Fuzzy蕴涵代数的素犹豫模糊滤子及其谱空间

doi:10.6040/j.issn.1671-9352.0.2022.343

摘要/Abstract

摘要：

运用犹豫模糊集和拓扑学的方法及原理对Fuzzy蕴涵代数的滤子问题做深入研究。首先，引入Fuzzy蕴涵代数的素犹豫模糊滤子概念并考察其性质特征，建立并证明了并半格Fuzzy蕴涵代数的素犹豫模糊滤子定理; 其次，在一个给定的Fuzzy蕴涵代数(X, →, 0)的全体素犹豫模糊滤子之集PHFil(X)上构造了一个拓扑τ，获得拓扑τ的一个基，证明拓扑空间(PHFil(X)，τ)是T₀空间。

关键词: 模糊逻辑, Fuzzy蕴涵代数, 素犹豫模糊滤子, 谱空间

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

中图分类号:

O141.1

刘春辉. Fuzzy蕴涵代数的素犹豫模糊滤子及其谱空间[J]. 《山东大学学报(理学版)》, 2023, 58(12): 108-117.

Chunhui LIU. Prime hesitant fuzzy filters in fuzzy implication algebras and its spectral spaces[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(12): 108-117.

图/表 1

表1

参考文献 33

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.

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