Soft Intersection-difference Product

Authors

  • Aslıhan SEZGİN * Amasya University
  • Zeynep Ay Amasya University

https://doi.org/10.48313/uda.vi.52

Abstract

Soft set theory constitutes a comprehensive mathematical apparatus for modeling and managing
uncertainty. Central to this theory are soft set operations and product constructions, which facilitate novel
methodologies for addressing problems characterized by parametric data. In the present study, we propose a new
product structure for soft sets whose parameter sets possess a group structure, termed the soft intersectiondifference
product. A rigorous investigation of its fundamental algebraic properties is conducted, encompassing
various soft subsets and notions of equality. The findings are anticipated to stimulate further scholarly inquiry,
potentially laying the groundwork for a nascent soft group theory derived from this construction. Given that the
development of soft algebraic structures fundamentally relies on well-defined soft set operations and products,
the study offers a substantial contribution to the theoretical advancement of soft set theory.

Keywords:

Soft set, soft subset, soft equality, soft intersection-difference product

References

  1. [1] D. A. Molodtsov, Soft set theory–first results, Computers and Mathematics with Applications 37 (4-5) (1999) 19–31.
  2. [2] L. A. Zadeh, Fuzzy sets, Information Control (8) (1965) 338–353.
  3. [3] P. K. Maji, A. R. Roy, R. Biswas, An application of soft sets in a decision making problem, Computers and Mathematics with Applications 44 (8-9) (2002) 1077–1083.
  4. [4] De-Gang Chen, E. C. C. Tsang, D. S. Yeung, Some notes on the parameterization reduction of soft sets, in: Proceedings of the 2003 International Conference on Machine Learning and Cybernetics, Xi’an, 2003, pp. 1442–1445.
  5. [5] De-Gang Chen, E. C. C. Tsang, X. Wang, The parametrization reduction of soft sets and its applications Computers and Mathematics with Applications 49 (5–6) (2005) 757–763.
  6. [6] Z. Xiao, L. Chen, B. Zhong, S. Ye, Recognition for soft information based on the theory of soft sets, In: J. Chen (ed.), IEEE proceedings of International Conference on Services Systems and Services Management, 2005, pp 1104–1106.
  7. [7] M. M. Mushrif, S. Sengupta, A. K. Ray, Texture classification using a novel, soft-set theory based classification algorithm, In: P. J. Narayanan, S. K. Nayar, H. T. Shum (Eds.), Computer Vision – ACCV 2006. Lecture Notes in Computer Science, vol 3851. Springer, Berlin, Heidelberg.
  8. [8] M. T. Herawan, M. M. Deris, A direct proof of every rough set is a soft set, Third Asia International Conference on Modelling & Simulation, Bundang, Indonesia, 2009, pp. 119–124.
  9. [9] M. T. Herawan, M. M. Deris, Soft decision making for patients suspected influenza, In: D. Taniar, O. Gervasi, B. Murgante, E. Pardede, B. O. Apduhan (Eds.), Computational Science and Its Applications -ICCSA 2010. Lecture Notes in Computer Science, vol 6018. Springer, Berlin, Heidelberg.
  10. [10] T. Herawan, Soft set-based decision making for patients suspected influenza-like illness, International Journal of Modern Physics: Conference Series 1 (1) (2005) 1–5.
  11. [11] N. Çağman, S. Enginoğlu, Soft set theory and uni-int decision making, European Journal of Operational Research 207 (2) (2010) 848–855.
  12. [12] N. Çağman, S. Enginoğlu, Soft matrix theory and its decision making, Computers and Mathematics with Applications 59 (10) (2010) 3308–3314.
  13. [13] X. Gong, Z. Xiao, X. Zhang, The bijective soft set with its operations, Computers and Mathematics with Applications 60 (8) (2010) 2270–2278.
  14. [14] Z. Xiao, K. Gong, S. Xia, Y. Zou, Exclusive disjunctive soft sets, Computers and Mathematics with Applications 59 (6) (2010) 2128–2137.
  15. [15] F. Feng, Y. Li, N. Çağman, Generalized uni-int decision making schemes based on choice value soft sets, European Journal of Operational Research 220 (1) (2012) 162–170.
  16. [16] Q. Feng, Y. Zhou, Soft discernibility matrix and its applications in decision making. Applied Soft Computing (24) (2014) 749–756.
  17. [17] A. Kharal, Soft approximations and uni-int decision making, The Scientific World Journal (4) (2014) 327408.
  18. [18] M. K. Dauda, M. Mamat, M. Y. Waziri, An application of soft set in decision making. Jurnal Teknologi 77 (13) (2015) 119–122.
  19. [19] V. Inthumathi, V. Chitra, S. Jayasree, The role of operators on soft set in decision making problems, International Journal of Computational and Applied Mathematics 12 (3) (2017) 899–910.
  20. [20] A. O. Atagün, H. Kamacı, O. Oktay, Reduced soft matrices and generalized products with applications in decision making, Neural Computing and Applications (29) (2018) 445–456.
  21. [21] H. Kamacı, K. Saltık, H. F. Akız, A. O. Atagün, Cardinality inverse soft matrix theory and its applications in multicriteria group decision making, Journal of Intelligent & Fuzzy Systems 34 (3) (2018) 2031–2049.
  22. [22] J. L. Yang, Y.Y. Yao, Semantics of soft sets and three-way decision with soft sets, Knowledge-Based Systems 194 (2020) 105538.
  23. [23] S. Petchimuthu, H. Garg, H. Kamacı, A. O. Atagün, The mean operators and generalized products of fuzzy soft matrices and their applications in MCGDM, Computational and Applied Mathematics, 39 (2) (2020) 1–32.
  24. [24] İ. Zorlutuna, Soft set-valued mappings and their application in decision making problems, Filomat 35 (5) (2021) 1725–1733.
  25. [25] P. K. Maji, R. Biswas, A. R. Roy, Soft set theory, Computers and Mathematics with Applications, 45 (1) (2003) 555–562.
  26. [26] D. Pei, D. Miao, From soft sets to information systems, in: X. Hu, Q. Liu, A. Skowron, T.Y. Lin, R.R. Yager, B. Zhang (Eds.), Proceedings of Granular Computing (2) IEEE (2005) 617–621.
  27. [27] M. I. Ali, F. Feng, X. Liu, W. K. Min, M. Shabir, On some new operations in soft set theory, Computers and Mathematics with Applications, 57 (9) (2009) 1547–1553.
  28. [28] C. F. Yang, A note on: “Soft set theory” [Computers & Mathematics with Applications 45 (2003), 4-5, 555–562], Computers and Mathematics with Applications. 56 (7) (2008) 1899–1900.
  29. [29] F. Feng, Y. M. Li, B. Davvaz, M. I. Ali, Soft sets combined with fuzzy sets and rough sets: a tentative approach, Soft Computing 14 (2010) 899–911.
  30. [30] Y. Jiang, Y. Tang, Q. Chen, J. Wang, S. Tang, Extending soft sets with description logics, Computers and Mathematics with Applications 59 (6) (2010) 2087–2096.
  31. [31] M. I. Ali, M. Shabir, M. Naz, Algebraic structures of soft sets associated with new operations, Computers and Mathematics with Applications. 61 (9) (2011) 2647–2654.
  32. [32] C. F. Yang, A note on soft set theory, Computers and mathematics with applications, 56 (7) (2008) 1899–1900.
  33. [33] I. J. Neog, D. K. Sut, A new approach to the theory of soft set, International Journal of Computer Applications 32 (2) (2011) 1–6.
  34. [34] L. Fu, Notes on soft set operations, ARPN Journal of Systems and Softwares 1 (6) (2011) 205–208.
  35. [35] X. Ge, S. Yang, Investigations on some operations of soft sets, World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences, 5 (3) (2011) 370–373.
  36. [36] D. Singh, I. A. Onyeozili, Some conceptual misunderstanding of the fundamentals of soft set theory, ARPN Journal of Systems and Softwares, 2 (9) (2012) 251–254.
  37. [37] D. Singh, I. A. Onyeozili, Some results on distributive and absorption properties on soft operations, IOSR Journal of Mathematics, 4 (2) (2012) 18–30.
  38. [38] D. Singh, I. A. Onyeozili, On some new properties on soft set operations, International Journal of Computer Applications. 59 (4) (2012) 39–44.
  39. [39] D. Singh, I. A. Onyeozili, Notes on soft matrices operations, ARPN Journal of Science and Technology. 2 (9) (2012) 861–869.
  40. [40] P. Zhu, Q. Wen, Operations on soft sets revisited, Journal of Applied Mathematics (2013) 1–7.
  41. [41] J. Sen, On algebraic structure of soft sets, Annals of Fuzzy Mathematics and Informatics, 7 (6) (2014) 1013–1020.
  42. [42] Ö. F. Eren, On operations of soft sets, Master’s Thesis Ondokuz Mayıs University (2019) Samsun.
  43. [43] N. S. Stojanovic, A new operation on soft sets: extended symmetric difference of soft sets, Military Technical Courier, 69 (4) (2021) 779–791.
  44. [44] A. Sezgin, N. Çağman, A. O. Atagün, F. N. Aybek, Complemental binary operations of sets and their application to group theory, Matrix Science Mathematic, 7(2) (2023) 114-121.
  45. [45] A. Sezgin, K. Dagtoros, Complementary soft binary piecewise symmetric difference operation: A novel soft set operation, Scientific Journal of Mehmet Akif Ersoy University, 6 (2) (2023) 31-45.
  46. [46] A. Sezgin, H. Çalışıcı, A comprehensive study on soft binary piecewise difference operation, Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B-Teorik Bilimler 12 (1) (2024) 32-54.
  47. [47] A. Sezgin, E. Yavuz, Soft binary piecewise plus operation: A new type of operation for soft sets, Uncertainty Discourse and Applications, 1 (1) (2024) 79-100.
  48. [48] A. Sezgin, M. Sarıalioğlu, A new product for soft sets with its decision-making: Soft star-product, Natural and Applied Sciences Journal 7 (1) (2024) 15-44.
  49. [49] A. Sezgin, E. Şenyiğit, New soft set operation: Complementary soft binary piecewise star operation, Big Data and Computing Visions, 5 (1) (2025) 52–73.
  50. [50] K. Y. Qin, Z. Y. Hong, On soft equality, Journal of Computational and Applied Mathematics, 234 (5) (2010) 1347–1355.
  51. [51] Y. B. Jun, X. Yang, A note on the paper “Combination of interval-valued fuzzy set and soft set” [Comput. Math. Appl. 58 (2009) 521–527], Computers and Mathematics with Applications, 61 (5) (2011) 1468–1470.
  52. [52] X. Y. Liu, F. F. Feng, Y. B. Jun, A note on generalized soft equal relations, Computers and Mathematics with Applications, 64 (4) (2012) 572–578.
  53. [53] F. Feng, L. Yongming, Soft subsets and soft product operations, Information Sciences (232) (2013) 44–57.
  54. [54] M. Abbas, B. Ali, S. Romaguer, On generalized soft equality and soft lattice structure, Filomat 28 (6) (2014) 1191–1203.
  55. [55] M. Abbas, M. I. Ali, S. Romaguera, Generalized operations in soft set theory via relaxed conditions on parameters, Filomat 31 (19) (2017) 5955–5964.
  56. [56] T. Alshami, Investigation and corrigendum to some results related to g-soft equality and g f-soft equality relations, Filomat 33 (11) (2019) 3375–3383.
  57. [57] T. Alshami, M. El-Shafei, T-soft equality relation, Turkish Journal of Mathematics, 44 (4) (2020) 1427–1441.
  58. [58] B. Ali, N. Saleem, N. Sundus, S. Khaleeq, M. Saeed, R. A. George, Contribution to the theory of soft sets via generalized relaxed operations, Matematics 10 (15) (2022) 26–36.
  59. [59] A. Sezgin, A. O. Atagün, N. Çağman, A complete study on and-product of soft sets, Sigma Journal of Engineering and Natural Sciences 43 (1) (2025) 1-14.
  60. [60] A. S. Sezer, A new view to ring theory via soft union rings, ideals and bi-ideals, Knowledge-Based Systems 36 (2012) 300-314.
  61. [61] A. Sezgin, A new approach to semigroup theory I: Soft union semigroups, ideals and bi-ideals, Algebra Letters 2016 (2016) 3 1–46.
  62. [62] E. Muştuoğlu, A. Sezgin, Z. K. Türk, Some characterizations on soft uni-groups and normal soft uni-groups, International Journal of Computer Applications, 155 (10) (2016) 1–8.
  63. [63] K. Kaygisiz, On soft int-groups, Annals of Fuzzy Mathematics and Informatics, 4 (2) (2012) 363-375.
  64. [64] A. S. Sezer, N. Çağman, A. O. Atagün, M. I. Ali, E. Türkmen, Soft intersection semigroups, ideals and bi-ideals; A new application on semigroup theory I, Filomat 29 (5) (2015) 917-946.
  65. [65] A. S. Sezer, N. Çağman, A. O. Atagün, M. I. Ali, E. Türkmen, A completely new view to soft intersection rings via soft uni-int product, Applied Soft Computing 54 (2017) 366-392.
  66. [66] İ. Durak, Soft union-intersection, soft intersection-symmetric difference product and the constructions of soft symmetrical difference groups, Unpublished Master’s Thesis Amasya University (2025) Amasya.
  67. [67] A. Khan, M. Izhar, A. Sezgin, Characterizations of Abel Grassmann’s groupoids by the properties of double-framed soft ideals, International Journal of Analysis and Applications, 15 (1) (2017) 62–74.
  68. [68] A. O. Atagün, A. Sezgin, Soft sets, soft semimodules and soft substructures of semimodules. Mathematical Sciences Letters, 4 (3) (2015) 235.
  69. [69] A. S. Sezer, A. O. Atagün, N. Çağman, N-group SI-action and its applications to N-Group Theory. Fasciculi Mathematici, 52 (2014) 139-153.
  70. [70] A. O. Atagün, A. Sezgin, Int-soft substructures of groups and semirings with applications. Applied Mathematics & Information Sciences, 11 (1) (2017) 105-113.
  71. [71] M. Gulistan, F. Feng, M. Khan, A. Sezgin, Characterizations of right weakly regular semigroups in terms of generalized cubic soft sets, Mathematics (6) (2018) 293.
  72. [72] A. S. Sezer, A. O. Atagün, N. Çağman. A new view to N-group theory: Soft N-groups, Fasciculi Mathematici, 51 (2013) 123-140.
  73. [73] C. Jana, M. Pal, F. Karaaslan, A. Sezgi̇n, (α, β)-Soft intersectional rings and ideals with their applications. New Mathematics and Natural Computation, 15 (02) (2019) 333–350.
  74. [74] A. O. Atagün, H. Kamacı, İ. Taştekin, A. Sezgin, P-properties in near-rings, Journal of Mathematical and Fundamental Sciences, 51 (2) (2019) 152–167.
  75. [75] A. Sezgin, M. Orbay, Analysis of semigroups with soft intersection ideals, Acta Universitatis Sapientiae, Mathematica, 14 (1) (2022) 166-210.
  76. [76] A. O. Atagün, A. Sezgin. A new view to near-ring theory: Soft near-rings. South East Asian Journal of Mathematics & Mathematical Sciences, 14 (3) (2018) 1-14.
  77. [77] T. Manikantan, P. Ramasany, A. Sezgin, Soft quasi-ideals of soft near-rings, Sigma Journal of Engineering and Natural Science 41 (3) (2023) 565–574.
  78. [78] K. Naeem, Soft set theory & soft sigma algebras, LAP LAMBERT Academic Publishing, 2017.
  79. [79] M. Riaz, K. Naeem, Novel concepts of soft sets with applications, Annals of Fuzzy Mathematics & Informatics 13 (2) (2017) 239–251.
  80. [80] M. Riaz, K. Naeem, Measurable soft mappings, Punjab University Journal of Mathematics 48 (2) (2016) 19–34.
  81. [81] S. Memiş, Another view on picture fuzzy soft sets and their product operations with soft decision-making, Journal of New Theory (38) (2022) 1–13.
  82. [82] K. Naeem, S. Memis, Picture fuzzy soft σ-algebra and picture fuzzy soft measure and their applications to multi-criteria decision-making. Granular Computing 8 (2) (2023) 397–410.
  83. [83] A. Sezgin, İ. Durak, Z. Ay, Some new classifications of soft subsets and soft equalities with soft symmetric difference-difference product, Journal of Amasya University the Institute of Sciences and Technology 6 (1) (20225) in press.

Published

2025-05-31

Issue

Articles in Press

Section

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How to Cite

SEZGİN, A., & Ay, Z. (2025). Soft Intersection-difference Product. Uncertainty Discourse and Applications. https://doi.org/10.48313/uda.vi.52

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