Soft Intersection-lambda Product of Groups

Abstract

Soft set theory constitutes a mathematically rigorous and algebraically expressive formalism for modeling systems permeated by epistemic indeterminacy, vagueness, and parameter-dependent variability—characteristics that are endemic to foundational problems in decision theory, engineering, economics, and the information sciences. Central to this framework is a broad spectrum of algebraic operations and binary product constructions that collectively impart a deep and intricate internal structure to the universe of soft sets, capable of encoding complex parametric interdependencies with high fidelity. Within this context, we propose and systematically examine a novel product, termed the soft intersection-lambda product, defined over soft sets whose parameter sets are endowed with a group-theoretic structure. The operation is rigorously axiomatized to ensure formal coherence with generalized notions of soft subsethood and soft equality, thereby preserving the algebraic integrity of the underlying system. A detailed algebraic analysis is conducted to investigate fundamental structural properties of the operation—including closure, associativity, commutativity, idempotency, the presence of identity and absorbing elements, and distributy over other soft set operations—as well as its interactions with the null and absolute soft sets. Furthermore, the proposed product is analytically compared with previously established soft binary operations within the taxonomy of soft subset classifications, yielding refined insights into their relative expressive power and mutual algebraic compatibility. Theoretical findings confirm that the product not only adheres to the structural constraints imposed by the group-parameterized domain but also engenders a formally consistent and well-behaved algebraic system on the collection of soft sets. Two principal algebraic implications emerge from this investigation: (i) the integration of the soft intersection-lambda product enhances the internal operational cohesion of soft set theory by embedding it within an axiomatically sound and operation-preserving environment, and (ii) the proposed product serves as a conceptual cornerstone for the development of a generalized soft group theory, wherein soft sets defined over group-structured parameter spaces emulate the axiomatic behavior of classical group-theoretic constructs under newly defined soft operations. Given that the algebraic maturation of soft set theory hinges upon the rigorous formulation of operations that satisfy semantically and structurally significant axioms, the present work represents a substantial advancement in the algebraic unification and generalization of the field. Beyond its theoretical import, the proposed operation offers practical utility in the construction of abstract algebra-based soft computational models, with far-reaching applications in multi-criteria decision-making systems, algebraically guided classification schemes, and uncertainty-aware data analysis frameworks over group-parameterized semantic spaces. Accordingly, the algebraic architecture developed herein significantly expands the foundational landscape of soft set theory and consolidates its relevance across both pure and applied mathematical domains.

Keywords

• Soft sets
• soft subsets
• soft equalities
• soft intersection-lambda product

Grupların Esnek Kesişim-lamda Çarpımı

Abstract

Esnek küme teorisi, epistemik belirsizlik, muğlaklık ve parametreye bağlı değişkenlik gibi özelliklerle kuşatılmış sistemleri modellemek için matematiksel olarak titiz ve cebirsel olarak ifade gücü yüksek bir formel yapı sunar. Bu özellikler, karar teorisi, mühendislik, ekonomi ve bilgi bilimlerindeki temel problemlerde yaygın olarak görülmektedir. Bu çerçevenin merkezinde, esnek kümeler evrenine derin ve karmaşık bir iç yapı kazandıran, karmaşık parametreler arası bağımlılıkları yüksek doğrulukla kodlayabilen çok çeşitli cebirsel işlemler ve ikili çarpım yapıları yer alır.Bu bağlamda, parametre kümeleri grup kuramı (grup teorisi) yapısıyla donatılmış esnek kümeler üzerinde tanımlanan ve esnek kesişim-lamda çarpımı olarak adlandırılan yeni bir çarpım işlemi önerilmekte ve sistematik olarak incelenmektedir. Bu işlem, esnek altkümelik ve esnek eşitlik kavramlarının genelleştirilmiş biçimleriyle biçimsel tutarlılığı sağlamak üzere aksiyomatik olarak tanımlanmış olup, altında yatan sistemin cebirsel bütünlüğünü koruyacak şekilde yapılandırılmıştır. Önerilen işlemin temel yapısal özelliklerini araştırmak amacıyla kapsamlı bir cebirsel analiz gerçekleştirilmiştir; bu analizde kapalılık, birleşme, değişme, idempotentlik, birim ve yutan elemanların varlığı ile diğer esnek küme işlemleri üzerindeki dağılım gibi özellikler ve bu işlemin boş ve evrensel (mutlak) esnek kümelerle olan etkileşimleri ele alınmıştır. Ayrıca, önerilen çarpım işlemi daha önce literatürde tanımlanmış esnek ikili işlemlerle karşılaştırılarak, ifade gücü ve cebirsel uyumluluk açısından derinlemesine analiz edilmiştir. Teorik bulgular, bu işlemin yalnızca grup-parametreli tanım alanının yapısal kısıtlamalarına uyum sağlamakla kalmayıp, aynı zamanda esnek kümeler kümesi üzerinde biçimsel olarak tutarlı ve iyi tanımlı bir cebirsel sistem oluşturduğunu da doğrulamaktadır. Bu çalışmadan iki temel cebirsel sonuç elde edilmiştir: (I) Esnek kesişim-lambda çarpımının entegrasyonu, esnek küme teorisinin içsel işlemsel tutarlılığını artırmakta ve onu aksiyomatik olarak sağlam, işlem koruyucu bir yapı içerisine yerleştirmektedir. (II) Önerilen çarpım, grup yapılı parametre alanları üzerinde tanımlı esnek kümelerin, klasik grup teorisindeki yapıları taklit eden yeni tanımlı esnek işlemler altında davranış göstermesini sağlayarak, genelleştirilmiş bir esnek grup teorisinin gelişimi için kavramsal bir temel sunmaktadır. Esnek küme teorisinin cebirsel olarak olgunlaşması, anlamsal ve yapısal olarak anlamlı aksiyomları karşılayan işlemlerin titizlikle formüle edilmesine bağlı olduğundan, bu çalışma, alanın cebirsel birleşimi ve genelleştirilmesi yönünde önemli bir ilerlemeyi temsil etmektedir. Teorik öneminin ötesinde, önerilen işlem, çok kriterli karar verme sistemlerinde, cebirsel olarak yönlendirilmiş sınıflandırma şemalarında ve grup-parametreli anlamsal uzaylarda belirsizlik duyarlı veri analiz çerçevelerinde kullanılabilecek soyut cebire dayalı esnek hesaplama modellerinin inşasında pratik bir yarar sunmaktadır. Böylece, burada geliştirilen cebirsel yapı, esnek küme teorisinin temellerini anlamlı şekilde genişletmekte ve hem teorik hem de uygulamalı matematik alanlarında önemini pekiştirmektedir.

Keywords

• Esnek kümeler
• esnek alt kümeler
• esnek eşitlikler
• esnek kesişim-lamda işlemi

References

1. Abbas, M., Ali, B. and Romaguera, S. (2014). On generalized soft equality and soft lattice structure. Filomat, 28(6), 1191-1203.
2. Abbas, M., Ali, M. I. and Romaguera, S. (2017). Generalized operations in soft set theory via relaxed conditions on parameters. Filomat, 31(19), 5955-5964.
3. Aktas, H. and Çağman, N. (2007). Soft sets and soft groups. Information Science, 177(13), 2726-2735.
4. Alcantud, J.C.R. and Khameneh, A.Z., Santos-García, G. and Akram, M. (2024). A systematic literature review of soft set theory. Neural Computing and Applications, 36, 8951–8975.
5. Ali, M. I., Feng, F., Liu, X., Min, W. K. and Shabir, M. (2009). On some new operations in soft set theory. Computers and Mathematics with Applications, 57(9) 1547-1553.
6. Ali, M. I., Mahmood, M., Rehman, M.U. and Aslam, M. F. (2015). On lattice ordered soft sets, Applied Soft Computing, 36, 499-505.
7. Ali, B., Saleem, N., Sundus, N., Khaleeq, S., Saeed, M. and George, R. (2022). A contribution to the theory of soft sets via generalized relaxed operations. Mathematics, 10(15), 26-36.
8. Ali, M. I., Shabir, M. and Naz, M. (2011). Algebraic structures of soft sets associated with new operations. Computers and Mathematics with Applications, 61(9), 2647-2654.

9. 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
10. Al-shami, T. M. and El-Shafei, M. (2020). T-soft equality relation. Turkish Journal of Mathematics, 44(4), 1427-1441.
11. Atagün, A.O., Kamacı, H., Taştekin, İ. and Sezgin, A. (2019). P-properties in near-rings. Journal of Mathematical and Fundamental Sciences, 51(2), 152-167.
12. Atagün, A. O. and Sezer, A. S. (2015). Soft sets, soft semimodules and soft substructures of semimodules. Mathematical Sciences Letters, 4(3), 235-242
13. Atagün, A. O. and Sezgin, A. (2015). Soft subnear-rings, soft ideals and soft N-subgroups of near-rings, Mathematical Sciences Letters, 7(1), 37-42.
14. Atagün, A.O. and Sezgin, A. (2017). Int-soft substructures of groups and semirings with applications, Applied Mathematics & Information Sciences, 11(1), 105-113.
15. Atagün, A. O. and 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.
16. Atagün, A. O. and Sezgin, A. (2022). More on prime, maximal and principal soft ideals of soft rings. New mathematics and natural computation, 18(1), 195-207.
17. Ay, Z. and Sezgin, A. (2025). Soft union-gamma product of groups, Journal of Studies in Advanced Technologies, 3 (2), in press
18. Çağman, N. and Enginoğlu, S. (2010). Soft set theory and uni-int decision making. European Journal of Operational Research, 207(2), 848-855.
19. Eren, Ö. F. and Çalışıcı, H. (2019). On some operations of soft sets. The Fourth International Conference on Computational Mathematics and Engineering Sciences.
20. Feng, F. and Li, Y. (2013). Soft subsets and soft product operations. Information Sciences, 232(20), 44-57
21. Feng, F., Li, Y. M., Davvaz, B. and Ali, M. I. (2010). Soft sets combined with fuzzy sets and rough sets: a tentative approach. Soft Computing, 14, 899-911.
22. Feng, F., Jun, Y. B. and Zhao, X. (2008). Soft semirings. Computers and Mathematics with Applications, 56(10), 2621-2628.
23. Fu, L. (2011). Notes on soft set operations, ARPN Journal of Systems and Software, 1, 205-208.
24. Ge, X. and Yang, S. (2011). Investigations on some operations of soft sets, World Academy of Science, Engineering and Technology, 75, 1113-1116.
25. Gulistan, M., Shahzad, M. (2014). On soft KU-algebras, Journal of Algebra, Number Theory: Advances and Applications, 11(1), 1-20.
26. Gulistan, M., Feng, F., Khan, M., and Sezgin, A. (2018). Characterizations of right weakly regular semigroups in terms of generalized cubic soft sets. Mathematics, No: 6, 293.
27. Jana, C., Pal, M., Karaaslan, F. and Sezgin, A. (2019). (α, β)-soft intersectional rings and ideals with their applications. New Mathematics and Natural Computation, 15(2), 333–350.
28. Jiang, Y., Tang, Y., Chen, Q., Wang, J. and Tang, S. (2010). Extending soft sets with description logics. Computers and Mathematics with Applications, 59(6), 2087-2096
29. Jun, Y. B. and Yang, X. (2011). A note on the paper combination of interval-valued fuzzy set and soft set. Computers and Mathematics with Applications, 61(5), 1468-1470.
30. Karaaslan, F. (2019). Some properties of AG*-groupoids and AG-bands under SI-product Operation. Journal of Intelligent and Fuzzy Systems, 36(1), 231-239.
31. Kaygisiz, K. (2012). On soft int-groups. Annals of Fuzzy Mathematics and Informatics, 4(2), 363–375.
32. Khan, M., Ilyas, F., Gulistan, M. and Anis, S. (2015). A study of soft AG-groupoids, Annals of Fuzzy Mathematics and Informatics, 9(4), 621–638.
33. Khan, A., Izhar, I., & 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.
34. Liu, X., Feng, F. and Jun, Y. B. (2012). A note on generalized soft equal relations. Computers and Mathematics with Applications, 64(4), 572-578.
35. Maji, P. K., Biswas, R. and Roy, A. R. (2003). Soft set theory. Computers and Mathematics with Application, 45(1), 555-562.
36. Mahmood, T., Waqas, A., and Rana, M. A.(2015). Soft intersectional ideals in ternary semiring. Science International, 27(5), 3929-3934.
37. Mahmood, T., Rehman, Z. U., and Sezgin, A. (2018). Lattice ordered soft near rings. Korean Journal of Mathematics, 26(3), 503-51
38. Manikantan, T., Ramasany, P., and Sezgin, A. (2023). Soft quasi-ideals of soft near-rings, Sigma Journal of Engineering and Natural Science, 41(3), 565-574.
39. 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.
40. Molodtsov, D. (1999). Soft set theory. Computers and Mathematics with Applications, 37(1), 19-31.
41. Muştuoğlu, E., Sezgin, A., and Türk, Z.K.(2016). Some characterizations on soft uni-groups and normal soft uni-groups. International Journal of Computer Applications, 155(10), 1-8
42. Neog, I.J and Sut, D.K. (2011). A new approach to the theory of softset, International Journal of Computer Applications, 32(2), 1-6.
43. Onyeozili, I. A. and Gwary T. M. (2014). A study of the fundamentals of soft set theory, International Journal of Scientific & Technology Research, 3(4), 132-143.
44. Özlü, Ş. and Sezgin, A. (2020). Soft covered ideals in semigroups. Acta Universitatis Sapientiae Mathematica, 12(2), 317-346
45. Pei, D. and Miao, D. (2005). From soft sets to information systems, In: Proceedings of Granular Computing (Eds: X. Hu, Q. Liu, A. Skowron, T. Y. Lin, R. R. Yager, B. Zhang) IEEE, 2, 617-621.
46. Riaz, M., Hashmi, M. R., Karaaslan, F., Sezgin, A., Shamiri, M. M. A. A. and Khalaf, M. M. (2023). Emerging trends in social networking systems and generation gap with neutrosophic crisp soft mapping. CMES-computer modeling in engineering and sciences, 136(2), 1759-1783
47. Qin, K. and Hong, Z. (2010). On soft equality. Journal of Computational and Applied Mathematics, 234(5), 1347-1355
48. Sen, J. (2014).On algebraic structure of soft sets. Annals of Fuzzy Mathematics and Informatics, 7(6), 1013-1020.
49. Sezer, A. S. (2012). A new view to ring theory via soft union rings, ideals and bi-ideals. Knowledge-Based Systems, 36, 300–314.
50. Sezer, A. S. and Atagün, A. O. (2016). A new kind of vector space: soft vector space, Southeast Asian Bulletin of Mathematics, 40(5), 753-770.
51. Sezer, A., Atagün, A. O. and Çağman, N. (2017). N-group SI-action and its applications to N-group theory, Fasciculi Mathematici, 52, 139-153.
52. Sezer, A., Atagün, A. O. and Çağman, N. (2013). A new view to N-group theory: soft N-groups, Fasciculi Mathematici, 51, 123-140.
53. Sezer, A. S., Çağman, N., Atagün, A. O., Ali, M. I. and Türkmen, E. (2015). Soft intersection semigroups, ideals and bi-Ideals; A New application on semigroup theory I. Filomat, 29(5), 917-946.
54. Sezer, A. S., Çağman, N. and Atagün, A. O. (2014). Soft intersection interior ideals, quasi-ideals and generalized bi-ideals; A new approach to semigroup theory II. J. Multiple-Valued Logic and Soft Computing, 23(1-2), 161-207.
55. Sezgin, A. (2016). A new approach to semigroup theory I: Soft union semigroups, ideals and bi-ideals. Algebra Letters, 2016, 3, 1-46.
56. Sezgin, A., Atagün, A. O and Çağman N. (2025a). A complete study on and-product of soft sets. Sigma Journal of Engineering and Natural Sciences, 43(1), 1−14.
57. Sezgin, A., Atagün, A. O., Çağman, N. and Demir, H. (2022). On near-rings with soft union ideals and applications. New Mathematics and Natural Computation, 18(2), 495-511.
58. Sezgin, A. and Aybek, F. N. (2023). A new soft set operation: Complementary soft binary piecewise gamma operation. Matrix Science Mathematic, 7(1), 27-45.
59. Sezgin, A., Aybek, F. and Atagün, A. O. (2023a). A new soft set operation: Complementary soft binary piecewise intersection operation. Black Sea Journal of Engineering and Science, 6(4), 330-346.
60. Sezgin, A., Aybek, F. and Güngör, N. B. (2023b). A new soft set operation: Complementary soft binary piecewise union operation. Acta Informatica Malaysia, 7(1), 38-53.
61. Sezgin, A. and Çağman, N. (2024). A new soft set operation: Complementary soft binary piecewise difference operation. Osmaniye Korkut Ata University Journal of the Institute of Science and Technology, 7(1), 1-37.
62. Sezgin, A. and Çağman, N. (2025). An extensive study on restricted and extended symmetric difference operations of soft sets, Utilitas Mathematica. in press.
63. Sezgin, A., Çağman, N. and Atagün, A. O. (2017). A completely new view to soft intersection rings via soft uni-int product, Applied Soft Computing, 54, 366-392.
64. Sezgin, A., Çağman, N., Atagün, A. O. and Aybek, F. (2023c). Complemental binary operations of sets antheir application to group theory. Matrix Science Mathematic, 7(2), 99-106.
65. Sezgin, A., Çağman, N., and Çıtak, F. (2019a). α-inclusions applied to group theory via soft set and logic. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(1), 334-352.
66. Sezgin, A. and Çalışıcı, H. (2024). A comprehensive study on soft binary piecewise difference operation, Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B- Teorik Bilimler, 12(1), 1-23.
67. Sezgin, A. and 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.
68. Sezgin, A. and Demirci, A. M. (2023). A new soft set operation: Complementary soft binary piecewise star operation. Ikonion Journal of Mathematics, 5(2), 24-52.
69. Sezgin, A., Durak, İ. and Ay, Z. (2025b). Some new classifications of soft subsets and soft equalities with soft symmetric difference-difference product of groups. Amesia, 6(1), 16-32.
70. Sezgin, A. and İlgin, A. (2024a). Soft intersection almost subsemigroups of semigroups. International Journal of Mathematics and Physics, 15(1), 13-20.
71. Sezgin, A. and İlgin, A. (2024b). Soft intersection almost ideals of semigroups. Journal of Innovative Engineering and Natural Science, 4(2), 466-481.
72. Sezgin, A. and Onur, B. (2024). Soft intersection almost bi-ideals of semigroups. Systemic Analytics, 2(1), 95-105
73. Sezgin, A., Onur, B. and İlgin, A. (2024a). Soft intersection almost tri-ideals of semigroups. SciNexuses, 1, 126-138.
74. Sezgin, A. and Orbay, M. (2022). Analysis of semigroups with soft intersection ideals, Acta Universitatis Sapientiae, Mathematica, 14(2), 166-210.
75. Sezgin, A. and Sarıalioğlu, M. (2024a). A new soft set operation: Complementary soft binary piecewise theta operation. Journal of Kadirli Faculty of Applied Sciences, 4(2), 325-357.
76. Sezgin, A. and Sarıalioğlu, M. (2024b). Complementary extended gamma operation: A new soft set operation, Natural and Applied Sciences Journal, 7(1),15-44.
77. Sezgin, A., Shahzad, A. and Mehmood, A. (2019b). A new operation on soft sets: Extended difference of soft sets. Journal of New Theory, 27, 33-42.
78. Sezgin, A. and 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.
79. Sezgin, A. and Yavuz, E. (2023a). A new soft set operation: Soft binary piecewise symmetric difference operation. Necmettin Erbakan University Journal of Science and Engineering, 5(2), 150-168.
80. Sezgin, A. and Yavuz, E. (2023b). A new soft set operation: Complementary soft binary piecewise lambda operation. Sinop University Journal of Natural Sciences, 8(2), 101-133.
81. Sezgin, A. and Yavuz, E. (2024). Soft binary piecewise plus operation: A new type of operation for soft sets, Uncertainty Discourse and Applications, 1(1), 79-100.
82. Sezgin, A., Yavuz, E. and Özlü, Ş. (2024b). Insight into soft binary piecewise lambda operation: a new operation for soft sets. Journal of Umm al-Qura University for Applied Sciences, 1-15.
83. Singh, D. and Onyeozili, I. A. (2012a). Notes on soft matrices operations. ARPN Journal of Science and Technology, 2(9), 861-869.
84. Singh, D. and Onyeozili, I. A.(2012b). On some new properties on soft set operations. International Journal of Computer Applications, 59(4), 39-44.
85. Singh, D. and Onyeozili, I. A. (2012c). Some results on distributive and absorption properties on soft operations. IOSR Journal of Mathematics (IOSR-JM), 4(2), 18-30.
86. Singh, D. and Onyeozili, I. A. (2012d). Some conceptual misunderstanding of the fundamentals of soft set theory. ARPN Journal of Systems and Software, 2(9), 251-254.
87. Stojanovic, N. S. (2021). A new operation on soft sets: Extended symmetric difference of soft sets. Military Technical Courier, 69(4), 779-791.
88. Tunçay, M. and Sezgin, A. (2016). Soft union ring and its applications to ring theory, International Journal of Computer Applications, 151(9), 7-13.
89. Ullah, A., Karaaslan, F. and Ahmad, I. (2018). Soft uni-abel-grassmann's groups. European Journal of Pure and Applied Mathematics, 11(2), 517-536.
90. Yang, C. F. (2008). A note on: Soft set theory. Computers and Mathematics with Applications, 56(7), 1899-1900.
91. Zadeh, L. A. (1965). Fuzzy sets. Information Control, 8(3), 338-353.
92. Zhu, P. and Wen, Q. (2013). Operations on soft sets revisited, Journal of Applied Mathematics, 2013, Article ID 105752, 7 pages