Soft symmetric difference complement-lambda product of groups
Abstract
The present study introduces the soft symmetric difference complement–lambda product of soft sets whose parameter sets are group. The key algebraic properties are investigted, its algebraic characteristics is analyzed in relation to identity, absorbing elements, null and absolute soft sets. Its broad usefulness in abstract algebraic modeling are further demonstrated by its smooth integration into soft inclusion hierarchies with generalized soft equalities.
Keywords:
Soft sets, Soft subsets, Soft equalities, Soft symmetric difference complement-lambdaReferences
- [1] Zadeh, L. A. (1965). Fuzzy sets. Information and control, 8(3), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
- [2] 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
- [3] Maji, P. K., Biswas, R., & Roy, A. R. (2003). Soft set theory. Computers & mathematics with applications, 45(4–5), 555–562. 10.1016/S0898-1221(03)00016-6
- [4] Pei, D., & Miao, D. (2005). From soft sets to information systems. 2005 Ieee international conference on granular computing (Vol. 2, pp. 617–621). IEEE. https://doi.org/10.1109/GRC.2005.1547365
- [5] 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
- [6] Yang, C. F. (2008). A note on “soft set theory.” Computers & mathematics with applications, 56(7), 1899–1900. https://doi.org/10.1016/j.camwa.2008.03.019
- [7] 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
- [8] Singh, D., & Onyeozili, I. A. (2012). Notes on soft matrices operations. ARPN journal of science and technology, 2(9), 861–869. https://doi.org/10.1142/9789814365147_0008
- [9] Zhu, P., & Wen, Q. (2013). Operations on soft sets revisited. Journal of applied mathematics, 2013(1), 105752. http://dx.doi.org/10.1155/2013/105752
- [10] Onyeozili, I. A., & Gwary, T. M. (2014). A study of the fundamentals of soft set theory. International journal of scientific & technology research, 3(4), 132–143. https://www.studocu.com/my/document/universiti-teknologi-mara/principles-of-entrepreneurship/a-study-of-the-fundamentals-of-soft-set-theory/19476726
- [11] Sen, J. (2014). On algebraic structure of soft sets. Annals of fuzzy mathematics and informatics, 7(6), 1013–1020. http://www.afmi.or.kr/papers/2014/Vol-07_No-06/AFMI-7-6(859-1020)/AFMI-7-6(1013-1020)-H-130711R1.pdf
- [12] 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
- [13] Ali, M. I., Shabir, M., & Naz, M. (2011). Algebraic structures of soft sets associated with new operations. Computers & mathematics with applications, 61(9), 2647–2654. https://doi.org/10.1016/j.camwa.2011.03.011
- [14] 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://doi.org/10.5120/3874-5415
- [15] Li, F. (2011). Notes on the soft operations. ARPN journal of systems and software, 1(6), 205–208. https://doi.org/10.1142/9789814365147_0008
- [16] Ge, X., & Yang, S. (2011). Investigations on some operations of soft sets. World academy of science engineering and technology, 5(3), 370–373. https://doi.org/10.5281/zenodo.1085167
- [17] 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
- [18] 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
- [19] 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
- [20] Sezgin, A., & Yavuz, E. (2023). A new soft set operation: Soft binary piecewise symmetric difference operation. Necmettin erbakan üniversitesi fen ve mühendislik bilimleri dergisi, 5(2), 189–208. https://doi.org/10.47112/neufmbd.2023.18
- [21] Stojanović, N. S. (2021). A new operation on soft sets: Extended symmetric difference of soft sets. Vojnotehnički glasnik/military technical courier, 69(4), 779–791. https://doi.org/10.5937/vojtehg69-33655
- [22] 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
- [23] Sezgin, A., Aybek, F. N., & Güngör, N. B. (2023). New soft set operation: Complementary soft binary piecewise union operation. Acta informatica malaysia, 7(1), 38–53. http://doi.org/10.26480/aim.01.2023.38.53
- [24] Sezgin, A., & Sarıalioğlu, M. (2024). Complementary extended gamma operation: A new soft set operation. Natural and applied sciences journal, 7(1), 15–44. https://doi.org/10.38061/idunas.1482044
- [25] Sezgin, A., Kökçü, H., & Atagün, A. O. (2025). A comprehensive study on restricted and extended intersection operations of soft sets. Natural and applied sciences journal, 8(1), 44-111. https://doi.org/10.38061/idunas.1613387
- [26] 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. http://doi.org/10.26480/msmk.02.2023.114.121
- [27] 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/pub/sjmakeu
- [28] 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
- [29] 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://uda.reapress.com/journal/article/download/26/35/104
- [30] 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
- [31] 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
- [32] Sezgi̇n, A., Atagün, A. O., & Cagan, N. (2025). A complete study on and-product of soft sets. Sigma journal of engineering and natural sciences, 43(1), 1–14. 10.14744/sigma.2025.00002
- [33] Sezgin, A., & Yavuz, E. (2023). A new soft set operation: Complementary soft binary piecewise lamda (λ) operation. Sinop üniversitesi fen bilimleri dergisi, 8(2), 101–133. https://doi.org/10.33484/sinopfbd.1320420
- [34] Feng, F., Jun, Y. B., & Zhao, X. (2008). Soft semirings. Computers & mathematics with applications, 56(10), 2621–2628. https://doi.org/10.1016/j.camwa.2008.05.011
- [35] 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
- [36] 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) 521--527]. Computers & mathematics with applications, 61(5), 1468–1470. https://doi.org/10.1016/j.camwa.2010.12.077
- [37] Liu, X., Feng, F., & Jun, Y. B. (2012). A note on generalized soft equal relations. Computers & mathematics with applications, 64(4), 572–578. https://doi.org/10.1016/j.camwa.2011.12.052
- [38] 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
- [39] Abbas, M., Ali, B., & Romaguera, S. (2014). On generalized soft equality and soft lattice structure. Filomat, 28(6), 1191–1203. http://hdl.handle.net/2263/43570
- [40] 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://www.jstor.org/stable/27381589
- [41] Al-shami, T. M. (2019). Investigation and corrigendum to some results related to g-soft equality and gf-soft equality relations. Filomat, 33(11), 3375--3383. https://doi.org/10.2298/FIL1911375A
- [42] 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
- [43] Kaygisiz, K. (2012). On soft int-groups. Annals of fuzzy mathematics and informatics, 4(2), 363–375. https://www.researchgate.net/publication/265000048_On_soft_int-groups
- [44] Mustuoglu, 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
- [45] Sezer, A. S., Çağman, 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
- [46] Sezgin, A., Çağman, N., & Atagün, A. O. (2017). A completely new view to soft intersection rings via soft uni-int product. Applied soft computing, 54, 366–392. https://doi.org/10.1016/j.asoc.2016.10.004
- [47] 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
- [48] Sezgin, A. (2016). A new approach to semigroup theory I: Soft union semigroups, ideals and bi-ideals. Algebra lett., 2016, 1–46. https://scik.org/index.php/abl/article/viewFile/2989/1473
- [49] Sezgin, A., Durak, Ibrahim, & 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
- [50] Atagün, A. O., & Sezgin, A. (2018). Soft subnear-rings, soft ideals and soft N-subgroups of near-rings. Mathematical sciences letters an international journal, 7(1), 37–42. http://doi.org/10.18576/msl/070106
- [51] Khan, A., Izhar, M., & Sezign, 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. https://doi.org/10.28924/2291-8639
- [52] 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), 1–14. https://rsmams.org/download/articles/2_14_3_1150608629_Paper 1 A new view to near ring theory Soft near rings.pdf
- [53] Manikantan, T., Ramasany, P., & Sezgin, A. (2023). Soft quasi-ideals of soft near-rings. Sigma journal of engineering and natural science, 41(3), 565–574. https://doi.org/10.14744/sigma.2023.00062
- [54] Naeem, K. (2017). Soft set theory & soft sigma algebras. LAP LAMBERT Academic Publishing. https://www.abebooks.com/9783330073050/Soft-Set-Theory-Sigma-Algebras-3330073055/plp
- [55] Riaz, M., Hashmi, M., Karaaslan, F., Sezgin, A., Mohammed, M., & Khalaf, M. (2023). Emerging trends in social networking systems and generation gap with neutrosophic crisp soft mapping. Computer modeling in engineering & sciences, 136(2), 1759. http://doi.org/10.32604/cmes.2023.023327
- [56] Memiş, S. (2022). Another view on picture fuzzy soft sets and their product operations with soft decision-making. Journal of new theory, 38(2022), 1–13. https://doi.org/10.53570/jnt.1037280
- [57] Tunçay, M., & Sezgin, A. (2016). Soft union ring and its applications to ring theory. International journal of computer applications, 151(9), 7–13. https://doi.org/10.5120/ijca2016911867
- [58] Çağman, N., Çıtak, F., & Aktaş, H. (2012). Soft int-group and its applications to group theory. Neural computing and applications, 21(1), 151–158. https://doi.org/10.1007/s00521-011-0752-x
- [59] Mahmood, T., Rehman, Z. U., & Sezgin, A. (2018). Lattice ordered soft near rings. Korean journal of mathematics, 26(3), 503–517. https://doi.org/10.11568/kjm.2018.26.3.503
- [60] Sezer, A. S., Çağman, N., & Atagün, A. O. (2015). Uni-soft substructures of groups. Annals of fuzzy mathematics and informatics, 9(2), 235–246. http://www.afmi.or.kr/papers/2015/Vol-09_No-02/PDF/AFMI-9-2(235-246)-H-140701R2.pdf
- [61] Sezer, A. S. (2014). Certain characterizations of LA-semigroups by soft sets. Journal of intelligent & fuzzy systems, 27(2), 1035–1046. https://doi.org/10.3233/IFS-131064
- [62] Atagün, A. O., & Sezer, A. S. (2015). Soft sets, soft semimodules and soft substructures of semimodules. Mathematical sciences letters, 4(3), 235. http://doi.org/10.12785/msl/040303
- [63] Sezgin, A. (2018). A new view on AG-groupoid theory via soft sets for uncertainty modeling. Filomat, 32(8), 2995–3030. https://doi.org/10.2298/FIL1808995S
- [64] Sezgin, A., Atagün, A. O., Çağman, N., & Demir, H. (2022). On near-rings with soft union ideals and applications. New mathematics and natural computation, 18(02), 495–511. https://doi.org/10.1142/S1793005722500247
- [65] Sezer, A. S., & Atagün, A. O. (2016). A new kind of vector space: soft vector space. Southeast asian bulletin of mathematics, 40(5), 753–770. https://avesis.bozok.edu.tr/yayin/24e24549-32e1-46c7-86d7-e64aaec0fb9a/a-new-kind-of-vector-space-soft-vector-space
- [66] Sezgin, A., & İlgin, A. (2024). Soft intersection almost subsemigroups of semigroups. International journal of mathematics and physics, 15(1), 13–20. https://doi.org/10.26577/ijmph.2024v15i1a2
- [67] Sezer, A. S., Atagün, A. O., & Çağman, N. (2014). N-group SI-action and its applications to N-Group Theory. Fasciculi mathematici, 52, 139–153. https://www.researchgate.net/profile/Aslihan-Sezgin-2/publication/263651539_N-group_SI-action_and_its_application_to_N-group_theory/links/54353a080cf2bf1f1f283279/N-group-SI-action-and-its-application-to-N-group-theory.pdf
- [68] Atagün, A. O., & Sezgin, A. (2017). Int-soft substructures of groups and semirings with applications. Applied mathematics & information sciences, 11(1), 105–113. http://doi.org/10.18576/amis/110113
- [69] 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/10.3390/math6120293
- [70] Sezer, A. S., Atagün, A. O., & Çağman, N. (2013). A new view to N-group theory: soft N-groups. Fasciculi mathematici, 51, 123–140. https://www.researchgate.net/profile/Aslihan-Sezgin-2/publication/263651532_A_new_view_to_N-group_theory-Soft_N-groups/links/0046353b68f17da045000000/A-new-view-to-N-group-theory-Soft-N-groups.pdf
- [71] Jana, C., Pal, M., Karaaslan, F., & Sezgi̇n, A. (2019). (α, β)-Soft intersectional rings and ideals with their applications. New mathematics and natural computation, 15(02), 333–350. https://doi.org/10.1142/S1793005719500182
- [72] Atagun, A., Kamaci, H. I., Tastekin, I., & Sezgin Sezer, A. (2019). P-properties in near-rings. Journal of mathematical and fundamental sciences, 51(2), 152–167. http://doi.org/10.5614/j.math.fund.sci.2019.51.2.5
- [73] Sezgin, A., & Orbay, M. (2022). Analysis of semigroups with soft intersection ideals. Acta universitatis sapientiae, mathematica, 14(1), 166–210. 10.2478/ausm-2022-0012
- [74] Durak, İ., & Sezgin, A. (2025). Soft symmetric difference-gamma product of groups. IKJM, 7(1), 1-17. https://www.researchgate.net/profile/Aslihan-Sezgin-2/publication/394106326_Soft_Symmetric_Difference-gamma_Product_of_Groups/links/68a842fa6327cf7b63d8b3d9/Soft-Symmetric-Difference-gamma-Product-of-Groups.pdf