Fuzzy MultiLinear Sets, Neutrosophic MultiLinear Sets, abd Plithogenic MultiLinear Sets

Takaaki Fujita; Arif Mehmood

doi:10.66972/iscis2120265

Authors

Takaaki Fujita Independent Researcher, Tokyo, Japan Author https://orcid.org/0000-0002-9380-386X
Arif Mehmood Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail Khan 29050, KPK, Pakistan Author https://orcid.org/0000-0003-3256-1104

DOI:

https://doi.org/10.66972/iscis2120265

Keywords:

Fuzzy MultiLinear Sets, Neutrosophic MultiLinear Sets, abd Plithogenic MultiLinear Sets

Abstract

To address uncertainty, vagueness, and imprecision in decision-making, various extensions of classical set theory have been developed. Examples of such sets include fuzzy sets, neutrosophic sets, and plithogenic sets. A Multilinear Set consists of binary variables constrained by multilinear equations, where each auxiliary variable represents the product of selected primary binary variables. Extended concepts of the Multilinear Set using frameworks such as Fuzzy Sets have not yet been explored. To fill this gap, this paper investigates and analyzes the structures of Fuzzy Multilinear Sets, Neutrosophic Multilinear Sets, and Plithogenic Multilinear Sets.

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References

Jech, T. (2003). Set theory: The third millennium edition, revised and expanded. Springer. https://doi.org/10.1007/3-540-44761-X

Zadeh, L. A. (1965). Fuzzy sets. Information and control, 8(3), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X

Zadeh, L. A. (1972). A fuzzy-set-theoretic interpretation of linguistic hedges.

Atanassov, K. T. (2020). Circular intuitionistic fuzzy sets. Journal of Intelligent & Fuzzy Systems, 39(5), 5981–5986. https://doi.org/10.3233/JIFS-189072

Smarandache, F. (1999). A unifying field in logics: Neutrosophic logic. In Philosophy (pp. 1–141). American Research Press.

Broumi, S., Talea, M., Bakali, A., & Smarandache, F. (2016). Single valued neutrosophic graphs. Journal of New theory, (10), 86–101. https://doi.org/10.5281/zenodo.50940

Broumi, S., Talea, M., Bakali, A., Smarandache, F., & Kumar, P. K. (2017). Shortest path problem on single valued neutrosophic graphs. 2017 international symposium on networks, computers and communications (ISNCC), 1–6. https://doi.org/10.1109/ISNCC.2017.8071993

Luqman, A., Akram, M., & Smarandache, F. (2019). Complex neutrosophic hypergraphs: New social network models. Algorithms, 12, 234. https://api.semanticscholar.org/CorpusID:208609884

Ali, M., & Smarandache, F. (2016). Complex neutrosophic set. Neural Computing and Applications, 28, 1817–1834. https://api.semanticscholar.org/CorpusID:21384716

Al-Sharqi, F., Al-Quran, A., et al. (2022). Similarity measures on interval-complex neutrosophic soft sets with applications to decision making and medical diagnosis under uncertainty. Neutrosophic Sets and Systems, 51, 495–515. https://doi.org/10.5281/zenodo.7135362

Deli, I., Ali, M., & Smarandache, F. (2015). Bipolar neutrosophic sets and their application based on multi-criteria decision making problems. 2015 International Conference on Advanced Mechatronic Systems (ICAMechS), 249–254. https://api.semanticscholar.org/CorpusID:22083124

Ulucay, V., Deli, I., & Sahin, M. (2018). Similarity measures of bipolar neutrosophic sets and their application to multiple criteria decision making. Neural Computing and Applications, 29, 739–748. https://api.semanticscholar.org/CorpusID:7947430

Ye, J., & Du, S. (2017). Some distances, similarity and entropy measures for interval-valued neutrosophic sets and their relationship. International Journal of Machine Learning and Cybernetics, 10, 347–355. https://api.semanticscholar.org/CorpusID:67788068

Thong, N. T., Smarandache, F., Hoa, N. D., Son, L. H., Lan, L. T. H., Giap, C. N., Son, D. T., & Long, H. V. (2020). A novel dynamic multi-criteria decision making method based on generalized dynamic interval-valued neutrosophic set. Symmetry, 12(4), 618. https://doi.org/10.3390/sym12040618

bin Mohammad Kamari, M. S., Rodzi, Z. B. M., Al-Obaidi, R., Al-Sharq, F., Al-Quran, A., et al. (2025). Deciphering the geometric bonferroni mean operator in pythagorean neutrosophic sets framework. Neutrosophic Sets and Systems, 75, 139–161. https://digitalrepository.unm.edu/nss_journal/vol75/iss1/7

Ismail, J. N., Rodzi, Z., Al-Sharqi, F., Al-Quran, A., Hashim, H., & Sulaiman, N. H. (2023). Algebraic operations on pythagorean neutrosophic sets (pns): Extending applicability and decision-making capabilities. International Journal of Neutrosophic Science (IJNS), 21(4). https://doi.org/10.54216/IJNS.210412

Fujita, T., & Smarandache, F. (2025). Some types of hyperneutrosophic set (2): Complex, single-valued triangular, fermatean, and linguistic sets.

Fujita, T. (2025). Advancing uncertain combinatorics through graphization, hyperization, and uncertainization: Fuzzy, neutrosophic, soft, rough, and beyond. Biblio Publishing.

Smarandache, F. (2018a). Plithogeny, plithogenic set, logic, probability, and statistics. https://api.semanticscholar.org/CorpusID:51980485

Abdelfattah, W. (2022). Variables selection procedure for the dea overall efficiency assessment based plithogenic sets and mathematical programming. International Journal of Scientific Research and Management. https://api.semanticscholar.org/CorpusID:250137956

Martin, N., Priya, R., & Smarandache, F. (2025). Generalized plithogenic sets in multi-attribute decision making. In Neutrosophic and plithogenic inventory models for applied mathematics (pp. 519–546). IGI Global Scientific Publishing.

Smarandache, F. (2018b). Plithogeny, plithogenic set, logic, probability, and statistics. arXiv preprint arXiv:1808.03948.

Zadeh, L. A. (1996). Fuzzy logic, neural networks, and soft computing. In Fuzzy sets, fuzzy logic, and fuzzy systems: Selected papers by lotfi a zadeh (pp. 775–782). World Scientific. https://doi.org/10.1142/9789814261302_0040

Smarandache, F. (1998). Neutrosophy: Neutrosophic probability, set, and logic: Analytic synthesis & synthetic analysis.

Del Pia, A., & Khajavirad, A. (2018a). On decomposability of multilinear sets. Mathematical Programming, 170(2), 387–415. https://doi.org/10.1007/s10107-017-1158-z

Del Pia, A., & Khajavirad, A. (2021). The running intersection relaxation of the multilinear polytope. Mathematics of Operations Research, 46(3), 1008–1037. https://doi.org/10.1287/moor.2021.1121

Kim, J., Richard, J.-P. P., & Tawarmalani, M. (2024). A reciprocity between tree ensemble optimization and multilinear optimization. Operations Research. https://doi.org/10.1287/opre.2022.0150

Chen, R., Dash, S., & Günlük, O. (2023). Multilinear sets with two monomials and cardinality constraints. Discrete Applied Mathematics, 324, 67–79. https://doi.org/10.1016/j.dam.2022.09.010

Avis, D., & Bremner, D. (1995). How good are convex hull algorithms? Proceedings of the eleventh annual symposium on Computational geometry, 20–28. https://dl.acm.org/doi/pdf/10.1145/220279.220282

Barber, C. B., Dobkin, D. P., & Huhdanpaa, H. (1996). The quickhull algorithm for convex hulls. ACM Transactions on Mathematical Software (TOMS), 22(4), 469–483. https://doi.org/10.1145/235815.235821

Melkman, A. A. (1987). On-line construction of the convex hull of a simple polyline. Information Processing Letters, 25(1), 11–12.

Del Pia, A., & Khajavirad, A. (2018b). The multilinear polytope for acyclic hypergraphs. SIAM Journal on Optimization, 28(2), 1049–1076.

Rachmad, Y. E. (2018). Cultural synergy theory.

Fluck, Z., & Lynch, A. W. (1999). Why do firms merge and then divest? A theory of financial synergy. The journal of business, 72(3), 319–346.

Latash, M. L. (2008). Synergy. Oxford University Press.

Castillo, O., & Melin, P. (2021). Interval type-3 fuzzy decision-making in material surface quality control. Virtual International Conference on Soft Computing, Optimization Theory and Applications, 157–169. https://doi.org/10.1007/978-981-19-6406-0_12

Jdid, M., Smarandache, F., & Broumi, S. (2023). Inspection assignment form for product quality control using neutrosophic logic. Infinite Study.

Perrot, N., Ioannou, I., Allais, I., Curt, C., Hossenlopp, J., & Trystram, G. (2006). Fuzzy concepts applied to food product quality control: A review. Fuzzy sets and systems, 157(9), 1145–1154.

Diestel, R. (2024). Graph theory. Springer (print edition); Reinhard Diestel (eBooks).

Bretto, A. (2013). Hypergraph theory. An introduction. Mathematical Engineering. Cham: Springer, 1. https://doi.org/10.1007/978-3-319-00080-0

Berge, C. (1984). Hypergraphs: Combinatorics of finite sets (Vol. 45). Elsevier.

Hila, K., Onar, S., Ersoy, B. A., & Davvaz, B. (2013). On generalized intuitionistic fuzzy subhyperalgebras of boolean hyperalgebras. Journal of Inequalities and Applications, 2013, 1–15. https://doi.org/10.1186/1029-242X-2013-501

Smarandache, F. (2020). Extension of hypergraph to n-superhypergraph and to plithogenic n-superhypergraph, and extension of hyperalgebra to n-ary (classical-/neutro-/anti-) hyperalgebra. Infinite Study. https://fs.unm.edu/nss8/index.php/111/article/view/198

Amable, N. H., De Salazar, E. E. V., Isaac, M. G. M., Sánchez, O. C. O., & Palma, J. M. S. (2025). Representation of motivational dynamics in school environments through plithogenic n-super hyper graphs with family participation. Neutrosophic Sets and Systems, 92, 570–583. https://doi.org/10.5281/zenodo.17239914

Hamidi, M., Smarandache, F., & Davneshvar, E. (2022). Spectrum of superhypergraphs via flows. Journal of Mathematics, 2022(1), 9158912. https://doi.org/10.1155/2022/9158912

Berrocal Villegas, S. M., Montalvo Fritas, W., Berrocal Villegas, C. R., Flores Fuentes Rivera, M. Y., Espejo Rivera, R., Bautista Puma, L. D., & Macazana Fernández, D. M. (2025). Using plithogenic n-superhypergraphs to assess the degree of relationship between information skills and digital competencies. Neutrosophic Sets and Systems, 84(1), 41. https://digitalrepository.unm.edu/nss_journal/vol84/iss1/41/

Ghods, M., Rostami, Z., & Smarandache, F. (2022). Introduction to neutrosophic restricted superhypergraphs and neutrosophic restricted superhypertrees and several of their properties. Neutrosophic Sets and Systems, 50, 480–487.

Al Tahan, M., & Davvaz, B. (2018). Weak chemical hyperstructures associated to electrochemical cells. Iranian Journal of Mathematical Chemistry, 9(1), 65–75. https://doi.org/10.22052/ijmc.2017.88790.1294

Vougioukli, S. (2020). Hyperoperations defined on sets of s-helix matrices.

Ruggero, M. S., & Vougiouklis, T. (2017). Hyperstructures in lie-santilli admissibility and isotheories. Ratio Mathematica, 33, 151. http://eiris.it/ratio_numeri/ratio_33_2017/ruggero_151-165.pdf

Fuzzy MultiLinear Sets, Neutrosophic MultiLinear Sets, abd Plithogenic MultiLinear Sets

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