lattices[J]. Journal of Shandong University: Natural Science, 2011, 46(8):73-76.
[14] 罗从文. De Morgan代数[M]. 北京: 北京理工大学出版社, 2005. LUO Congwen. De Morgan algebras[M]. Beijing: Beijing Institute of Technology Press, 2005.
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