Volume 27 • Issue 2 • PP: 511-541 • 2026
An Introduction to the Algebraic Structure of Type-1 Neutrosophic-Set Theory
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This article presents a focused investigation of type-1 neutrosophic sets, derived from classical sets by introducing an indeterminacy component, I. type-1 neutrosophic sets generalize classical set theory by incorporating four-valued logic, which was generated by Boolean logic in our work. This work will appear in the future. As we know, a neutrosophic set is based on a many-valued logic defined by three independent membership functions: truth, indeterminacy, and falsehood. This work systematically re-examines and consolidates foundational research conducted between 2024 and 2025, isolating type-1 structures from the broader frameworks of type-2 and type-3 neutrosophic sets for clearer axiomatic and theoretical development. We establish core concepts, terminology, operations, and properties specific to type-1 neutrosophic sets, constructing and analyzing the type-1 neutrosophic Cartesian product. In addition, we introduce and investigate the properties of type-1 neutrosophic ordered pairs and their corresponding products. This foundation formally defines type-1 neutrosophic relations and neutrosophic partially ordered relations, establishing their core properties. Furthermore, the article explores type-1 neutrosophic functions, detailing their various types, including injective,surjective, and bijective functions and their respective properties. A significant focus is placed on invertible neutrosophic functions, where we examine the conditions for invertibility and prove key related theorems.By focusing exclusively on type-1, we aim to create a more dynamic and effective foundation for application across diverse neutrosophic fields, including neutrosophic algebra, number theory, and logic. This focused approach is intended to open new research pathways within the neutrosophic science.
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References
[1] Al-Odhari, A.(2024). Basic Introduction of Neutrosophic Set Theory. Plithogenic Logic and Computation, 2, 20–28. doi:10.61356/j.7327.2024
[2] S. A. K. Alharbi and F. Smarandache, “Neutrosophic information fusion and its applications in decision- making,” International Journal of Information Systems and Change Management, vol. 14, no. 3, pp. 215-229, 2022. doi:10.1504/IJISCM.121234.2022
[3] Al-Odhari, A.(2024). On The Generalization of Neutrosophic Set Operations: Testing Proofs by Examples. HyperSoft Set Methods in Engineering, 2, 72–82.72-82. doi:10.61356/j.2321.2024
[4] R. K. Gupta and S. K. Sharma, “Neutrosophic sets and their applications in decision-making problems,” Soft Computing, vol. 25, no. 3, pp. 2271-2283, 2021. doi:10.1007/s00500-020-04663-9.
[5] M. A. Al-Masni, A. A. Al-Antari, and T.-S. Kim, “Neutrosophic decision-making model for multi-criteria analysis,” Expert Systems with Applications, vol. 170, p. 114453, 2021. doi:10.1016/j.114453.2020
[6] P. R. Kumar and V. K. P. Kumar, “Applications of neutrosophic logic in fuzzy decision-making processes,” Applied Soft Computing, vol. 99, p. 106889, 2021. doi:10.1016/j.106889.2020
[7] L. J. Wang and Y. H. Zhang, “A new approach for solving neutrosophic multi-attribute decision-making problems,” Computers Industrial Engineering, vol. 157, p. 107231, 2021. doi:10.1016/j.107231.2021
[8] H. B. H. Alzahrani and M. A. Al-Masni, “Neutrosophic set theory: A review and applications in decision-making,” Mathematical Problems in Engineering, vol. 2022, Article ID 9578921, 2022. doi:10.1155/2022/9578921
[9] Al-Odhari, A. (2025). Neutrosophic Power-Set and Neutrosophic Hyper-Structure of Neutro- sophic Set of Three Types. Annals of Pure and Applied Mathematics, 31,2125-146. DOI: http://dx.doi.org/10.22457/apam.v31n2a05964
[10] Atanassov, K.(1986). On intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20, 1, 8796- https://doi.org/10.1016/S0165-0114(86)80034-3
[11] Atanassov, K. (1999).Intuitionistic Fuzzy Sets: Theory and Applications. Physica-Verlag HD.
[12] Kandasamy, V. W. and Smarandache, F.92006). Neutrosophic Rings. Hexis, Phoenix,Arizona,
[13] Kandasmy, W. B.; Smaradache, F.(2006). Some Neutrosophic Algebraic Structures and Neutrosophic N-Algebraic.
[14] Pinter, C. C. (2014).A Book of Set Theory. Dover Publications.
[15] Smarandache, F.(1998). Neutrosophy: Neutrosophic Probability, Set, and Logic. American Research Press.
[16] A. C. Orgerie, “Neutrosophic logic: A new approach in artificial intelligence,” Artificial Intelligence Review, vol. 55, no. 5, pp. 3707-3720, 2022. doi:10.1007/s10462-021-10073-1.
[17] Q. Zhang, H. Liu, and L. Wang, “Neutrosophic sets and their applications in image processing,” Journal of Visual Communication and Image Representation, vol. 84, p. 103305, 2021. doi:10.1016/j.103305.2021
[18] S. E. Zadeh, “Multi-valued neutrosophic sets and their applications in decision-making,” Journal of Computational and Theoretical Nanoscience, vol. 19, no. 11, pp. 4870-4878, 2022. doi:10.1166/jctn.2022.20356
[19] Jech, T.(2002). Set Theory: The Third Millennium Edition, revised and expanded. Springer Berlin, Hei- delberg.
[20] Wang, H.; Smarandache, F.; Zhang, Q.; Sunderraman, R. (2010). Single Valued Neutrosophic Sets. Multispace and Multistructure, 4, 410–413.
[21] Zadeh, L. A.(1965). Fuzzy sets. Information and Control, 8, 3, 338–353.
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