Tri-valued Neutrosophic Soft Structures with Graphics and Machine-Learning-Assisted Visualizations

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DOI: https://doi.org/10.28924/2291-8639-24-2026-144
Published: May 11, 2026
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Jamil J. Hamja, Diana Amin Mohammad Mahmoud, Raed Hatamleh, Haitham Qawaqneh, Shoaib Arbab, Alaa M. Abd El-latif, Mohammed Mamoun Ahmed Abubakr, Arif Mehmood, Cris L. Armada, Tri-valued Neutrosophic Soft Structures with Graphics and Machine-Learning-Assisted Visualizations, Int. J. Anal. Appl., 24 (2026), 144.

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Jamil J. Hamja, Diana Amin Mohammad Mahmoud, Raed Hatamleh, Haitham Qawaqneh, Shoaib Arbab, Alaa M. Abd El-latif, Mohammed Mamoun Ahmed Abubakr, Arif Mehmood, Cris L. Armada

Abstract

The notion of the tri-valued neutrosophic soft set (TVNSS), its basic operations, and examples of each operation are first explained in this paper as worked problems that are subsequently displayed as understandable graphics. Based on this, we further offer the tri-valued neutrosophic topological space (TVNSTS), outline its fundamental functions, and provide instances once more, illustrating the outcomes with excellent graphical representations. We also employ the chosen machine-learning and statistical techniques to provide comparative visualizations in order to improve and triangulate these constructs. Next, we use these tools to depict patient-disease similarity at complementary angles on a 5x5 grid. A 3D bar chart (grouped into a 3D) and a cool-to-warm Heatmaps both agree on the single best match, P2 -D1 = 0.9107, which we identify and emphasize. Additionally, the corner highs at P1D5 and P3D5 are steady, the majority of the patients have high scores at the region, and there is a noticeable high ridge at D4. Likewise, P2D3, P4D4, and P5D5 are among the strong pairs in the 0.87–0.91 range, whereas P4D1 and P2D5 are near the lower end of the scale. With no change in narrative, min-max normalization to [0,1] then saturates the contrast: row-wise maxima are apparent at P1 -D5, P3 -D4, P4 -D4, and P5 -D5, while P2 -D1 is the global maximum at 1.000. The signal is strong on metrics and scales when a Pearson correlation representation in a red-yellow-green palette regularly follows the cosine pattern and, after normalization, isolates the same dominant cells. Finally, PCA with k-means is used to describe the form of these visuals: in the raw matrix, two small clusters, {P2, P3, P5} and {P1, P4}, form. These clusters are primarily concentrated on PC1 at about 81% variance; during normalization, P2 is an outlier that is enjoyable on its own, and the remaining four are tightly clustered by PC1 and PC2, which account for about 94% of the variation. When taken together, these associated opinions show a repeated high ridge along D4, a solid unification of classification, and a single definitive assignment of the patient and condition.

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References

  1. F. Smarandache, Neutrosophy: Neutrosophic Probability, Set, and Logic: Analytic Synthesis & Synthetic Analysis, American Research Press, 1998.
  2. F. Smarandache, A Unifying Field in Logics. Neutrosophy: Neutrosophic Probability, Set, and Logic, American Research Press, Rehoboth, 1999.
  3. H. Wang, F. Smarandache, Y.Q. Zhang, R. Sunderraman, Single valued neutrosophic sets, in: F. Smarandache (Ed.), Multispace & Multistructure. Neutrosophic Transdisciplinarity, Vol. 4, North-European Scientific Publishers, Hanko, Finland, 2010, pp. 410–413.
  4. S. Alkhazaleh, B. Batiha, A. Al-khateeb, H. Zureigat, A. Al-shboul, K. Batiha, Possibility Fermatean Neutrosophic Soft Set, Int. J. Neutrosophic Sci. 24 (2024), 105-125. https://doi.org/10.54216/IJNS.240408.
  5. R.K. Mohanty, B.K. Tripathy, A New Approach to Neutrosophic Soft Sets and Their Application in Decision Making, Neutrosophic Sets Syst. 60 (2023), 159-174.
  6. M. Parimala, M. Karthika, F. Smarandache, A Review of Fuzzy Soft Topological Spaces, Intuitionistic Fuzzy Soft Topological Spaces and Neutrosophic Soft Topological Spaces, Int. J. Neutrosophic Sci. 10 (2020), 96–104.
  7. P. Liu, L. Shi, The Generalized Hybrid Weighted Average Operator Based on Interval Neutrosophic Hesitant Set and Its Application to Multiple Attribute Decision Making, Neural Comput. Appl. 26 (2014), 457-471. https://doi.org/10.1007/s00521-014-1736-4.
  8. C. Kahraman, I. Otay, Eds., Fuzzy Multi-Criteria Decision-Making Using Neutrosophic Sets, Springer, 2019. https://doi.org/10.1007/978-3-030-00045-5.
  9. Y. Guo, H. Cheng, New Neutrosophic Approach to Image Segmentation, Pattern Recognit. 42 (2009), 587-595. https://doi.org/10.1016/j.patcog.2008.10.002.
  10. M. Abdel-Basset, M. Saleh, A. Gamal, F. Smarandache, An Approach of TOPSIS Technique for Developing Supplier Selection with Group Decision Making Under Type-2 Neutrosophic Number, Appl. Soft Comput. 77 (2019), 438-452. https://doi.org/10.1016/j.asoc.2019.01.035.
  11. F. Smarandache, Refined Literal Indeterminacy and the Multiplication Law of Sub-Indeterminacies, Neutrosophic Sets Syst. 9 (2015), 58–63.
  12. I. Deli, Refined Neutrosophic Sets and Refined Neutrosophic Soft Sets: Theory and Applications, in: Handbook of Research on Generalized and Hybrid Set Structures and Applications for Soft Computing, IGI Global, 2016: pp. 321--343. https://doi.org/10.4018/978-1-4666-9798-0.ch016.
  13. U. Solang, J. Ye, Refined Simplified Neutrosophic Similarity Measures Based on Trigonometric Function and Their Application in Construction Project Decision-Making, J. Soft Comput. Civ. Eng. 2 (2018), 1-12. https://doi.org/10.22115/scce.2018.126129.1056.
  14. R. Sahin, M. Yigider, A Multi-Criteria Neutrosophic Group Decision Making Method Based TOPSIS for Supplier Selection, Appl. Math. Inf. Sci. 10 (2016), 1843-1852. https://doi.org/10.18576/amis/100525.
  15. M. Teodorescu, D. Ionescu, Florentin Smarandache & Ştefan Vlăduţescu: Neutrosophic Emergences and Incidences in Communication and Information - Book Review, Int. Lett. Soc. Humanist. Sci. 38 (2014), 94-99. https://doi.org/10.18052/www.scipress.com/ILSHS.38.94.
  16. M.M. Saeed, R.H. Hatamleh, A.M.A. El-latif, A. Al-Husban, T. Fujita, et al., Separation Axioms in Quadri-Partition Neutrosophic Soft Topological Spaces, Eur. J. Pure Appl. Math. 18 (2025), 6324. https://doi.org/10.29020/nybg.ejpam.v18i3.6324.
  17. F. Smarandache, Neutrosophic Set – A Generalization of the Intuitionistic Fuzzy Sets, J. Pure Appl. Math. 24 (2005), 287-297.
  18. D. Molodtsov, Soft Set Theory—First Results, Comput. Math. Appl. 37 (1999), 19-31. https://doi.org/10.1016/S0898-1221(99)00056-5.
  19. P.K. Maji, Neutrosophic Soft Set, Ann. Fuzzy Math. Inform. 5 (2013), 157–168.
  20. T. Bera, N.K. Mahapatra, Introduction to Neutrosophic Soft Topological Space, OPSEARCH 54 (2017), 841-867. https://doi.org/10.1007/s12597-017-0308-7.
  21. T.Y. Ozturk, C.G. Aras, S. Bayramov, A New Approach to Operations on Neutrosophic Soft Sets and to Neutrosophic Soft Topological Spaces, Commun. Math. Appl. 10 (2019), 481–493. https://doi.org/10.26713/cma.v10i3.1068.
  22. T. Fujita, The Hyperfuzzy VIKOR and Hyperfuzzy DEMATEL Methods for Multi-Criteria Decision-Making, Spectr. Decis. Mak. Appl. 3 (2026), 292-315. https://doi.org/10.31181/sdmap31202654.
  23. M.E.M. Abdalla, A. Uzair, A. Ishtiaq, M. Tahir, M. Kamran, Algebraic Structures and Practical Implications of Interval-Valued Fermatean Neutrosophic Super HyperSoft Sets in Healthcare, Spectr. Oper. Res. 2 (2025), 240-259. https://doi.org/10.31181/sor21202523.
  24. R. Gul, An Extension of VIKOR Approach for MCDM Using Bipolar Fuzzy Preference Δ-Covering Based Bipolar Fuzzy Rough Set Model, Spectr. Oper. Res. 2 (2025), 72-91. https://doi.org/10.31181/sor21202511.
  25. A. Jaleel, T. Mahmood, D. Pamucar, M. Iftikhar, Optimizing Computer Engineering Problems Using CODAS Method with Bipolar Complex Fuzzy Soft Frank Aggregation Operators, Contemp. Math. 6 (2025), 5145–5171.
  26. S. Goswami, D. Pamucar, Application of Fuzzy MCDM in Selecting Eco-Friendly Materials for Electric Vehicle Interiors, J. Appl. Eng. Sci. 23 (2025), 504-522. https://doi.org/10.5937/jaes0-57423.
  27. A. Tarafdar, A. Shaikh, D. Bhowmik, P. Majumder, D. Pamucar, et al., Integration of Data-Driven T-Spherical Fuzzy Mathematical Models for Evaluation of Electric Vehicles: Response to Electric Vehicle Market Demands, Renew. Sustain. Energy Rev. 223 (2025), 116008. https://doi.org/10.1016/j.rser.2025.116008.
  28. M. Palanikumar, N. Kausar, D. Pamucar, V. Simic, Optimizing Industrial Robot Selection Using Novel Trigonometric Pythagorean Fuzzy Normal Aggregation Operators, Complex Intell. Syst. 11 (2025), 453. https://doi.org/10.1007/s40747-025-02083-5.
  29. F. Lilik, L. Solecki, B. Sziová, L.T. Kóczy, S. Nagy, On Wavelet Based Enhancing Possibilities of Fuzzy Classification Methods, in: P. Kulczycki, J. Kacprzyk, L.T. Kóczy, R. Mesiar, R. Wisniewski (Eds.), Information Technology, Systems Research, and Computational Physics, Springer, Cham, 2020: pp. 56–73. https://doi.org/10.1007/978-3-030-18058-4_5.
  30. A. Tormási, L.T. Kóczy, Comparing the Efficiency of a Fuzzy Single-Stroke Character Recognizer with Various Parameter Values, in: S. Greco, B. Bouchon-Meunier, G. Coletti, M. Fedrizzi, B. Matarazzo, R.R. Yager (Eds.), Advances on Computational Intelligence, Springer, Berlin, Heidelberg, 2012: pp. 260–269. https://doi.org/10.1007/978-3-642-31709-5_27.
  31. D. Göndöcs, V. Dörfler, AI in Medical Diagnosis: AI Prediction & Human Judgment, Artif. Intell. Med. 149 (2024), 102769. https://doi.org/10.1016/j.artmed.2024.102769.
  32. A. Ballagi, L.T. Koczy, Decision Making in Multi-Robot Cooperation by Fuzzy Signature Sets, in: International Conference on Fuzzy Systems, IEEE, 2010, pp. 1-8. https://doi.org/10.1109/FUZZY.2010.5584821.
  33. A.A. Abubaker, R. Hatamleh, K. Matarneh, A. Al-Husban, On the Numerical Solutions for Some Neutrosophic Singular Boundary Value Problems by Using (LPM) Polynomials, Int. J. Neutrosophic Sci. 25 (2025), 197-205. https://doi.org/10.54216/IJNS.250217.
  34. A. Ahmad, R. Hatamleh, K. Matarneh, A. Al-Husban, On the Irreversible K-Threshold Conversion Number for Some Graph Products and Neutrosophic Graphs, Int. J. Neutrosophic Sci. 25 (2025), 183–196. https://doi.org/10.54216/IJNS.250216.
  35. R. Hatamleh, A. Al-Husban, K. Sundareswari, G. Balaj, M. Palanikumar, Complex Tangent Trigonometric Approach Applied to (γ, τ)-Rung Fuzzy Set Using Weighted Averaging, Geometric Operators and Its Extension, Commun. Appl. Nonlinear Anal. 32 (2024), 133-144. https://doi.org/10.52783/cana.v32.2978.
  36. R. Hatamleh, A. Hazaymeh, On Some Topological Spaces Based on Symbolic n-Plithogenic Intervals, Int. J. Neutrosophic Sci. 25 (2025), 23–37. https://doi.org/10.54216/IJNS.250102.
  37. H. Qawaqneh, Fractional Analytic Solutions and Fixed Point Results with Some Applications, Adv. Fixed Point Theory, 14 (2024), 1. https://doi.org/10.28919/afpt/8279.
  38. H. Qawaqneh, M.S.M. Noorani, H. Aydi, A. Zraiqat, A.H. Ansari, On Fixed Point Results in Partial b-Metric Spaces, J. Funct. Spaces 2021 (2021), 8769190. https://doi.org/10.1155/2021/8769190.
  39. H. Qawaqneh, M.S.M. Noorani, H. Aydi, Some New Characterizations and Results for Fuzzy Contractions in Fuzzy b-Metric Spaces and Applications, AIMS Math. 8 (2023), 6682-6696. https://doi.org/10.3934/math.2023338.
  40. H. Qawaqneh, J. Manafian, M. Alharthi, Y. Alrashedi, Stability Analysis, Modulation Instability, and Beta-Time Fractional Exact Soliton Solutions to the Van Der Waals Equation, Mathematics 12 (2024), 2257. https://doi.org/10.3390/math12142257.
  41. H. Qawaqneh, New Functions for Fixed Point Results in Metric Spaces with Some Applications, Indian J. Math. 66 (2024), 55–84.
  42. H. Qawaqneh, H.A. Hammad, H. Aydi, Exploring New Geometric Contraction Mappings and Their Applications in Fractional Metric Spaces, AIMS Math. 9 (2024), 521-541. https://doi.org/10.3934/math.2024028.
  43. M. Elbes, T. Kanan, M. Alia, M. Ziad, COVD-19 Detection Platform from X-Ray Images Using Deep Learning, Int. J. Adv. Soft Comput. Appl. 14 (2022), 197-211. https://doi.org/10.15849/IJASCA.220328.13.
  44. T. Kanan, M. Elbes, K.A. Maria, M. Alia, Exploring the Potential of IoT-Based Learning Environments in Education, Int. J. Adv. Soft Comput. Appl. 15 (2023), 166-178.
  45. I.M. Batiha, S.A. Njadat, R.M. Batyha, A. Zraiqat, A. Dababneh, et al., Design Fractional-Order PID Controllers for Single-Joint Robot Arm Model, Int. J. Adv. Soft Comput. Appl. 14 (2022), 97-114. https://doi.org/10.15849/IJASCA.220720.07.
  46. A.M.A. El-latif, M.H. Alqahtani, Novel Categories of Supra Soft Continuous Maps via New Soft Operators, AIMS Math. 9 (2024), 7449-7470. https://doi.org/10.3934/math.2024361.
  47. A.M. Abd El-latif, M.H. Alqahtani, F.A. Gharib, Strictly Wider Class of Soft Sets via Supra Soft Δ-Closure Operator, Int. J. Anal. Appl. 22 (2024), 47. https://doi.org/10.28924/2291-8639-22-2024-47.
  48. A. Al-Omari, M.H. Alqahtani, Some Operators in Soft Primal Spaces, AIMS Math. 9 (2024), 10756-10774. https://doi.org/10.3934/math.2024525.
  49. A.M. Abd El-latif, A.A. Azzam, R. Abu-Gdairi, M. Aldawood, M.H. Alqahtani, New Versions of Maps and Connected Spaces via Supra Soft Sd-Operators, PLOS ONE 19 (2024), e0304042. https://doi.org/10.1371/journal.pone.0304042.
  50. A.M.A. El-latif, A.A. Azzam, R. Abu-Gdairi, M.H. Alqahtani, G.M. Abd-Elhamed, Applications on Soft Somewhere Dense Sets, J. Interdiscip. Math. 27 (2024), 1679-1699. https://doi.org/10.47974/JIM-2007.
  51. M.H. Alqahtani, Neutrosophic Primal Structure with Closure and Proximity Operators, AIMS Math. 11 (2025), 644-660. https://doi.org/10.3934/math.2026028.
  52. M.H. Alqahtani, Operators and Separation Axioms Within the Framework of Diving Topological Spaces, AIMS Math. 10 (2025), 25253-25273. https://doi.org/10.3934/math.20251118.
  53. I. Ibedou, S.E. Abbas, M.H. Alqahtani, Defining New Structures on a Universal Set: Diving Structures and Floating Structures, Mathematics 13 (2025), 1859. https://doi.org/10.3390/math13111859.
  54. M.H. Alqahtani, M. Tkachenko, Normality and N-Factorizable Topological Groups, Topol. Appl. 373 (2025), 109497. https://doi.org/10.1016/j.topol.2025.109497.
  55. S.A. El-Sheikh, A.M. Abd El-latif, Decompositions of Some Types of Supra Soft Sets and Soft Continuity, Int. J. Math. Trends Technol. 9 (2014), 37-56. https://doi.org/10.14445/22315373/IJMTT-V9P504.
  56. A.M. Abd El-latif, Specific Types of Lindelöfness and Compactness Based on Novel Supra Soft Operator, AIMS Math. 10 (2025), 8144-8164. https://doi.org/10.3934/math.2025374.
  57. A.M. Abd El-latif, M.H. Alqahtani, New Soft Operators Related to Supra Soft δ-Open Sets and Applications, AIMS Math. 9 (2024), 3076-3096. https://doi.org/10.3934/math.2024150.
  58. M.H. Alqahtani, A.M. Abd El-latif, Separation Axioms via Novel Operators in the Frame of Topological Spaces and Applications, AIMS Math. 9 (2024), 14213-14227. https://doi.org/10.3934/math.2024690.
  59. Z.A. Ameen, M.H. Alqahtani, Baire Category Soft Sets and Their Symmetric Local Properties, Symmetry 15 (2023), 1810. https://doi.org/10.3390/sym15101810.
  60. M.H. Alqahtani, O.F. Alghamdi, Z.A. Ameen, Nodecness of Soft Generalized Topological Spaces, Int. J. Anal. Appl. 22 (2024), 149. https://doi.org/10.28924/2291-8639-22-2024-149.
  61. Z.A. Ameen, M.H. Alqahtani, Some Classes of Soft Functions Defined by Soft Open Sets Modulo Soft Sets of the First Category, Mathematics 11 (2023), 4368. https://doi.org/10.3390/math11204368.
  62. Z.A. Ameen, M.H. Alqahtani, Congruence Representations via Soft Ideals in Soft Topological Spaces, Axioms 12 (2023), 1015. https://doi.org/10.3390/axioms12111015.
  63. M.H. Alqahtani, H.Y. Saleh, A Novel Class of Separation Axioms, Compactness, and Continuity via C-Open Sets, Mathematics 11 (2023), 4729. https://doi.org/10.3390/math11234729.
  64. R. Hatamleh, N. Odat, H.A. Abujabal, F. Khan, A.M. Khattak, et al., Fermatean Double-Valued Neutrosophic Soft Topological Spaces, Eur. J. Pure Appl. Math. 18 (2025), 6483. https://doi.org/10.29020/nybg.ejpam.v18i3.6483.
  65. A.M.A. El-latif, On Soft Supra Compactness in Supra Soft Topological Spaces, Tbilisi Math. J. 11 (2018), 169–178. https://doi.org/10.32513/tbilisi/1524276038.