Soft intersection-difference product of groups

Authors

  • Aslıhan Sezgin * Department of Mathematics and Science Education, Faculty of Education, Amasya University, Amasya, Türkiye. https://orcid.org/0000-0002-1519-7294
  • Zeynep Ay Department of Mathematics, Graduate School of Natural and Applied Sciences, Amasya University, Amasya, Türkiye.

https://doi.org/10.48313/uda.v2i1.55

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 intersection-difference 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 sets, Soft subsets, Soft equalities, Soft intersection-difference product

References

  1. [1] Molodtsov, D. (1999). Soft set theory—first results. Computers & mathematics with applications, 37(4-5), 19-31. https://doi.org/10.1016/S0898-1221(99)00056-5
  2. [2] Zadeh, L. A. (1965). Fuzzy sets. Information and control, 8(3), 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
  3. [3] Maji, P. K., Roy, A. R., & Biswas, R. (2002). An application of soft sets in a decision making problem. Computers and mathematics with applications, 44(8–9), 1077–1083. https://doi.org/10.1016/S0898-1221(02)00216-X
  4. [4] Chen, D. G., Tsang, E. C. C., & Yeung, D. S. (2003). Some notes on the parameterization reduction of soft sets. Proceedings of the 2003 international conference on machine learning and cybernetics (IEEE cat. no. 03ex693) (Vol. 3, pp. 1442–1445). IEEE. https://doi.org/10.1109/ICMLC.2003.1259720
  5. [5] Chen, D., Tsang, E. C. C., Yeung, D. S., & Wang, X. (2005). The parameterization reduction of soft sets and its applications. Computers and mathematics with applications, 49(5–6), 757–763. https://doi.org/10.1016/j.camwa.2004.10.036
  6. [6] Xiao, Z., Chen, L., Zhong, B., & Ye, S. (2005). Recognition for soft information based on the theory of soft sets. 2005 international conference on services systems and services management, proceedings of icsssm'05 (Vol. 2, pp. 1104–1106). IEEE. https://doi.org/10.1109/ICSSSM.2005.1500166
  7. [7] Mushrif, M. M., Sengupta, S., & Ray, A. K. (2006). Texture classification using a novel, soft-set theory based classification algorithm. Lecture notes in computer science (including subseries lecture notes in artificial intelligence and lecture notes in bioinformatics) (Vol. 3851 LNCS, pp. 246–254). Springer. https://doi.org/10.1007/11612032_26
  8. [8] Herawan, T., & Deris, M. M. (2009). A direct proof of every rough set is a soft set. Proceedings-2009 3rd Asia international conference on modelling and simulation, ams 2009 (pp. 119–124). IEEE. https://doi.org/10.1109/AMS.2009.148
  9. [9] Herawan, T., & Deris, M. M. (2010). Soft decision making for patients suspected influenza. Lecture notes in computer science (including subseries lecture notes in artificial intelligence and lecture notes in bioinformatics) (Vol. 6018 LNCS, pp. 405–418). Springer. https://doi.org/10.1007/978-3-642-12179-1_34
  10. [10] Herawan, T. (2012). Soft set-based decision making for patients suspected influenza-like illness. International journal of modern physics: Conference series (Vol. 9, pp. 259-270). World Scientific Publishing Company. https://doi.org/10.1142/s2010194512005302
  11. [11] Çağman, N., & Enginoğlu, S. (2010). Soft set theory and uni–int decision making. European journal of operational research, 207(2), 848-855. https://doi.org/10.1016/j.ejor.2010.05.004
  12. [12] Çağman, N., & Enginoğlu, S. (2010). Soft matrix theory and its decision making. Computers and mathematics with applications, 59(10), 3308–3314. https://doi.org/10.1016/j.camwa.2010.03.015
  13. [13] Gong, K., Xiao, Z., & Zhang, X. (2010). The bijective soft set with its operations. Computers and mathematics with applications, 60(8), 2270–2278. https://doi.org/10.1016/j.camwa.2010.08.017
  14. [14] Xiao, Z., Gong, K., Xia, S., & Zou, Y. (2010). Exclusive disjunctive soft sets. Computers and mathematics with applications, 59(6), 2128–2137. https://doi.org/10.1016/j.camwa.2009.12.018
  15. [15] Feng, F., Li, Y., & Çaǧman, N. (2012). Generalized uni-int decision making schemes based on choice value soft sets. European journal of operational research, 220(1), 162–170. https://doi.org/10.1016/j.ejor.2012.01.015
  16. [16] Feng, Q., & Zhou, Y. (2014). Soft discernibility matrix and its applications in decision making. Applied soft computing journal, 24, 749–756. https://doi.org/10.1016/j.asoc.2014.08.042
  17. [17] Kharal, A. (2014). Soft approximations and uni-int decision making. The scientific world journal, 2014(1), 327408. https://doi.org/10.1155/2014/327408
  18. [18] Dauda, M. K., Mamat, M., & Waziri, M. Y. (2015). An application of soft set in decision making. Jurnal teknologi (sciences & engineering), 77(13), 119–122. https://journals.utm.my/jurnalteknologi/issue/view/226
  19. [19] Inthumathi, V., Chitra, V., & Jayasree, S. (2017). The role of operators on soft sets in decision making problems. International journal of computational and applied mathematics, 12(3), 899–910. http://www.ripublication.com
  20. [20] Atagün, A. O., Kamacı, H., & Oktay, O. (2018). Reduced soft matrices and generalized products with applications in decision making. Neural computing and applications, 29(9), 445–456. https://doi.org/10.1007/s00521-016-2542-y
  21. [21] Kamaci, H., Saltik, K., Fulya Akiz, H., & Osman Atagün, A. (2018). Cardinality inverse soft matrix theory and its applications in multicriteria group decision making. Journal of intelligent and fuzzy systems, 34(3), 2031–2049. https://doi.org/10.3233/JIFS-17876
  22. [22] Yang, J., & Yao, Y. (2020). Semantics of soft sets and three-way decision with soft sets. Knowledge-based systems, 194, 105538. https://doi.org/10.1016/j.knosys.2020.105538
  23. [23] Petchimuthu, S., Garg, H., Kamacı, H., & Atagün, A. O. (2020). The mean operators and generalized products of fuzzy soft matrices and their applications in MCGDM. Computational and applied mathematics, 39(2), 1–32. https://doi.org/10.1007/s40314-020-1083-2
  24. [24] Zorlutuna, İ. (2021). Soft set-valued mappings and their application in decision making problems. Filomat, 35(5), 1725–1733. https://doi.org/10.2298/FIL2105725Z
  25. [25] Maji, P. K., Biswas, R., & Roy, A. R. (2003). Soft set theory. Computers and mathematics with applications, 45(4–5), 555–562. https://doi.org/10.1016/S0898-1221(03)00016-6
  26. [26] Pei, D., & Miao, D. (2005). From soft sets to information systems. 2005 IEEE international conference on granular computing (Vol. 2005, pp. 617–621). IEEE. https://doi.org/10.1109/GRC.2005.1547365
  27. [27] Ali, M. I., Feng, F., Liu, X., Min, W. K., & Shabir, M. (2009). On some new operations in soft set theory. Computers & mathematics with applications, 57(9), 1547–1553. https://doi.org/10.1016/j.camwa.2008.11.009
  28. [28] Yang, C. F. (2008). A note on "soft set theory" [comput. math. appl. 45 (4--5)(2003) 555--562]. Computers & mathematics with applications, 56(7), 1899–1900. https://doi.org/10.1016/j.camwa.2008.03.019
  29. [29] Feng, F., Li, C., Davvaz, B., & Ali, M. I. (2010). Soft sets combined with fuzzy sets and rough sets: A tentative approach. Soft computing, 14(9), 899–911. https://doi.org/10.1007/s00500-009-0465-6
  30. [30] Jiang, Y., Tang, Y., Chen, Q., Wang, J., & Tang, S. (2010). Extending soft sets with description logics. Computers & mathematics with applications, 59(6), 2087–2096. https://doi.org/10.1016/j.camwa.2009.12.014
  31. [31] Ali, M. I., Shabir, M., & Naz, M. (2011). Algebraic structures of soft sets associated with new operations. Computers and mathematics with applications, 61(9), 2647–2654. https://doi.org/10.1016/j.camwa.2011.03.011
  32. [32] Neog, T. J., & Sut, D. K. (2011). A new approach to the theory of soft sets. International journal of computer applications, 32(2), 1–6. https://d1wqtxts1xzle7.cloudfront.net/53823653/2011-new_approach_soft_set-libre.pdf?1499755677=&response-contentdisposition=inline%3B+filename%3DA_New_Approach_to_the_Theory_of_Soft_Set.pdf&Expires=1740299428&Signature=abUw8rHu4pPoWo9GGYn8lFITB64pDE61N9d
  33. [33] Li, F. (2011). Notes on the soft operations. ARPN journal of systems and software, 1(6), 205–208. https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=5e028fe2a1df00303f8012ac465fa114611788d5
  34. [34] Ge, X., & Yang, S. (2011). Investigations on some operations of soft sets. World academy of science, engineering and technology, 51, 1112–1115. https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=0f185b915f3223c39847c632a2ea34d1e64a0083
  35. [35] Singh, D., & Onyeozili, I. A. (2012). Some conceptual misunderstandings of the fundamentals of soft set theory. ARPN journal of systems and software, 2(9), 251–254. https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=92f0b823a431a365680bc6c0f1b12dd6bb4f8d30
  36. [36] Singh, D., & Onyeozili, I. A. (2012). Some results on distributive and absorption properties on soft operations. IOSR journal of mathematics, 4(2), 18–30. https://www.iosrjournals.org/iosr-jm/papers/Vol4-issue2/C0421830.pdf
  37. [37] Singh, D., & A. Onyeozili, I. (2012). On some new properties of soft set operations. International journal of computer applications, 59(4), 39–44. https://doi.org/10.5120/9538-3975
  38. [38] Singh, D., & Onyeozili, I. A. (2012). Notes on soft matrices operations. ARPN journal of science and technology, 2(9), 861–869. http://www.ejournalofscience.org
  39. [39] Zhu, P., & Wen, Q. (2013). Operations on soft sets revisited. Journal of applied mathematics, 2013(1), 105752. https://doi.org/10.1155/2013/105752
  40. [40] Sen, J. (2014). On algebraic structure of soft sets. Annals of fuzzy mathematics and informatics, 7(6), 1013–1020. https://B2n.ir/ws4845
  41. [41] Sezgin, A., & Atagün, A. O. (2011). On operations of soft sets. Computers & mathematics with applications, 61(5), 1457–1467. https://doi.org/10.1016/j.camwa.2011.01.018
  42. [42] Stojanović, N. (2021). A new operation on soft sets: extended symmetric difference of soft sets. Vojnotehnicki glasnik, 69(4), 779–791. https://doi.org/10.5937/vojtehg69-33655
  43. [43] Sezgin, A., Çağman, N., Atagün, A. O., & Aybek, F. N. (2023). Complemental binary operations of sets and their application to group theory. Matrix science mathematic, 7(2), 114–121. https://doi.org/10.26480/msmk.02.2023.114.121
  44. [44] Sezgin, A., & Dagtoros, K. (2023). Complementary soft binary piecewise symmetric difference operation: A novel soft set operation. Scientific journal of mehmet akif ersoy university, 6(2), 31–45. https://dergipark.org.tr/en/pub/sjmakeu/issue/82332/1365021
  45. [45] Sezgin, A., & Çalışıcı, H. (2024). A comprehensive study on soft binary piecewise difference operation. Eskişehir teknik üniversitesi bilim ve teknoloji dergisi b-teorik bilimler, 12(1), 32-54. https://doi.org/10.20290/estubtdb.1356881
  46. [46] Sezgin, A., & Yavuz, E. (2024). Soft binary piecewise plus operation: A new type of operation for soft sets. Uncertainty discourse and applications, 1(1), 79–100. https://doi.org/10.48313/uda.v1i1.26
  47. [47] Sezgin, A., & Şenyiğit, E. (2025). A new product for soft sets with its decision-making: Soft star-product. Big data and computing visions, 5(1), 52–73. https://doi.org/10.22105/bdcv.2024.492834.1221
  48. [48] Sezgin, A., & Demirci, A. M. (2023). A new soft set operation: Complementary soft binary piecewise star (*) operation. Ikonion journal of mathematics, 5(2), 24–52. https://doi.org/10.54286/ikjm.1304566
  49. [49] Qin, K., & Hong, Z. (2010). On soft equality. Journal of computational and applied mathematics, 234(5), 1347–1355. https://doi.org/10.1016/j.cam.2010.02.028
  50. [50] Jun, Y. B., & Yang, X. (2011). A note on the paper "combination of interval-valued fuzzy set and soft set" [Comput. Math. Appl. 58 (2009) 521527]. Computers and mathematics with applications, 61(5), 1468–1470. https://doi.org/10.1016/j.camwa.2010.12.077
  51. [51] Liu, X., Feng, F., & Jun, Y. B. (2012). A note on generalized soft equal relations. Computers and mathematics with applications, 64(4), 572–578. https://doi.org/10.1016/j.camwa.2011.12.052
  52. [52] Feng, F., & Li, Y. (2013). Soft subsets and soft product operations. Information sciences, 232, 44–57. https://doi.org/10.1016/j.ins.2013.01.001
  53. [53] Abbas, M., Ali, B., & Romaguera, S. (2014). On generalized soft equality and soft lattice structure. Filomat, 28(6), 1191–1203. https://doi.org/10.2298/FIL1406191A
  54. [54] Abbas, M., Ali, M. I., & Romaguera, S. (2017). Generalized operations in soft set theory via relaxed conditions on parameters. Filomat, 31(19), 5955–5964. https://doi.org/10.2298/FIL1719955A
  55. [55] Al-Shami, T. M. (2019). Investigation and corrigendum to some results related to g-soft equality and g f-soft equality relations. Filomat, 33(11), 3375–3383. https://doi.org/10.2298/FIL1911375A
  56. [56] Alshami, T., & El-Shafei, M. (2020). $ T $-soft equality relation. Turkish journal of mathematics, 44(4), 1427–1441. https://doi.org/10.3906/mat-2005-117
  57. [57] Ali, B., Saleem, N., Sundus, N., Khaleeq, S., Saeed, M., & George, R. (2022). A contribution to the theory of soft sets via generalized relaxed operations. Mathematics, 10(15), 2636. https://doi.org/10.3390/math10152636
  58. [58] Sezgin, A., Atagün, A. O., & Çağman, N. (2025). A complete study on and-product of soft sets. Sigma journal of engineering and natural sciences, 43(1), 1–14. https://doi.org/10.14744/sigma.2025.00002
  59. [59] Sezer, A. S. (2012). A new view to ring theory via soft union rings, ideals and bi-ideals. Knowledge-based systems, 36, 300–314. https://doi.org/10.1016/j.knosys.2012.04.031
  60. [60] Sezgin, A. (2016). A new approach to semigroup theory I: Soft union semigroups, ideals and bi-ideals. Algebra letters, 2016(3), 1–46. https://scik.org/index.php/abl/article/view/2989
  61. [61] Muştuoğlu, E., Sezgin, A., & Türk, Z. K. (2016). Some characterizations on soft uni-groups and normal soft uni-groups. International journal of computer applications, 155(10), 1–8. https://doi.org/10.5120/ijca2016912412
  62. [62] Kaygisiz, K. (2012). On soft int-groups. Annals of fuzzy mathematics and informatics, 4(2), 365–375. http://www.afmi.or.kr@fmihttp//www.kyungmoon.com
  63. [63] Sezer, A. S., Çagman, N., Atagün, A. O., Ali, M. I., & Türkmen, E. (2015). Soft intersection semigroups, ideals and bi-ideals; A new application on semigroup theory I. Filomat, 29(5), 917–946. https://doi.org/10.2298/FIL1505917S
  64. [64] Sezgin, A., Çağman, N., & Atagün, A. O. (2017). A completely new view to soft intersection rings via soft uni-int product. Applied soft computing journal, 54, 366–392. https://doi.org/10.1016/j.asoc.2016.10.004
  65. [65] Sezgin, A., Durak, İ., & Ay, Z. (2025). Some new classifications of soft subsets and soft equalities with soft symmetric difference-difference product of groups. Amesia, 6(1), 16-32. https://doi.org/10.54559/amesia.1730014
  66. [66] Khan, A., Izhar, M., & Sezgin, A. (2017). Characterizations of abel grassmann's groupoids by the properties of their double-framed soft ideals. International journal of analysis and applications, 15(1), 62–74. http://www.etamaths.com
  67. [67] Atagün, A. O., & Sezer, A. S. (2015). Soft sets, soft semimodules and soft substructures of semimodules. Mathematical sciences letters, 4(3), 235–242. https://B2n.ir/n85398
  68. [68] Sezer, A. S., Atagün, A. O., & Çagman, N. (2014). N-group SI-action and its applications to N-Group Theory. Fasciculi mathematici, 54, 139–153. https://B2n.ir/n19479
  69. [69] Atagün, A. O., & Sezgin, A. (2017). Int-soft substructures of groups and semirings with applications. Applied mathematics and information sciences, 11(1), 105–113. https://doi.org/10.54559/jauist.158924210.18576/amis/110113
  70. [70] Gulistan, M., Feng, F., Khan, M., & Sezgin, A. (2018). Characterizations of right weakly regular semigroups in terms of generalized cubic soft sets. Mathematics, 6(12), 293. https://doi.org/10.54559/jauist.158924210.3390/math6120293
  71. [71] Sezer, A. S., Atagün, A. O., & Çağman, N. (2013). A new view to n-group theory: Soft N-groups. Fasciculi mathematici, 51(51), 123–140. https://B2n.ir/m81412
  72. [72] Jana, C., Pal, M., Karaaslan, F., & Sezgin, A. (2019). (α, β)-soft intersectional rings and ideals with their applications. New mathematics and natural computation, 15(2), 333–350. https://doi.org/10.1142/S1793005719500182
  73. [73] Atagün, A. O., Kamacı, H., Taştekin, İ., & Sezgin, A. (2019). P-properties in Near-rings. Journal of mathematical and fundamental sciences, 51(2), 152–167. https://dx.doi.org/10.5614/j.math.fund.sci.2019.51.2.5
  74. [74] Sezgin, A., & Orbay, M. (2022). Analysis of semigroups with soft intersection ideals. Acta universitatis sapientiae, mathematica, 14(1), 166–210. https://doi.org/10.2478/ausm-2022-0012
  75. [75] Atagün, A. O., & Sezgin, A. (2018). A new view to near-ring theory: Soft near-rings. South east Asian journal of mathematics & mathematical sciences, 14(3), 19-32. https://rsmams.org/journals/articleinfo.php?articleid=313&tag=seajmams
  76. [76] Manikantan, T., Ramasamy, P., & Sezgin, A. (2023). Soft Quasi-ideals of soft near-rings. Sigma, 41(3), 565–574. https://doi.org/10.14744/sigma.2023.00062
  77. [77] Naeem, K. (2017). Soft set theory & soft sigma algebras. LAP LAMBERT Academic Publishing. https://www.amazon.com/Soft-Set-Theory-Sigma-Algebras/dp/3330073055
  78. [78] Riaz, M., Naeem, K., & Ahmad, M. O. (2017). Novel concepts of soft sets with applications. Annals of fuzzy mathematics and informatics, 13(2), 239-251. https://doi.org/10.30948/afmi.2017.13.2.239
  79. [79] Sezgin, A., Yavuz, E., & Özlü, Ş. (2024). Insight into soft binary piecewise lambda operation: a new operation for soft sets. Journal of umm al-qura university for applied sciences, 1-15. https://doi.org/10.1007/s43994-024-00187-1
  80. [80] Memiş, S. (2022). Another view on picture fuzzy soft sets and their product operations with soft decision-making. Journal of new theory, (38), 1–13. https://doi.org/10.53570/jnt.1037280
  81. [81] Naeem, K., & Memiş, S. (2023). Picture fuzzy soft σ -algebra and picture fuzzy soft measure and their applications to multi-criteria decision-making. Granular computing, 8(2), 397–410. https://doi.org/10.1007/s41066-022-00333-2
  82. [82] Sezgin, A., Aybek, F., & Güngör, N. B. (2023). A new soft set operation: Complementary soft binary piecewise union operation. Acta informatica malaysia, 7(1), 38–53. https://actainformaticamalaysia.com/archives/AIM/1aim2023/1aim2023-38-53.pdf
  83. [83] Sezgin, A., & Durak, İ. (2025). Soft intersetion-symmetric difference product of groups. Matrix science mathematic, 9(2), 49-55. http://doi.org/10.26480/msmk.02.2025.49.55
  84. [84] Ay, Z. & Sezgin, A. (2025). Soft union-plus product of groups. International journal of mathematics, statistics, and computer science, 3, 365-376. https://doi.org/10.59543/ijmscs.v3i.14961

Downloads

Published

2025-03-20

Issue

Vol. 2 No. 1 (2025): Uncertainty Discourse and Applications

Section

Articles

How to Cite

Sezgin, A. ., & Ay, Z. . (2025). Soft intersection-difference product of groups. Uncertainty Discourse and Applications, 2(1), 45-60. https://doi.org/10.48313/uda.v2i1.55

Similar Articles

You may also start an advanced similarity search for this article.