JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2019, Vol. 54 ›› Issue (2): 57-65.doi: 10.6040/j.issn.1671-9352.0.2018.228

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Expand fuzzy filters in Heyting algebras

LIU Chun-hui   

  1. Department of Mathematics and Statistics, Chifeng University, Chifeng 024001, Inner Mongolia, China
  • Published:2019-02-25

Abstract: 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.

Key words: fuzzy logic, Heyting algebra, fuzzy filter, expand fuzzy filter, invariant fuzzy filter

CLC Number: 

  • O141.1
[1] VICKERS S. Topology via logic[M]. Cambridge: Cambridge University Press, 1989.
[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 Mathematics, 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 Mathematics: 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 Zhiwei, ZHENG Chongyou. Heyting algebras and fuzzy implication algebras[J]. Journal of Mathematics, 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 Mathematics, 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 Mathematica 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]. Xian: 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]. Xian: Northwest 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 Mathematics 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 Mathematics 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.
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[2] PENG Jia-yin. Falling fuzzy filters on residuated lattices [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(2): 52-64.
[3] LIANG Ying, CUI Yan-li, WU Hong-bo. Properties of residuated lattice of deduction systems set algebra in BL system [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(11): 65-70.
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regular residuated lattices [J]. J4, 2013, 48(12): 52-56.
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[13] LI Ling-ling, WU Hong-bo*. BR0-distributivity and its generalization [J]. J4, 2012, 47(2): 93-97.
[14] ZHOU Jian-ren, WU Hong-bo*. The regularness of WBR0-algebras and relationship with other logic algebras [J]. J4, 2012, 47(2): 86-92.
[15] QIN Xue-cheng, LIU Chun-hui*. The lattice of fuzzy ⊙-ideals in regular residuated lattices [J]. J4, 2011, 46(8): 73-76.
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