A Comprehensive Study of Neutrosophic SuperHyper BCI-Semigroups and their Algebraic Significance
الموضوعات : Transactions on Fuzzy Sets and Systems
Ajoy Kanti Das
1
,
Rajat DAS
2
,
Suman Das
3
,
Bijoy Krishna Debnath
4
,
Carlos Granados
5
,
Bimal Shil
6
,
Rakhal Das
7
1 - Department of Mathematics, Tripura University, Agartala, India.
2 - Department of Mathematics, Tripura University, Agartala,Tripura, India.
3 - Department of Education (ITEP), NIT Agartala, Agartala,Tripura, India.
4 - Department of Applied Sciences, School of Engineering, Tezpur University, Assam, India.
5 - Department of Mathematics, Universidad de Antioquia, Medellin, Colombia.
6 - Department of Statistics, Tripura University, Agartala, India.
7 - Department of Mathematics, The ICFAI University Tripura, Agartala, India.
الکلمات المفتاحية: Neutrosophic, SuperHyperStructure, SuperHyperAlgebra, SuperHyper Operations, BCI-Algebras, Semigroups, Ideal.,
ملخص المقالة :
This paper introduces and explores the concept of neutrosophic superhyper BCI-semigroups, an advanced algebraic structure integrating neutrosophic logic, superhyper operations, and BCI-algebras into a semigroup framework. Neutrosophic superhyper BCI-semigroups facilitate the handling of indeterminate and conflicting information by incorporating neutrosophic triplets for each element, denoting degrees of truth, indeterminacy, and falsity. The study defines essential properties such as closure, associativity, BCI identity, and neutrosophic membership and provides illustrative examples to demonstrate these properties in practical scenarios. Further, the paper delves into the characteristics of idempotent and commutative elements, homomorphisms, ideals, subsemigroups, and quotient sets within the context of neutrosophic superhyper BCI-semigroups. Key theorems are presented to establish foundational principles and behaviors of these structures, highlighting their theoretical implications and potential for practical applications.
[1] Smarandache F. A unifying field in logics: Neutrosophic logic. Neutrosophy, neutrosophic set, neutrosophic probability and statistics. Rehoboth: American Research Press; 1998. https://philpapers.org/rec/SMAAUF-3
[2] Smarandache F. SuperHyperAlgebra and neutrosophic SuperHyperAlgebra. In Nidus Idearum. Scilogs, II: de rerum consecration, 2nd ed. Brussels: Pons Press; 2016. https://digitalrepository.unm.edu/math-fsp/208
[3] Smarandache F. Foundation of SuperHyperStructure & neutrosophic SuperHyperStructure (review paper). Neutrosophic Sets and Systems. 2024; 63: 367-381. DOI: https://doi.org/10.5281/zenodo.10535374
[4] Smarandache F. A unifying field in logics: Neutrosophic logic. Neutrosophy, neutrosophic set, neutrosophic probability and statistics. Rehoboth: American Research Press; 1998. https://philpapers.org/rec/SMAAUF-3
[5] Smarandache F. Hyperuncertain, superuncertain, and superhyper uncertain sets/logics/probabilities/statistics. Critical Review. 2017; 29: 10-19.
[6] Smarandache F. N-SuperHyperGraph and plithogenic N-SuperHyperGraph, in Nidus Idearum, Vol. 7, 3rd ed. Brussels: Pons Press; 2019. https://fs.unm.edu/NidusIdearum7-ed3.pdf
[7] Smarandache F. Extension of HyperGraph to n-SuperHyperGraph and to plithogenic n SuperHyperGraph, and extension of HyperAlgebra to n-ary (Classical-/Neutro-/Anti-) HyperAlgebra. Neutrosophic Sets and Systems. 2020; 33: 290-296. DOI: https://doi.org/10.5281/zenodo.3783103
[8] Smarandache F. Introduction to the n-SuperHyperGraph-the most general form of graph today. Neutrosophic Sets and Systems. 2022; 48: 483485. DOI: https://doi.org/10.5281/zenodo.6096894
[9] Smarandache F. Foundation of revolutionary topologies: An overview, examples, trend analysis, research issues, challenges, and future directions. Neutrosophic Systems with Applications. 2024; 13: 45-66. DOI: https://doi.org/10.61356/j.nswa.2024.125
[10] Smarandache F. Foundation of the SuperHyperSoft set and the fuzzy extension SuperHyperSoft set: A new vision. Neutrosophic Systems with Applications. 2023; 11: 48-51. DOI: https://doi.org/10.61356/j.nswa.2023.95
[11] Smarandache, F. The SuperHyperFunction and the Neutrosophic SuperHyperFunction. Neutrosophic Sets and Systems. 2022; 49: 594-600. DOI: https://doi.org/10.5281/zenodo.6466524
[12] Smarandache F. Introduction to SuperHyperAlgebra and Neutrosophic SuperHyperAlgebra. Journal of Algebraic Hyperstructures and Logical Algebras. 2022; 3(2): 17-24. DOI: https://doi.org/10.52547/HATEF.JAHLA.3.2.2
[13] Smarandache, F. Neutrosophic set is a generalization of intuitionistic fuzzy set, inconsistent intuitionistic fuzzy set (picture fuzzy set, ternary fuzzy set), Pythagorean fuzzy set (Atanassovs intuitionistic fuzzy set of second type), q-rung orthopair fuzzy set, spherical fuzzy set, and n-hyperspherical fuzzy set, while neu trosophication is a generalization of regret theory, grey system theory, and three-ways decision. Journal of New Theory 2019; 29: 01-35.
[14] Zadeh LA. Fuzzy sets. Information and Control. 1965; 8(3): 33853. DOI: https://doi.org/10.1016/S0019-9958(65)90241-X
[15] Atanassov KT. Intuitionistic fuzzy sets. Fuzzy Sets and Systems. 1986; 20: 8796. DOI: https://doi.org/10.1016/S0165-0114(86)80034-3
[16] Gorzaczany MB. A method of inference in approximate reasoning based on interval valued fuzzy sets. Fuzzy Sets and Systems. 1987; 21: 117. DOI: https://doi.org/10.1016/0165-0114(87)90148-5
[17] Torra V. Hesitant fuzzy sets. International Journal of Intelligent Systems. 2010; 25(6): 529539. DOI: https://doi.org/10.1002/int.20418
[18] Yager RR. Pythagorean fuzzy subsets. In: Proceedings of the Joint IFSA World Congress and NAFIPS Congress and NAFIPS Annual Meeting, 2013; 5761.DOI: https://doi.org/10.1109/IFSANAFIPS.2013.6608375
[19] Kutlu GF, Kahraman C. Spherical fuzzy sets and spherical fuzzy TOPSIS method. J. Journal of Intelligent and Fuzzy Systems 2019; 36(1); 337352. DOI: https://doi.org/10.3233/JIFS-181401
[20] Smarandache F. NeutroAlgebra is a generalization of partial algebra. International Journal of Neutrosophic Science. 2020; 2: 8-17.DOI: https://doi.org/10.54216/IJNS.020103
[21] Agboola AAA, Ibrahim MA, Adeleke EO. Elementary Examination of NeutroAlgebras and antiAlgebras viz-a-viz the Classical Number Systems. International Journal of Neutrosophic Science. 2020; 4: 16-19. DOI: https://doi.org/10.54216/IJNS.040102
[22] Xin XL. Hyper BCI-Algebras. Discussiones Mathematicae-General Algebra and Applications. 2006; 26(1): 519. DOI: https://doi.org/10.7151/dmgaa.1102
[23] Kim HS, Kim YH. On BE-algebras. Science and Mathematics in Japan. 2007; 66(1): 113-116.
[24] Rezaei A, Smarandache F. On Neutro-BE-algebras and Anti-BE-algebras. International Journal of Neutrosophic Science. 2020; 4: 8-15. DOI: https://doi.org/10.5281/zenodo.3751862
[25] Ozturk MA, Jun YB. Commutative ideals of BCI-Algebras using MBJ-neutrosophic structures. Mathematics. 2021; 9(17): 1-13. DOI: https://doi.org/10.3390/math9172047
[26] Hamidi M. On neutro-d-subalgebras. Journal of Algebraic Hyperstructures and Logical Algebras. 2021; 2(2): 1323. DOI: https://doi.org/10.52547/HATEF.JAHLA.2.2.13
[27] Hamidi M. On superhyper BCK-Algebras. Neutrosophic Sets and Systems. 2023; 55: 580-588.
[28] Das S, Das R, Pramanik S. Neutro algebra and neutro group. In F. Smarandache, M. Al-Tahan (Eds.) Theory and applications of neutroalgebras as generalizations of classical algebras, IGI Global: Hershey, PA, USA, 2022; pp 141-154. DOI: https://doi.org/10.4018/978-1-6684-3495-6.ch009
[29] Smarandache F, Al-Tahan M. (Eds.). Theory and applications of neutroalgebras as generalizations of classical algebras. IGI Global: Hershey, PA, USA, 2022; page 333. DOI: https://doi.org/10.4018/978- 1-6684-3495-6
[30] Das S, Das R, Shil B, Tripathy BC. Lie-algebra of single-valued pentapartitioned neutrosophic set. Neutrosophic Sets and Systems. 2022; 51: 157-171.
[31] Das R, Das S, Granados C, Hasan AK. Neutrosophic d-Filter of d-Algebra. Iraqi Journal of Science. 2023; 65(2): 855-864. DOI: https://doi.org/10.24996/ijs.2023.64.2.31
[32] Muhiuddin G, Abughazalah N, Mahboob A, et al. Hyperstructure theory applied to BF Algebras. Symmetry. 2023; 15(5):1-11. DOI: https://doi.org/10.3390/sym15051106
[33] Muhiuddin G, Elavarasan B, Porselvi K, et al. Properties of neutrosophic N-Ideals in subtraction semigroups. Neutrosophic Sets and Systems. 2023; 60: 120-130.
[34] Santhakumar S, Sumathi IR, Mahalakshmi J. A novel approach to the algebraic structure of neutrosophic SuperHyperAlgebra. Neutrosophic Sets and Systems. 2023; 60: 100-110.
[35] Smarandache F, Bordbar H. Neutrosophic N-Structures applied to BCK/BCI-Algebras. Information. 2023; 8(4): 128-140. DOI: https://doi.org/10.3390/info8040128
[36] Yang E, Roh EH, Jun YB. Single valued neutrosophic ordered subalgebras of ordered BCI-algebras. Neutrosophic Sets and Systems. 2023; 60: 150-160.
[37] Rahmati M, Hamidi M. Extension of G algebras to SuperHyper G algebras. Neutrosophic Sets and Systems. 2023; 55: 557-567.
[38] Smarandache F, Pramanik S, editors. New trends in neutrosophic theory and applications. Pons Editions, Brussels; 2016. DOI: https://doi.org/10.48550/arXiv.1611.08555
[39] Smarandache F, Pramanik S, editors. New trends in neutrosophic theory and applications, vol 2. Pons Editions, Brussels; 2018.
[40] Broumi S, Bakali A, Talea M, Smarandache F, Uluay V, Sahin S, Dey A, Dhar M, Tan RP, de Oliveira A, Pramanik S. Neutrosophic sets: An overview. In: Smarandache F, Pramanik S, editors. New trends in neutrosophic theory and applications, Pons Editions, Brussels; 2018; 2: 403-434. DOI:
https://doi.org/10.5281/zenodo.1237940
[41] Pramanik S, Mallick R, Dasgupta A. Contributions of selected Indian researchers to multi-attribute decision making in neutrosophic environment. Neutrosophic Sets and Systems. 2018; 20: 108-131.
[42] Otay I, Kahraman C. A state-of-the-art review of neutrosophic sets and theory. In: Kahraman, C.; Otay, I. (eds) Fuzzy multi-criteria decision-making using neutrosophic sets. studies in fuzziness and soft computing, vol 369. Springer, Cham, 2019. DOI: https://doi.org/10.1007/978-3-030-00045-5 1
[43] Peng X, Dai J. A bibliometric analysis of neutrosophic set: Two decades review from 1998 to 2017. Artificial Intelligence Review. 2020; 53(1): 199-255. DOI: https://doi.org/10.1007/s10462-018-9652-0
[44] Pramanik S. Rough neutrosophic set: An overview. In F. Smarandache, and S. Broumi, Eds.), Neutrosophic theories in communication, management and information technology, New York. Nova Science Publishers. 2020; 275-311.
[45] Pramanik S. Single-valued neutrosophic set: An overview. In: Rezaei N, editor. Transdisciplinarity. Integrated Science. 2022; 5: 563-608. DOI: https://doi.org/10.1007/978-3-030-94651-7 26.
[46] Delcea C, Domenteanu A, Ioanas C, Vargas VM, Ciucu-Durnoi AN. (2023). Quantifying neutrosophic research: A bibliometric study. Axioms. 2023; 12(12): 1083. DOI: https://doi.org/10.3390/axioms12121083
[47] Boole G. An investigation of the laws of thought.Prometheus Books. Cambridge University Press. 1854. DOI: https://doi.org/10.1017/CBO9780511693090
[48] Das AK. On partially included intuitionistic fuzzy rough relations. Afrika Matematika. 2016; 27: 993- 1001. DOI: https://doi.org/10.1007/s13370-016-0395-2
[49] Das AK, Granados C. Preference intuitionistic fuzzy rough relation and its theoretical approach. Journal of the Indian Mathematical Society. 2023; 90(3-4): 199-212.
[50] Mukherjee A, Das AK. Relations on intuitionistic fuzzy soft multi-sets. In: Kim K, editor.Science and Applications. Lecture Notes in Electrical Engineering. Berlin, Heidelberg: Springer; 2015; 339: 607-614. DOI: https://doi.org/10.1007/978-3-662-46578-3 71.
[51] Mukherjee A, Das AK. Essentials of fuzzy soft multisets: Theory and applications. Springer Nature Singapore Pte Ltd. 2023; 9 : 1-151. DOI: https://doi.org/10.1007/978-981-19-2760-7
[52] Das AK. Weighted fuzzy soft multiset and decision-making. International Journal of Machine Learning and Cybernetics. 2018; 9: 787-794. DOI: https://doi.org/10.1007/s13042-016-0607-y
[53] Mukherjee A, Das AK. Einstein operations on fuzzy soft multisets and decision making. Boletim Da Sociedade Paranaense De Matematica. 2022; 40: 1-10. DOI: https://doi.org/10.5269/Bspm.32546
