基于模糊等式的Ω-BCK代数

Ω-BCK Algebras Based on Fuzzy Equalities

摘要

在模糊等式的框架下,引入Ω-BCK代数的概念.首先,将BCK代数中的清晰等式用模糊等式取代,给出Ω-BCK代数的定义;其次,刻画Ω-BCK代数的子结构,并证明Ω-BCK子代数判定的充分必要条件;再次,利用模糊同余关系得到Ω-BCK代数与BCK代数的关系;最后,将BCK代数的偏序关系进行推广,定义Ω-BCK代数的E-偏序关系.

Abstract

We introduced the concept of Ω-BCK algebras in the framework of fuzzy equalities. Firstly, we replaced the crisp equalities in BCK algebras with fuzzy equalities and gave the definition of Ω-BCK algebras. Secondly, we characterized the substructures of Ω-BCK algebras and proved the necessary and sufficient conditions for the determination of Ω-BCK subalgebras. Thirdly, by using fuzzy congruence relations, we obtained the relationship between Ω-BCK algebras and BCK algebras. Finally, we generalized the partial order relation of BCK algebras and defined the E-partial order relation of Ω-BCK algebras.

关键词

Ω-BCK代数 / 完备格 / 模糊等式 / 模糊恒等式 / 模糊子代数 / E-偏序关系

Key words

Ω-BCK algebra / complete lattice / fuzzy equality / fuzzy identity / fuzzy subalgebra / E-partial order relation

参考文献

[1] ZADEH L A. Fuzzy Sets[J]. Information and Control, 1965, 8(3): 338-353.

[2] GOGUEN J A. L-Fuzzy Sets[J]. Journal of Mathematical Analysis and Applications, 1967, 18(1): 145-174.

[3] FOURMAN M P, SCOTT D S. Sheaves and Logic[M]//Applications of Sheaves. Berlin: Springer, 1979: 302-401.

[4] HÖHLE U. Quotients with Respect to Similarity Relations[J]. Fuzzy Sets and Systems, 1988, 27(1): 31-44.

[5] DEMIRCI M. L-Equivalence Relations on L-Fuzzy Sets, L-Partitions of L-Fuzzy Sets and Their One-to-One Connections[J]. International Journal of Approximate Reasoning, 2019, 111: 21-34.

[6] DEMIRCI M. Vague Groups[J]. Journal of Mathematical Analysis and Applications, 1999, 230(1): 142-156.

[7] BĚLOHLÁVEK R, VYCHODIL V. Algebras with Fuzzy Equalities[J]. Fuzzy Sets and Systems, 2006, 157(2): 161-201.

[8] ŠEŠELJA B, TEPAVČEVIĆ A. Fuzzy Identities[C]// 2009 IEEE International Conference on Fuzzy Systems. [S. l.]: IEEE, 2009: 1660-1664.

[9] BUDIMIROVIĆ B, BUDIMIROVIĆ V, ŠEŠELJA B, et al. Fuzzy Equational Classes Are Fuzzy Varieties[J]. Iranian Journal of Fuzzy Systems, 2013, 10(4): 1-18.

[10] BUDIMIROVIĆ B, BUDIMIROVIĆ V, ŠEŠELJA B, et al. Fuzzy Identities with Application to Fuzzy Semigroups[J]. Information Sciences, 2014, 266: 148-159.

[11] ŠEŠELJA B, TEPAVČEVIĆ A. Ω-Algebras[C]// 2015 IEEE Symposium Series on Computational Intelligence. [S. l.]: IEEE, 2015: 971-975.

[12] BUDIMIROVIĆ B, BUDIMIROVIĆ V, ŠEŠELJA B, et al. E-Fuzzy Groups[J]. Fuzzy Sets and Systems, 2016, 289: 94-112.

[13] ŠEŠELJA B, TEPAVČEVIĆ A. L-E-Fuzzy Lattices[J]. International Journal of Fuzzy Systems, 2015, 17(3): 366-374.

[14] JIMENEZ J, SERRANO M L, ŠEŠELJA B, et al. Omega-Rings[J]. Fuzzy Sets and Systems, 2023, 455(C): 183-197.

[15] 周鑫, 刘淼. 基于L-值泛代数的L-值模[J]. 山东大学学报(理学版), 2023, 58(3): 48-54.

[16] (ZHOU X, LIU M. L-Valued Modules Based on L-Valued Universal Algebras[J]. Journal of Shandong University(Natural Science), 2023, 58(3):48-54.)

[17] TCHOFFO FOKA S V, TONGA M. A Note on the Algebraicity of L-Fuzzy Subalgebras in Universal Algebra[J]. Soft Computing, 2020, 24(2): 895-899.

[18] 周鑫, 刘淼. 范畴视角下的L-值群结构[J]. 模糊系统与数学, 2024, 38(6): 55-61.

[19] (ZHOU X, LIU M. Structures of L -Valued Groups from Category Theory Perspective[J]. Fuzzy Systems and Mathematics, 2024, 38 (6):55-61.)

[20] ZHI Y, LI Q G, ZHOU X N. Fuzzy Green’s Relations and Its Applications in E-Fuzzy Semigroups[J]. Fuzzy Sets and Systems, 2025, 517: 109482-1-109482-21.

[21] ZHI Y, ZHOU X N, LI Q G. Green’s Relations in L-E-Fuzzy Skew Lattices[J]. Soft Computing, 2022, 26(14): 6481-6494.

[22] ISÉKI K. An Algebra Related with a Propositional Calculus[J]. Proceedings of the Japan Academy, 1966, 42(1): 26-29.

[23] 汪国军, 姜秀燕. 阶n≤5有条件(S)的真BCI代数结构[J]. 吉林大学学报(理学版), 2006, 44(6): 913-915.

[24] (WANG G J, JIANG X Y. The Structure of Proper BCI-Algebras with Order n $\leqslant$ 5 and Condition(S)[J]. Journal of Jilin University(Science Edition), 2006, 44(6):913-915.)

[25] CIUNGU L C. Results in L-Algebras[J]. Algebra Universalis, 2021, 82(1): 1-16.

[26] AALY KOLOGANI M. Relations between L-Algebras and Other Logical Algebras[J]. Journal of Algebraic Hyperstructures and Logical Algebras, 2023, 4(1): 27-46.

[27] XI O G. Fuzzy BCK-Agebras[J]. Mathematica Japonica, 1991, 36(5): 935-942.

[28] JUN Y B, MENG J, XIN X L. On Fuzzy BCK-Filters[J]. The Korean Journal of Computational and Applied Mathematics, 1998, 5(1): 91-97.

[29] AHMED M A, AMHEDE A. Fuzzy BCK-Algebras[J]. Journal of Applied Mathematics and Physics, 2020, 8(5): 927-932.

[30] 刘春辉. BCI/BCK代数的反模糊(闭)理想格[J]. 模糊系统与数学, 2025, 39(1): 21-32.

[31] (LIU C H. Lattice of Anti-fuzzy(Closed)Ideals in a Civen BCI/BCK-Algebra[J]. Fuzzy Systems and Mathematics, 2025, 39(1): 21-32.)

基金资助

国家自然科学基金(12471021)

伊犁师范大学提升学科综合实力专项项目(2022XKZY08)

伊犁师范大学科研创新项目(YSD2025CX01)

伊犁师范大学高级别培育项目(YSPY2022011)