Volume 26 • Issue 3 • PP: 26-48 • 2025
An Investigation of Complex Linear Diophantine Fuzzy Ideals in BCK-Algebras
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© 2025 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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A complex linear Diophantine fuzzy (CLDF) set extends a linear Diophantine fuzzy set (LDFS) by handling uncertainty with complex-valued membership degrees within a unit disc. In this paper, we combine the notions of LDFS, BCK-algebra, and complex fuzzy set (CFS) to preface and elaborate the innovative concepts of CLDF subalgebras (CLDF − Subs), CLDF ideals (CLDF − Ids), CLDF implicative ideals (CLDF − IIds), and CLDF positive implicative ideals (CLDF − PIIds) in BCK-algebras, and probe their fundamental characteristics. These new notations of certain kinds of algebraic substructures in BCK-algebras serve as a bridge among CLDFS, crisp set, and BCK-algebra and also demonstrate the influence of the CLDFS on a BCK-algebra. Moreover, we examine some illustrative examples and prevalent features of these innovative notions in detail. Finally, characterizations of these intricate fuzzy structures are given, and related results for ideals, implicative ideals, and positive implicative ideals in the view of CLDFSs are obtained.
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References
[1] Y. Imai, K. Is´eki, On axiom systems of propositional calculi, XIV. Proc. Jpn. Acad. 42(1) (1966), 19–21.
[2] K. Is´eki, An algebra related with a propositional calculus, Proc. Jpn. Acad. 42(1) (1966), 26–29.
[3] K. Is´eki, S. Tanaka, An introduction to the theory of BCK-algebras, Math. Japon. 3(1978), 1–26.
[4] L. A. Zadeh, Fuzzy sets, Inf. Control. 8(3) (1965), 338–353.
[5] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20(1) (1986), 87–96.
[6] R. R. Yager, Pythagorean membership grades in multicriteria decision making, IEEE Trans. Fuzzy Syst. 22(4) (2014), 958–965.
[7] P. Liu, P. Wang, Some q-rung orthopair fuzzy aggregation operators and their applications to multiple- attribute decision making, Int. J. Intell. Syst. 33(20) (2018), 259–280.
[8] M. I. Ali, Another view on q-rung orthopair fuzzy sets I,nt. J. Intell. Syst. 33(11) (2018), 2139–2153.
[9] M. Riaz, M. R. Hashmi, Linear Diophantine fuzzy set and its applications towards multi-attribute decision making problems, J. Intell. Fuzzy Syst. 37(4) (2019), 5417–5439.
[10] A. Aydogdu, S. Gul, T. Alniak, New information measures for linear Diophantine fuzzy sets and their ap- plications with LDF-ARAS on data storage system selection problem, Expert Systems with Applications, 252(A) (2024), 124135.
[11] A. O. Almagrabi, S. Abdullah, M. Shams, Y. D. Al-Otaibi, S. Ashraf, A new approach to q-linear Dio- phantine fuzzy emergency decision support system for COVID 19, J. Ambient. Intell. Humaniz. Comput. 13(2022), 1687–1713.
[12] A. Al-Quran, T-spherical linear Diophantine fuzzy aggregation operators for multiple attribute decision- making, AIMS Mathematics, bf8(5) (2023), 12257–12286.
[13] D. Ramot, M. Friedman, G. Langholz, A. Kandel, Complex fuzzy logic, IEEE Trans Fuzzy Syst. 11(4) (2003), 450–461.
[14] D. Ramot, R. Milo, M. Friedman, A. Kandel, Complex fuzzy sets, IEEE Trans. Fuzzy Syst. 10(20) (2002), 171–186.
[15] A. M. D. J. S. Alkouri, A. R. Salleh, Complex intuitionistic fuzzy sets, In AIP Conference Proceedings; American Institute of Physics: College Park, MD, USA. 1482(1) (2012), 464–470.
[16] K. Ullah, T. Mahmood, Z. Ali, N. Jan, On some distance measures of complex pythagorean fuzzy sets and their applications in pattern recognition, Complex Intell. Syst. 6 (2020), 15–27.
[17] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35(30) (1971), 512–517.
[18] O. G. Xi, Fuzzy BCK-algebras, Math. Jpn. 36 (1991), 935–942.
[19] B. B. Ahmad, Fuzzy BCI-algebras, J. Fuzzy Math. 1 (1993), 445–452.
[20] Y. B. Jun, H. M. Kim, Intuitionistic fuzzy ideals of BCK-algebras, Int. J. Math. Math. Sci. 24 (2000), 839–849.
[21] M. T. Nirmala, D. Jayalakshmi, G. Subbiah, Applications of pythagorean fuzzy subalgebras of BCK/BCI- algebra, Afr. J. Bio. Sc. 6(12) (2024), 3001–3013.
[22] G. Muhiuddin, M. Al-Tahan, A. Mahboob, S. Hoskova-Mayerova, S. Al-Kaseasbeh, Linear Diophantine fuzzy set theory applied to BCK/BCI-algebras, Mathematics. 10 (2022), 2138.
[23] H. Kamaci, Linear Diophantine fuzzy algebraic structures, J. Ambient Intell. Humaniz. Comput. 12(11) (2021), 10353–10373.
[24] F. Yousafzai, M. D. Zia, M.U.I. Khan, M. M. Khalaf, R. Ismail, Linear Diophantine fuzzy sets over complex fuzzy information with applications in information theory, Ain Shams Engineering Journal, 15(1) (2024), 102327.
[25] H. Kamaci, Complex linear diophantine fuzzy sets and their cosine similarity measures with applications, Complex Intell. Syst. 8(2) (2022), 1281–1305.
[26] G. Hao, F. Yousafzai, M. D. Zia, M. U. I. Khan, M. Irfan, K. Hila, Complex linear diophantine fuzzy sets over AG-groupoids with applications in civil engineering, Symmetry, 15(1) (2023), 74.
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