On the Structure and Properties of Hyperfuzzy and SuperhyperfuzzyIntervals

Authors

https://doi.org/10.22105/scfa.vi.70

Abstract

A fuzzy set assigns to each element in a universe a membership degree in the interval [0, 1], effectively modeling imprecision and vagueness. A hyperfuzzy set extends this concept by mapping each element to a
nonempty subset of [0, 1], thereby representing multiple possible membership degrees and capturing both uncertainty and variability. An m, n-superhyperfuzzy set further generalizes this structure by assigning to
each nonempty element of the mth and the nth power-set hierarchy a nonempty subset of [0, 1], enabling the modeling of hierarchical and nested uncertainty. A fuzzy interval is a special kind of fuzzy set on R
that is both normal and convex, with each α-cut forming a nonempty closed interval, thus providing a precise framework for modeling uncertain numerical values. In this paper, we introduce and define the notions of Hyperfuzzy Interval and SuperHyperfuzzy Interval, and provide a brief investigation of their mathematical properties.

Keywords:

Set Theory, HyperFuzzy Interval, Fuzzy Interval, HyperFuzzy Set, Superhyperfuzzy Set

References

  1. [1] Lotfi A Zadeh. Fuzzy sets. Information and control, 8(3):338–353, 1965.
  2. [2] Hans-Jürgen Zimmermann. Fuzzy set theory—and its applications. Springer Science & Business Media, 2011.
  3. [3] H-J Zimmermann. Fuzzy set theory and mathematical programming. Fuzzy sets theory and applications, pages 99–114, 1986.
  4. [4] Kostaq Hila, Serkan Onar, Bayram Ali Ersoy, and Bijan Davvaz. On generalized intuitionistic fuzzy subhyperalgebras of boolean
  5. [5] hyperalgebras. Journal of Inequalities and Applications, 2013:1–15, 2013.
  6. [6] Muhammad Akram, Bijan Davvaz, and Feng Feng. Intuitionistic fuzzy soft k-algebras. Mathematics in Computer Science,
  7. [7] :353–365, 2013.
  8. [8] Madeleine Al Tahan, Saba Al-Kaseasbeh, and Bijan Davvaz. Neutrosophic quadruple hv-modules and their fundamental module.
  9. [9] Neutrosophic Sets and Systems, 72:304–325, 2024.
  10. [10] Mustafa Hasan Hadi and LAA Al-Swidi. The neutrosophic axial set theory. Neutrosophic Sets and Systems, vol. 51/2022: An
  11. [11] International Journal in Information Science and Engineering, page 295, 2022.
  12. [12] Wen-Ran Zhang. Bipolar fuzzy sets. 1997.
  13. [13] Muhammad Akram. Bipolar fuzzy graphs. Information sciences, 181(24):5548–5564, 2011.
  14. [14] Vicenç Torra and Yasuo Narukawa. On hesitant fuzzy sets and decision. In 2009 IEEE international conference on fuzzy
  15. [15] systems, pages 1378–1382. IEEE, 2009.
  16. [16] Vicenç Torra. Hesitant fuzzy sets. International journal of intelligent systems, 25(6):529–539, 2010.
  17. [17] Libor Svadlenka, Vladimir Simic, Momcilo Dobrodolac, Dragana Lazarevic, and Gordana Todorovic. Picture fuzzy decisionmaking approach for sustainable last-mile delivery. IEEE Access, 8:209393–209414, 2020.
  18. [18] Bui Cong Cuong and Vladik Kreinovich. Picture fuzzy sets-a new concept for computational intelligence problems. In 2013
  19. [19] third world congress on information and communication technologies (WICT 2013), pages 1–6. IEEE, 2013.
  20. [20] An Lu and Wilfred Ng. Vague sets or intuitionistic fuzzy sets for handling vague data: which one is better? In International
  21. [21] conference on conceptual modeling, pages 401–416. Springer, 2005.
  22. [22] W-L Gau and Daniel J Buehrer. Vague sets. IEEE transactions on systems, man, and cybernetics, 23(2):610–614, 1993.
  23. [23] Shawkat Alkhazaleh. Plithogenic soft set. Neutrosophic Sets and Systems, 33:16, 2020.
  24. [24] Takaaki Fujita and Florentin Smarandache. A review of the hierarchy of plithogenic, neutrosophic, and fuzzy graphs: Survey
  25. [25] and applications. In Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy,
  26. [26] Neutrosophic, Soft, Rough, and Beyond (Second Volume). Biblio Publishing, 2024.
  27. [27] Takaaki Fujita. Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy,
  28. [28] Neutrosophic, Soft, Rough, and Beyond. Biblio Publishing, 2025.
  29. [29] Florentin Smarandache. Hyperuncertain, superuncertain, and superhyperuncertain sets/logics/probabilities/statistics. Critical
  30. [30] Review, XIV, 2017.
  31. [31] Young Bae Jun, Kul Hur, and Kyoung Ja Lee. Hyperfuzzy subalgebras of bck/bci-algebras. Annals of Fuzzy Mathematics and
  32. [32] Informatics, 2017.
  33. [33] Takaaki Fujita and Florentin Smarandache. Examples of fuzzy sets, hyperfuzzy sets, and superhyperfuzzy sets in climate change
  34. [34] and the proposal of several new concepts. Climate Change Reports, 2:1–18, 2025.
  35. [35] Jayanta Ghosh and Tapas Kumar Samanta. Hyperfuzzy sets and hyperfuzzy group. Int. J. Adv. Sci. Technol, 41:27–37, 2012.
  36. [36] Z Nazari and B Mosapour. The entropy of hyperfuzzy sets. Journal of Dynamical Systems and Geometric Theories, 16(2):173–185,
  37. [37] Takaaki Fujita. Some types of hyperneutrosophic set (3): Dynamic, quadripartitioned, pentapartitioned, heptapartitioned,
  38. [38] m-polar. Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic,
  39. [39] Soft, Rough, and Beyond, 2025.
  40. [40] Takaaki Fujita. Some types of hyperneutrosophic set (2): Complex, single-valued triangular, fermatean, and linguistic sets.
  41. [41] Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft,
  42. [42] Rough, and Beyond, 2025.
  43. [43] Takaaki Fujita. Exploring concepts of hyperfuzzy, hyperneutrosophic, and hyperplithogenic sets (i). Infinite Study, 2025.
  44. [44] Takaaki Fujita and Florentin Smarandache. A concise introduction to hyperfuzzy, hyperneutrosophic, hyperplithogenic, hypersoft,
  45. [45] and hyperrough sets with practical examples. Neutrosophic Sets and Systems, 80:609–631, 2025.
  46. [46] Amory Bisserier, Reda Boukezzoula, and Sylvie Galichet. A revisited approach to linear fuzzy regression using trapezoidal
  47. [47] fuzzy intervals. Information Sciences, 180(19):3653–3673, 2010.
  48. [48] Marina T Mizukoshi, Tiago M Costa, Yurilev Chalco-Cano, and Weldon A Lodwick. A formalization of constraint interval: A
  49. [49] precussor to fuzzy interval analysis. Fuzzy Sets and Systems, 482:108910, 2024.
  50. [50] Jérôme Fortin, Didier Dubois, and Hélene Fargier. Gradual numbers and their application to fuzzy interval analysis. IEEE
  51. [51] Transactions on fuzzy systems, 16(2):388–402, 2008.
  52. [52] Seok-Zun Song, Seon Jeong Kim, and Young Bae Jun. Hyperfuzzy ideals in bck/bci-algebras. Mathematics, 5(4):81, 2017.
  53. [53] Florentin Smarandache. Foundation of superhyperstructure & neutrosophic superhyperstructure. Neutrosophic Sets and Systems,
  54. [54] (1):21, 2024.
  55. [55] Takaaki Fujita and Florentin Smarandache. Neutrosophic TwoFold SuperhyperAlgebra and Anti SuperhyperAlgebra. Infinite
  56. [56] Study, 2025.
  57. [57] Ajoy Kanti Das, Rajat Das, Suman Das, Bijoy Krishna Debnath, Carlos Granados, Bimal Shil, and Rakhal Das. A comprehensive
  58. [58] study of neutrosophic superhyper bci-semigroups and their algebraic significance. Transactions on Fuzzy Sets and Systems,
  59. [59] (2):80, 2025.
  60. [60] Takaaki Fujita. Chemical hyperstructures, superhyperstructures, and shv-structures: Toward a generalized framework for
  61. [61] hierarchical chemical modeling. 2025.
  62. [62] Lotfi A Zadeh. Fuzzy logic, neural networks, and soft computing. In Fuzzy sets, fuzzy logic, and fuzzy systems: selected papers
  63. [63] by Lotfi A Zadeh, pages 775–782. World Scientific, 1996.
  64. [64] Takaaki Fujita. Short survey on the hierarchical uncertainty of fuzzy, neutrosophic, and plithogenic sets. Advancing Uncertain
  65. [65] Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond,
  66. [66] page 285, 2025.
  67. [67] Didier Dubois, Etienne Kerre, Radko Mesiar, and Henri Prade. Fuzzy interval analysis. In Fundamentals of fuzzy sets, pages
  68. [68] –581. Springer, 2000.
  69. [69] Didier Dubois, Hélene Fargier, and Jérôme Fortin. The empirical variance of a set of fuzzy intervals. In The 14th IEEE
  70. [70] International Conference on Fuzzy Systems, 2005. FUZZ’05., pages 885–890. IEEE, 2005.
  71. [71] İsmail Özcan and Sırma Zeynep Alparslan Gök. On the fuzzy interval equal surplus sharing solutions. Kybernetes, 51(9):2753–
  72. [72] , 2022.
  73. [73] Weldon A Lodwick and Elizabeth A Untiedt. A comparison of interval analysis using constraint interval arithmetic and fuzzy
  74. [74] interval analysis using gradual numbers. In NAFIPS 2008-2008 Annual Meeting of the North American Fuzzy Information
  75. [75] Processing Society, pages 1–6. IEEE, 2008.
  76. [76] Nivetha Martin. Plithogenic swara-topsis decision making on food processing methods with different normalization techniques.
  77. [77] Advances in Decision Making, 69, 2022.
  78. [78] P Sathya, Nivetha Martin, and Florentine Smarandache. Plithogenic forest hypersoft sets in plithogenic contradiction based
  79. [79] multi-criteria decision making. Neutrosophic Sets and Systems, 73:668–693, 2024.
  80. [80] Fazeelat Sultana, Muhammad Gulistan, Mumtaz Ali, Naveed Yaqoob, Muhammad Khan, Tabasam Rashid, and Tauseef Ahmed.
  81. [81] A study of plithogenic graphs: applications in spreading coronavirus disease (covid-19) globally. Journal of ambient intelligence
  82. [82] and humanized computing, 14(10):13139–13159, 2023.
  83. [83] Mayada Abualhomos, Abdallah Shihadeh, Ahmad A Abubaker, Khaled Al-Husban, Takaaki Fujita, Ahmed Atallah Alsaraireh,
  84. [84] Mutaz Shatnawi, and Abdallah Al-Husban. Unified framework for type-n extensions of fuzzy, neutrosophic, and plithogenic
  85. [85] offsets: Definitions and interconnections. Journal of Fuzzy Extension and Applications, 2025.
  86. [86] Zdzisław Pawlak. Rough sets. International journal of computer & information sciences, 11:341–356, 1982.
  87. [87] Zdzisław Pawlak. Rough sets: Theoretical aspects of reasoning about data, volume 9. Springer Science & Business Media, 2012.
  88. [88] Jonathan L Gross, Jay Yellen, and Mark Anderson. Graph theory and its applications. Chapman and Hall/CRC, 2018.
  89. [89] Alain Bretto. Hypergraph theory. An introduction. Mathematical Engineering. Cham: Springer, 1, 2013.
  90. [90] Claude Berge. Hypergraphs: combinatorics of finite sets, volume 45. Elsevier, 1984.
  91. [91] Eduardo Martín Campoverde Valencia, Jessica Paola Chuisaca Vásquez, and Francisco Ángel Becerra Lois. Multineutrosophic
  92. [92] analysis of the relationship between survival and business growth in the manufacturing sector of azuay province, 2020–2023,
  93. [93] using plithogenic n-superhypergraphs. Neutrosophic Sets and Systems, 84(1):28, 2025.
  94. [94] Min Huang, Fenghua Li, et al. Optimizing ai-driven digital resources in vocational english learning using plithogenic nsuperhypergraph structures for adaptive content recommendation. Neutrosophic Sets and Systems, 88:283–295, 2025.
  95. [95] Takaaki Fujita and Arkan A Ghaib. Toward a unified theory of brain hypergraphs and symptom hypernetworks in medicine
  96. [96] and neuroscience. Advances in Research, 26(3):522–565, 2025.
  97. [97] Takaaki Fujita. Directed acyclic superhypergraphs (dash): A general framework for hierarchical dependency modeling.
  98. [98] Neutrosophic Knowledge, 6:72–86, 2025.
  99. [99] Takaaki Fujita and Florentin Smarandache. A concise study of some superhypergraph classes. Neutrosophic Sets and Systems,
  100. [100] :548–593, 2024.

Published

2025-08-06

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Articles in Press

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How to Cite

On the Structure and Properties of Hyperfuzzy and SuperhyperfuzzyIntervals. (2025). Soft Computing Fusion With Applications . https://doi.org/10.22105/scfa.vi.70

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