JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (12): 108-117.doi: 10.6040/j.issn.1671-9352.0.2022.343

Previous Articles     Next Articles

Prime hesitant fuzzy filters in fuzzy implication algebras and its spectral spaces

Chunhui LIU()   

  1. School of Educational Science, Chifeng University, Chifeng 024001, Inner Mongolia, China
  • Received:2022-06-23 Online:2023-12-20 Published:2023-12-19

Abstract:

In this paper, filters problem of fuzzy implication algebras is further studied by using the method and principle of hesitant fuzzy sets and topology. Firstly, the notion of prime hesitant fuzzy filter in fuzzy implication algebras is introduced and its properties are investigated. The prime hesitant fuzzy filter theorem is established and proved of upper semilattice fuzzy implication algebras. Secondly, a topology τ is constructed on the set PHFil (X) consisting of all prime hesitant fuzzy filters in a given fuzzy implication algebra (X, →, 0), a base is obtained of the topology τ and it is proved that the topological space (PHFil(X), τ) is a T0 space.

Key words: fuzzy logic, fuzzy implication algebra, prime hesitant fuzzy filter, spectral space

CLC Number: 

  • O141.1

Table 1

Definition of the implicative operator → on X"

0 a b 1
0 1 1 1 1
a 0 1 1 1
b 0 b 1 1
1 0 a b 1
1 王国俊. 非经典数理逻辑与近似推理[M]. 北京: 科学出版社, 2003.
WANG Guojun . Non-classical mathematical logic and approximate reasoning[M]. Beijing: Science Press, 2003.
2 CHANG C C . Algebraic analysis of many value logic[J]. Transactions of the American Methodical Society, 1958, 87 (1): 467- 490.
3 XU Yang , RUAN Da , QIN Keyun , et al. Lattice-valued logic[M]. Berlin: Springer, 2004.
4 WANG Guojun , ZHOU Hongjun . Introduction to mathematical logic and resolution principle[M]. 2nd ed Beijing: Science Press, 2009.
5 HAJEK P . Metamathematics of fuzzy logic[M]. Dordrecht: Kluwer Academic Publishers, 1998.
6 HAJEK P . Basic fuzzy logic and BL-algebras[J]. Soft Computing, 1998, 2 (3): 124- 128.
doi: 10.1007/s005000050043
7 ESTEVA F , GODO L . Monoidalt-norm based logic: towards a logic for left-continuous t-norms[J]. Fuzzy Sets and Systems, 2001, 124 (3): 271- 288.
doi: 10.1016/S0165-0114(01)00098-7
8 吴望名. Fuzzy蕴涵代数[J]. 模糊系统与数学, 1990, 4(1): 56-64.
WU Wangming. Fuzzy implication algebras[M]. Fuzzy Systems and Mathematics, 1990, 4(1): 56-64.
9 刘练珍, 王国俊. Fuzzy蕴涵代数与MV代数[J]. 模糊系统与数学, 1998, 12 (1): 20- 25.
LIU Lianzhen , WANG Guojun . Fuzzy implication algebras and MV algebras[J]. Fuzzy Systems and Mathematics, 1998, 12 (1): 20- 25.
10 吴达. 交换Fuzzy蕴涵代数[J]. 模糊系统与数学, 1999, 13 (1): 27- 30.
WU Da . Commutative fuzzy implication algebras[J]. Fuzzy Systems and Mathematics, 1999, 13 (1): 27- 30.
11 刘春辉. 模糊蕴涵代数及其滤子理论[M]. 赤峰: 内蒙古科学技术出版社, 2020.
LIU Chunhui . Fuzzy implication algebras and its filters theory[M]. Chifeng: Inner Mongolia Science & Technology Press, 2020.
12 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.
ZHU Yiquan , CAO Xiwang . On PFI-algebras and residuated lattices[J]. Advances in Mathematics, 2006, 35 (2): 223- 231.
13 刘春辉, 徐罗山. Fuzzy蕴涵代数的MP滤子[J]. 模糊系统与数学, 2009, 23 (2): 1- 6.
LIU Chunhui , XU Luoshan . MP-filters in fuzzy implication algebras[J]. Fuzzy Systems and Mathematics, 2009, 23 (2): 1- 6.
14 裴道武, 王三民, 王瑞. 模糊蕴涵格理论[J]. 高校应用数学学报(A辑), 2011, 26 (3): 343- 354.
PEI Daowu , WANG Sanyang , WANG Rui . Theory of fuzzy implication lattices[J]. Applied Mathematics A Journal of Chinese Universities (Ser. A), 2011, 26 (3): 343- 354.
15 PEI Daowu . A survey of fuzzy implication algebras and their axiom atization[J]. International Journal of Approximate Reasoning, 2014, 55, 1643- 1658.
doi: 10.1016/j.ijar.2014.05.008
16 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.
doi: 10.6040/j.issn.1671-9352.0.2012.418
LIU Chunhui . Filters theory in fuzzy implication algebras[J]. Journal of Shandong University (Natural Science), 2013, 48 (9): 73- 77.
doi: 10.6040/j.issn.1671-9352.0.2012.418
17 LIU Chunhui , XU Luoshan . Prime MP filter spaces of fuzzy implication algebras[J]. Chinese Quarterly Journal of Mathematics, 2012, 27 (2): 246- 253.
18 ZADEH L A . Fuzzy sets[J]. Information Control, 1965, 8 (3): 338- 353.
doi: 10.1016/S0019-9958(65)90241-X
19 TORRA V, NARUKAWA Y. On hesitant fuzzy sets and decision[C]//The 18th IEEE International Conference on Fuzzy Systems. Jeju Island, Korea: IEEE, 2009: 1378-1382.
20 TORRA V . Hesitant fuzzy sets[J]. International Journal of Intelligent Systems, 2010, 25 (6): 529- 539.
21 VERMA R , SHARMA B D . New operations over hesitant fuzzy sets[J]. Fuzzy Informations and Engineering, 2013, 2 (5): 129- 146.
22 XU Zeshui , XIA Meng . Distance and similarity measures for hesitant fuzzy sets[J]. Informations Sciences, 2011, 181 (11): 2128- 2138.
doi: 10.1016/j.ins.2011.01.028
23 XU Zeshui . Hesitant fuzzy sets theory[M]. New York: Springer, 2013.
24 JUN Y B , SONG S Z . Hesitant fuzzy prefilters and filters of EQ-algebras[J]. Applied Mathematicla Science, 2015, 9 (11): 515- 532.
25 JUN Y B , SUN S A . Hesitant fuzzy set theory applied to BCK/BCI-algebras[J]. Journal of Computational Analysis and Applications, 2016, 20 (4): 635- 646.
26 刘春辉. 非对合剩余格的犹豫模糊理想[J]. 山东大学学报(理学版), 2021, 56 (2): 7- 16.
doi: 10.6040/j.issn.1671-9352.0.2020.418
LIU Chunhui . Hesitant fuzzy ideals in non-involutive residuated lattices[J]. Journal of Shandong University (Natural Science), 2021, 56 (2): 7- 16.
doi: 10.6040/j.issn.1671-9352.0.2020.418
27 刘春辉. 非对合剩余格的犹豫模糊理想理论[J]. 模糊系统与数学, 2021, 35 (5): 15- 35.
LIU Chunhui . Theory of hesitant fuzzy ideals in non-involutive residuated lattices[J]. Fuzzy Systems and Mathematics, 2021, 35 (5): 15- 35.
28 刘春辉, 姜志廷, 何涛. BE代数的犹豫模糊滤子和犹豫模糊同余关系[J]. 宁夏大学学报(自然科学版), 2019, 40 (1): 20- 26.
LIU Chunhui , JIANG Zhiting , HE Tao . Hesitant fuzzy filters and hesitant fuzzy congruence relations in BE-algebers[J]. Journal of Ningxia University (Natural Science Edition), 2019, 40 (1): 20- 26.
29 彭家寅. FI代数的犹豫模糊滤子理论[J]. 模糊系统与数学, 2018, 32 (3): 1- 15.
PENG Jiayin . Hesitant fuzzy filters theory of FI-lgebras[J]. Fuzzy Systems and Mathematics, 2018, 32 (3): 1- 15.
30 刘春辉. Fuzzy蕴涵代数的犹豫模糊滤子[J]. 模糊系统与数学, 2018, 32 (3): 29- 37.
LIU Chunhui . Hesitant fuzzy filters in fuzzy implication algebras[J]. Fuzzy Systems and Mathematics, 2018, 31 (3): 29- 37.
31 刘春辉, 何涛. FI代数的犹豫模糊滤子与犹豫模糊同余关系[J]. 数学的实践与认识, 2018, 48 (7): 266- 272.
LIU Chunhui , HE Tao . Hesitant fuzzy filters and hesitant fuzzy congruence relations in FI-algebras[J]. Mathematics in Practice and Theory, 2018, 48 (7): 266- 272.
32 刘春辉. Fuzzy蕴涵代数的犹豫模糊滤子格[J]. 模糊系统与数学, 2021, 35 (4): 30- 39.
LIU Chunhui . The lattice of hesitant fuzzy filters in a given fuzzy implication algebra[J]. Fuzzy Systems and Mathematics, 2021, 35 (4): 30- 39.
33 徐罗山, 毛徐新, 何青玉. 应用拓扑学基础[M]. 北京: 科学出版社, 2021.
XU Luoshan , MAO Xuxin , HE Qingyu . Fundamentals of applied topology[M]. Beijing: Science Press, 2021.
[1] LIU Chun-hui. Hesitant fuzzy ideals in non-involutive residuated lattices [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(2): 7-16.
[2] LIU Chun-hui, ZHANG Hai-yan, LI Yu-mao. Lattice of bipolar fuzzy ideals in negative non-involutive residuated lattices [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(9): 29-35.
[3] LIU Chun-hui, LI Yu-mao, ZHANG Hai-yan. Bipolar fuzzy ideals in negative non-involutive residuated lattices [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(5): 88-98.
[4] LIU Chun-hui. Expand fuzzy filters in Heyting algebras [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(2): 57-65.
[5] LIU Chun-hui. Bipolar fuzzy prime ideals in negative non-involutive residuated lattices [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(11): 71-80.
[6] . Representative forms of commutative BR0-algebras on a set by implication operator [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(6): 86-94.
[7] 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.
[8] LIU Chun-hui. Lattice of(,∨(-overq))-fuzzy filters in a BL-algebra [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(10): 104-110.
[9] QIAO Xi-min, WU Hong-bo. 〈,(-overQ)〉-fuzzy filter and its characterization of the non-commutative residual lattices on the interval sets [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(2): 102-107.
[10] ZHOU Jian-ren, WU Hong-bo. A schematic extension of IMTL logic system [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(12): 28-34.
[11] KOU Hai-yan, WU Hong-bo. Wajsberg's form of MTL algebras with applications [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(02): 75-82.
[12] LIU Chun-hui. Fuzzy ultra ⊙-ideals in regular residuated lattices [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(12): 87-94.
[13] LIU Chun-hui1,2. Prime fuzzy ⊙ideals and its topological properties of
regular residuated lattices
[J]. J4, 2013, 48(12): 52-56.
[14] LIU Chun-hui1,2. Lattice of fuzzy filter in a Heyting algebra [J]. J4, 2013, 48(12): 57-60.
[15] LIU Chun-hui1,2. Theory of filters in Fuzzy implication algebras [J]. J4, 2013, 48(09): 73-77.
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed   
[1] YANG Jun. Characterization and structural control of metalbased nanomaterials[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2013, 48(1): 1 -22 .
[2] HE Hai-lun, CHEN Xiu-lan* . Circular dichroism detection of the effects of denaturants and buffers on the conformation of cold-adapted protease MCP-01 and  mesophilic protease BP01[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2013, 48(1): 23 -29 .
[3] ZHAO Jun1, ZHAO Jing2, FAN Ting-jun1*, YUAN Wen-peng1,3, ZHANG Zheng1, CONG Ri-shan1. Purification and anti-tumor activity examination of water-soluble asterosaponin from Asterias rollestoni Bell[J]. J4, 2013, 48(1): 30 -35 .
[4] SUN Xiao-ting1, JIN Lan2*. Application of DOSY in oligosaccharide mixture analysis[J]. J4, 2013, 48(1): 43 -45 .
[5] LUO Si-te, LU Li-qian, CUI Ruo-fei, ZHOU Wei-wei, LI Zeng-yong*. Monte-Carlo simulation of photons transmission at alcohol wavelength in  skin tissue and design of fiber optic probe[J]. J4, 2013, 48(1): 46 -50 .
[6] YANG Lun, XU Zheng-gang, WANG Hui*, CHEN Qi-mei, CHEN Wei, HU Yan-xia, SHI Yuan, ZHU Hong-lei, ZENG Yong-qing*. Silence of PID1 gene expression using RNA interference in C2C12 cell line[J]. J4, 2013, 48(1): 36 -42 .
[7] MAO Ai-qin1,2, YANG Ming-jun2, 3, YU Hai-yun2, ZHANG Pin1, PAN Ren-ming1*. Study on thermal decomposition mechanism of  pentafluoroethane fire extinguishing agent[J]. J4, 2013, 48(1): 51 -55 .
[8] YANG Ying, JIANG Long*, SUO Xin-li. Choquet integral representation of premium functional and related properties on capacity space[J]. J4, 2013, 48(1): 78 -82 .
[9] LI Yong-ming1, DING Li-wang2. The r-th moment consistency of estimators for a semi-parametric regression model for positively associated errors[J]. J4, 2013, 48(1): 83 -88 .
[10] DONG Wei-wei. A new method of DEA efficiency ranking for decision making units with independent subsystems[J]. J4, 2013, 48(1): 89 -92 .