Essential And 0-Essential Soft Intersection Ideals of Semigroups

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DOI: https://doi.org/10.28924/2291-8639-24-2026-217
Published: Jul 17, 2026
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Aslıhan Sezgin, Aleyna İlgin, Thiti Gaketem, Essential And 0-Essential Soft Intersection Ideals of Semigroups, Int. J. Anal. Appl., 24 (2026), 217.

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Aslıhan Sezgin, Aleyna İlgin, Thiti Gaketem

Abstract

In this paper, we define the essential and 0-essential soft intersection ideals of semigroups and examine in terms of some soft set operations. Our study shows that, although every ideal of a semigroup with zero is an essential ideal, this is not the case as regards essential soft intersection ideal. Moreover, our fundamental theorem, which states that if an ideal of a semigroup is an essential ideal, then its soft characteristic function is an essential soft intersection ideal, and vice versa, enables us to bridge the gap between semigroup theory and soft set theory. Additionally, we showed that if an ideal of a semigroup is an essential ideal, then its support set is an essential ideal. Furthermore, we introduce the concept of nontrivial soft intersection ideal of a semigroup to define the concept of 0-essential soft intersection ideal of a semigroup. Another fundamental theorem that allows us to bridge the gap between semigroup theory and soft set theory states that if a nonzero ideal of a semigroup with zero is a 0-essential ideal, then its soft characteristic function is a 0-essential soft intersection ideal, and vice versa. Besides, we showed that if an ideal of a semigroup with zero is a 0-essential ideal, then its support set is a 0-essential ideal. Furthermore, we define the concepts of minimal (prime, semiprime) essential and 0-essential soft intersection ideals of a semigroup. Moreover, we demonstrated the connection between (0-) essential ideals and (0-) essential soft intersection ideals of a semigroup corresponding with minimality, primeness, and semiprimeness.

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References

  1. R.A. Good, D.R. Hughes, Associated Groups for a Semigroup, Bull. Amer. Math. Soc. 58 (1952), 624-625.
  2. O. Steinfeld, Uher die Quasi Ideals, Publ. Math. Debrecen 4 (1956), 262-275.
  3. S. Lajos, (m;k;n)-Ideals in Semigroups, Notes on Semigroups II, Karl Marx University of Economics Department of Mathematics, Budapest, pp. 12–19, (1976).
  4. G. Szasz, Interior Ideals in Semigroups, Notes on Semigroups IV, Karl Marx University of Economics Department of Mathematics, Budapest, pp. 1-7, (1977).
  5. G. Szasz, Remark on Interior Ideals of Semigroups, Stud. Sci. Math. Hung. 16 (1981), 61–63.
  6. D. Molodtsov, Soft Set Theory—First Results, Comput. Math. Appl. 37 (1999), 19-31. https://doi.org/10.1016/s0898-1221(99)00056-5.
  7. F. Feng, Y.B. Jun, X. Zhao, Soft Semirings, Comput. Math. Appl. 56 (2008), 2621-2628. https://doi.org/10.1016/j.camwa.2008.05.011.
  8. N. Cagman, S. Enginoglu, Soft Set Theory and Uni–Int Decision Making, Eur. J. Oper. Res. 207 (2010), 848-855. https://doi.org/10.1016/j.ejor.2010.05.004.
  9. P.K. Maji, R. Biswas, A.R. Roy, Soft Set Theory, Comput. Math. Appl. 45 (2003), 555-562. https://doi.org/10.1016/s0898-1221(03)00016-6.
  10. D. Pei, D. Miao, From Soft Sets to Information Systems, in: 2005 IEEE International Conference on Granular Computing, IEEE, 2005, pp. 617-621 Vol. 2. https://doi.org/10.1109/grc.2005.1547365.
  11. M.I. Ali, F. Feng, X. Liu, W.K. Min, M. Shabir, On Some New Operations in Soft Set Theory, Comput. Math. Appl. 57 (2009), 1547-1553. https://doi.org/10.1016/j.camwa.2008.11.009.
  12. C.-F. Yang, A Note on “Soft Set Theory” [Comput. Math. Appl. 45 (4–5) (2003) 555–562], Comput. Math. Appl. 56 (2008), 1899-1900. https://doi.org/10.1016/j.camwa.2008.03.019.
  13. F. Feng, C. Li, B. Davvaz, M.I. Ali, Soft Sets Combined with Fuzzy Sets and Rough Sets: A Tentative Approach, Soft Comput. 14 (2009), 899-911. https://doi.org/10.1007/s00500-009-0465-6.
  14. Y. Jiang, Y. Tang, Q. Chen, J. Wang, S. Tang, Extending Soft Sets with Description Logics, Comput. Math. Appl. 59 (2010), 2087-2096. https://doi.org/10.1016/j.camwa.2009.12.014.
  15. M.I. Ali, M. Shabir, M. Naz, Algebraic Structures of Soft Sets Associated with New Operations, Comput. Math. Appl. 61 (2011), 2647-2654. https://doi.org/10.1016/j.camwa.2011.03.011.
  16. T.J. Neog, D.K. Sut, A New Approach to the Theory of Soft Sets, Int. J. Comput. Appl. 32 (2011), 1-6. https://doi.org/10.5120/3874-5415.
  17. L. Fu, Notes on Soft Set Operations, ARPN J. Syst. Softw. 1 (2011), 205–208.
  18. X. Ge, S. Yang, Investigations on Some Operations of Soft Sets, World Acad. Sci. Eng. Technol. 75 (2011), 1113–1116.
  19. D. Singh, I.A. Onyeozili, Notes on Soft Matrices Operations, ARPN J. Sci. Technol. 2 (2012), 861–869.
  20. D. Singh, I.A. Onyeozili, On Some New Properties of Soft Set Operations, Int. J. Comput. Appl. 59 (2012), 39-44. https://doi.org/10.5120/9538-3975.
  21. D. Singh, Some Results on Distributive and Absorption Properties of Soft Operations, IOSR J. Math. 4 (2012), 18-30. https://doi.org/10.9790/5728-0421830.
  22. D. Singh, I.A. Onyeozili, Some Conceptual Misunderstanding of the Fundamentals of Soft Set Theory, ARPN J. Syst. Softw. 2 (2012), 251–254.
  23. P. Zhu, Q. Wen, Operations on Soft Sets Revisited, J. Appl. Math. 2013 (2013), 105752. https://doi.org/10.1155/2013/105752.
  24. I.A. Onyeozili, T.M. Gwary, A Study of the Fundamentals of Soft Set Theory, Int. J. Sci. Technol. Res. 3 (2014), 132–143.
  25. J. Sen, On Algebraic Structure of Soft Sets, Ann. Fuzzy Math. Inform. 7 (2014), 1013–1020.
  26. Ö.F. Eren, H. Çalışıcı, On Some Operations of Soft Sets, in: The Fourth International Conference on Computational Mathematics and Engineering Sciences (CMES-2019), Antalya, Türkiye, 2019.
  27. N. Stojanović, A New Operation on Soft Sets: 791 Extended Symmetric Difference of Soft Sets, Vojnoteh. Glasn. 69 (2021), 779-791. https://doi.org/10.5937/vojtehg69-33655.
  28. A. Sezgin, F.N. Aybek, A.O. Atagün, A New Soft Set Operation: Complementary Soft Binary Piecewise Intersection (∩) Operation, Black Sea J. Eng. Sci. 6 (2023), 330-346. https://doi.org/10.34248/bsengineering.1319873.
  29. A. Sezgin, F. Nur Aybek, N. Bilgili Güngör, A New Soft Set Operation: Complementary Soft Binary Piecewise Union (∪) Operation, Acta Inform. Malays. 7 (2023), 38-53. https://doi.org/10.26480/aim.01.2023.38.53.
  30. A. Sezgin, F. Nur Aybek, A New Soft Set Operation: Complementary Soft Binary Piecewise Gamma (γ) Operation, Matrix Sci. Math. 7 (2023), 27-45. https://doi.org/10.26480/msmk.01.2023.27.45.
  31. A. Sezgin, K. Dagtoros, Complementary Soft Binary Piecewise Symmetric Difference Operation: A Novel Soft Set Operation, Sci. J. Mehmet Akif Ersoy Univ. 6 (2023), 31-45.
  32. A. Sezgin, A.M. Demirci, A New Soft Set Operation: Complementary Soft Binary Piecewise Star (*) Operation, Ikonion J. Math. 5 (2023), 24-52. https://doi.org/10.54286/ikjm.1304566.
  33. A. Sezgin, H. Çalişici, A Comprehensive Study on Soft Binary Piecewise Difference Operation, Eskişehir Tek. Üniv. Bilim Teknol. Derg. B Teor. Bilimler 12 (2024), 32-54. https://doi.org/10.20290/estubtdb.1356881.
  34. A. Sezgin, E. Yavuz, A New Soft Set Operation: Soft Binary Piecewise Symmetric Difference Operation, Necmettin Erbakan Univ. J. Sci. Eng. 5 (2023), 189-208. https://doi.org/10.47112/neufmbd.2023.18.
  35. A. Sezgin, E. Yavuz, A New Soft Set Operation: Complementary Soft Binary Piecewise λ Operation, Sinop Univ. Fen Bilim. Derg. 8 (2023), 101-133. https://doi.org/10.33484/sinopfbd.1320420.
  36. A. Sezgin, E. Yavuz, Soft Binary Piecewise Plus Operation: A New Type of Operation for Soft Sets, Uncertainty Discourse Appl. 1 (2024), 79–100. https://doi.org/10.48313/uda.v1i1.26.
  37. A. Sezgin, N. Cagman, A New Soft Set Operation: Complementary Soft Binary Piecewise Difference () Operation, Osmaniye Korkut Ata Univ. Fen Bilimleri Enst. Derg. 7 (2024), 58-94. https://doi.org/10.47495/okufbed.1308379.
  38. A. Sezgin, N. Çağman, An Extensive Study on Restricted and Extended Symmetric Difference Operations of Soft Sets, Util. Math. 127 (2026), 3-34. https://doi.org/10.61091/um127-01.
  39. A. Sezgin, M. Sarıalioğlu, Complementary Extended Gamma Operation: A New Soft Set Operation, Nat. Appl. Sci. J. 7 (2024), 15-44. https://doi.org/10.38061/idunas.1482044.
  40. A. Sezgin, M. Sarialioglu, A New Soft Set Operation: Complementary Soft Binary Piecewise Theta (θ) Operation, Kadirli Uygulamali Bilimler Fakültesi Dergisi 4 (2024), 325-357.
  41. A. Sezgin, E. Senyigit, A New Product for Soft Sets with Its Decision-Making: Soft Star-Product, Big Data Comput. Vis. 5 (2025), 52–73. https://doi.org/10.22105/bdcv.2024.492834.1221.
  42. K. Qin, Z. Hong, On Soft Equality, J. Comput. Appl. Math. 234 (2010), 1347-1355. https://doi.org/10.1016/j.cam.2010.02.028.
  43. Y.B. Jun and X. Yang, A Note on The Paper Combination of Interval-Valued Fuzzy Set and Soft Set, Comput. Math. Appl. 61 (2011), 1468–1470. https://doi.org/10.1016/j.camwa.2010.12.077.
  44. X. Liu, F. Feng, Y.B. Jun, A Note on Generalized Soft Equal Relations, Comput. Math. Appl. 64 (2012), 572-578. https://doi.org/10.1016/j.camwa.2011.12.052.
  45. F. Feng, Y. Li, Soft Subsets and Soft Product Operations, Inf. Sci. 232 (2013), 44-57. https://doi.org/10.1016/j.ins.2013.01.001.
  46. M. Abbas, B. Ali, S. Romaguera, On Generalized Soft Equality and Soft Lattice Structure, Filomat 28 (2014), 1191-1203. https://doi.org/10.2298/FIL1406191A.
  47. M. Abbas, M. Ali, S. Romaguera, Generalized Operations in Soft Set Theory via Relaxed Conditions on Parameters, Filomat 31 (2017), 5955-5964. https://doi.org/10.2298/FIL1719955A.
  48. T. Al-Shami, Investigation and Corrigendum to Some Results Related to g-Soft Equality and gF-Soft Equality Relations, Filomat 33 (2019), 3375-3383. https://doi.org/10.2298/FIL1911375A.
  49. T.M. Al-Shami, M.E. El-Shafei, T-Soft Equality Relation, Turk. J. Math. 44 (2020), 1427-1441. https://doi.org/10.3906/mat-2005-117.
  50. A. Sezgin Sezer, A New View to Ring Theory via Soft Union Rings, Ideals and Bi-Ideals, Knowl. Based Syst. 36 (2012), 300-314. https://doi.org/10.1016/j.knosys.2012.04.031.
  51. A. Sezgin, A New Approach to Semigroup Theory I: Soft Union Semigroups, Ideals and Bi-ideals, Algebr. Lett. (2016), 3.
  52. E. Mustuoglu, A. Sezgin, Z. Kaya, Some Characterizations on Soft Uni-groups and Normal Soft Uni-groups, Int. J. Comput. Appl. 155 (2016), 1-8. https://doi.org/10.5120/ijca2016912412.
  53. K. Kaygisiz, On Soft Int-groups, Ann. Fuzzy Math. Inform. 4 (2012), 363–375.
  54. A.S. Sezer, N. Cagman, A.O. Atagun, Soft Intersection Interior Ideals, Quasi-ideals and Generalized Bi-ideals: A New Approach to Semigroup Theory II, J. Mult.-Valued Logic Soft Comput. 23 (2014), 161–207.
  55. A. Sezgin, N. Cagman, A.O. Atagun, A Completely New View to Soft Intersection Rings via Soft Uni-Int Product, Appl. Soft Comput. 54 (2017), 366-392. https://doi.org/10.1016/j.asoc.2016.10.004.
  56. A. Sezer, N. Cagman, O. Atagu, M.A. Irfan, E. Turkmen, Soft Intersection Semigroups, Ideals and Bi-ideals: A New Application on Semigroup Theory I, Filomat 29 (2015), 917-946. https://doi.org/10.2298/FIL1505917S.
  57. A. Sezgin, A. İlgün, Soft Intersection Almost Subsemigroups of Semigroups, Int. J. Math. Phys. 15 (2024), 13-20. https://doi.org/10.26577/ijmph.2024v15i1a2.
  58. A. Sezgin, A. Ilgin, Soft Intersection Almost Ideals of Semigroups, J. Innov. Eng. Nat. Sci. 4 (2024), 466-481. https://doi.org/10.61112/jiens.1464344.
  59. A. Sezgin, A. İlgin, Soft Intersection Almost Bi-interior Ideals of Semigroups, J. Nat. Appl. Sci. Pak. 6 (2024), 1619-1638.
  60. A. Sezgin, A. Ilgin, Soft Intersection Almost Bi-Quasi Ideals of Semigroups, Soft Comput. Fusion Appl. 1 (2024), 27-42.
  61. A. Sezgin, A. Ilgin, Soft Intersection Almost Weak-Interior Ideals of Semigroups: A Theoretical Study, J. Nat. Sci. Math. UT 9 (2024), 372-385. https://doi.org/10.62792/ut.jnsm.v9.i17-18.p2834.
  62. A. Sezgin, B. Onur, Soft Intersection Almost Bi-Ideals of Semigroups, Syst. Anal. 2 (2024), 95-105. https://doi.org/10.31181/sa21202415.
  63. A. Sezgin, F.Z. Kocakaya, A. Ilgin, Soft Intersection Almost Quasi-Interior Ideals of Semigroups, Eskişehir Tek. Üniv. Bilim Teknol. Derg. B Teor. Bilimler 12 (2024), 81-99. https://doi.org/10.20290/estubtdb.1473840.
  64. A. Sezgin, Z.H. Bas, A. Ilgin, Soft Intersection Almost Bi-Quasi-Interior Ideals of Semigroups, J. Fuzzy Ext. Appl. 6 (2025), 43-58. https://doi.org/10.22105/jfea.2024.452790.1445.
  65. A. Sezgin, B. Onur, A. Ilgin, Soft Intersection Almost Tri-Ideals of Semigroups, SciNexuses 1 (2024), 126-138. https://doi.org/10.61356/j.scin.2024.1414.
  66. A. Sezgin, A. Ilgin, A.O. Atagun, Soft Intersection Almost Tri-bi-ideals of Semigroups, Sci. Technol. Asia 29 (2024), 1-13.
  67. A. Sezgin, F.Z. Kocakaya, Soft Intersection Almost Quasi-ideals of Semigroups, Songklanakarin J. Sci. Technol. 47 (2025), 430-437.
  68. A. Sezgin, Z.H. Baş, Soft-Int Almost Interior Ideals for Semigroups, Inf. Sci. Appl. 4 (2024), 25-36. https://doi.org/10.61356/j.iswa.2024.4374.
  69. U. Medhi, K.K. Rajkhowa, L.K. Barthakur, H.K. Saikia, On Fuzzy Essential Ideals of Rings, Adv. Fuzzy Sets Syst. 3 (2008), 287–299.
  70. S. Baupradist, B. Chemat, K. Palanivel, R. Chinram, Essential Ideals and Essential Fuzzy Ideals in Semigroups, J. Discrete Math. Sci. Cryptogr. 24 (2020), 223-233. https://doi.org/10.1080/09720529.2020.1816643.
  71. R. Chinram, T. Gaketem, Essential (m, n)-Ideal and Essential Fuzzy (m, n)-Ideals in Semigroups, ICIC Express Lett. 15 (2021), 1037–1044.
  72. N. Panpetch, T. Muangngao, T. Gaketem, Some Essential Bi-ideals and Essential Fuzzy Bi-ideals in a Semigroup, J. Math. Comput. Sci. 28 (2022), 326-334. https://doi.org/10.22436/jmcs.028.04.02.
  73. T. Gaketem, A. Iampan, On Essential Bi-Ideals and Essential Fuzzy Bi-Ideals in Semigroups, Int. J. Appl. Math. 36 (2023), 483–496. https://doi.org/10.12732/ijam.v36i4.4.
  74. A. Sezgin, N. Çağman, A.O. Atagün, F.N. Aybek, Complemental Binary Operations of Sets and Their Application to Group Theory, Matrix Sci. Math. 7 (2023), 114-121. https://doi.org/10.26480/msmk.02.2023.114.121.
  75. H. Aktaş, N. Çağman, Soft Sets and Soft Groups, Inf. Sci. 177 (2007), 2726-2735. https://doi.org/10.1016/j.ins.2006.12.008.
  76. J.C.R. Alcantud, A.Z. Khameneh, G. Santos-García, M. Akram, A Systematic Literature Review of Soft Set Theory, Neural Comput. Appl. 36 (2024), 8951-8975. https://doi.org/10.1007/s00521-024-09552-x.
  77. M.I. Ali, T. Mahmood, M.M.U. Rehman, M.F. Aslam, On Lattice Ordered Soft Sets, Appl. Soft Comput. 36 (2015), 499-505. https://doi.org/10.1016/j.asoc.2015.05.052.
  78. B. Ali, N. Saleem, N. Sundus, S. Khaleeq, M. Saeed, et al., A Contribution to the Theory of Soft Sets via Generalized Relaxed Operations, Mathematics 10 (2022), 2636. https://doi.org/10.3390/math10152636.
  79. A.O. Atagun, H. Kamaci, I. Tastekin, A. Sezgin, P-Properties in Near-Rings, J. Math. Fundam. Sci. 51 (2019), 152-167. https://doi.org/10.5614/j.math.fund.sci.2019.51.2.5.
  80. A.O. Atagün, A.S. Sezer, Soft Sets, Soft Semimodules and Soft Substructures of Semimodules, Math. Sci. Lett. 4 (2015), 235–242.
  81. A.O. Atagun, A. Sezgin, Soft Subnear-Rings, Soft Ideals and Soft N-Subgroups of Near-Rings, Math. Sci. Lett. 7 (2018), 37-42. https://doi.org/10.18576/msl/070106.
  82. A.O. Atagun, A. Sezgin, Int-Soft Substructures of Groups and Semirings with Applications, Appl. Math. Inf. Sci. 11 (2017), 105-113. https://doi.org/10.18576/amis/110113.
  83. A.O. Atagün, A. Sezgin, A New View to Near-ring Theory: Soft Near-rings, South East Asian J. Math. Math. Sci. 14 (2018), 1–14.
  84. A.O. Atagün, A. Sezgi̇n, More on Prime, Maximal and Principal Soft Ideals of Soft Rings, New Math. Nat. Comput. 18 (2021), 195-207. https://doi.org/10.1142/S1793005722500119.
  85. M. Gulistan, M. Shahzad, On Soft KU-Algebras, J. Algebra Number Theory: Adv. Appl. 11 (2014), 1–20.
  86. 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. https://doi.org/10.3390/math6120293.
  87. A. Sezgin, A New View on AG-Groupoid Theory via Soft Sets for Uncertainty Modeling, Filomat 32 (2018), 2995-3030. https://doi.org/10.2298/FIL1808995S.
  88. C. Jana, M. Pal, F. Karaaslan, A. Sezgin, (α, β)-Soft Intersectional Rings and Ideals with Their Applications, New Math. Nat. Comput. 15 (2019), 333–350.
  89. F. Karaaslan, Some Properties of AG*-Groupoids and AG-Bands under SI-Product Operation, J. Intell. Fuzzy Syst. 36 (2018), 231-239. https://doi.org/10.3233/JIFS-181208.
  90. A. Khan, I. Izhar, A. Sezgin, Characterizations of Abel Grassmann's Groupoids by the Properties of Their Double-framed Soft Ideals, Int. J. Anal. Appl. 15 (2017), 62–74.
  91. T. Mahmood, A. Waqas, M.A. Rana, Soft Intersectional Ideals in Ternary Semiring, Sci. Int. 27 (2015), 3929–3934.
  92. T. Mahmood, Z.U. Rehman, A. Sezgin, Lattice Ordered Soft Near Rings, Korean J. Math. 26 (2018), 503–517.
  93. T. Manikantan, P. Ramasany, A. Sezgin, Soft Quasi-ideals of Soft Near-rings, Sigma J. Eng. Nat. Sci. 41 (2023), 565–574. https://doi.org/10.14744/sigma.2023.00062.
  94. S. Memiş, Another View on Picture Fuzzy Soft Sets and Their Product Operations with Soft Decision-Making, J. New Theory 38 (2022), 1-13. https://doi.org/10.53570/jnt.1037280.
  95. Ş. Özlü, A. Sezgin, Soft Covered Ideals in Semigroups, Acta Univ. Sapientiae Math. 12 (2020), 317-346. https://doi.org/10.2478/ausm-2020-0023.
  96. M. Riaz, M. Raza Hashmi, F. Karaaslan, A. Sezgin, M.M.A. Al Shamiri, et al., Emerging Trends in Social Networking Systems and Generation Gap with Neutrosophic Crisp Soft Mapping, Comput. Model. Eng. Sci. 136 (2023), 1759-1783. https://doi.org/10.32604/cmes.2023.023327.
  97. A.S. Sezer, A.O. Atagün, A New Kind of Vector Space: Soft Vector Space, Southeast Asian Bull. Math. 40 (2016), 753–770.
  98. A.S. Sezer, A.O. Atagün, N. Çağman, N-Group SI-Action and Its Applications to N-Group Theory, Fascic. Math. 52 (2017), 139–153.
  99. A.S. Sezer, A.O. Atagün, N. Çağman, A New View to N-group Theory: Soft N-groups, Fasc. Math. 51 (2013), 123–140.
  100. A.S. Sezer, N. Cagman, A.O. Atagun, Soft Intersection Interior Ideals, Quasi-ideals and Generalized Bi-ideals: a New Approach to Semigroup Theory II, J. Mult.-Valued Logic Soft Comput. 23 (2014), 161–207.
  101. A. Sezgin, N. Çagman, F. Çitak, α-Inclusions Applied to Group Theory via Soft Set and Logic, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68 (2018), 334-352. https://doi.org/10.31801/cfsuasmas.420457.
  102. A. Sezgin, A. Shahzad, A. Mehmood, A New Operation on Soft Sets: Extended Difference of Soft Sets, J. New Theory 27 (2019), 33–42.
  103. A. Sezgin, A.O. Atagün, N. Çağman, H. Demir, On Near-rings with Soft Union Ideals and Applications, New Math. Nat. Comput. 18 (2022), 495–511.
  104. A. Sezgin, M. Orbay, Analysis of Semigroups with Soft Intersection Ideals, Acta Univ. Sapientiae Math. 14 (2022), 166-210. https://doi.org/10.2478/ausm-2022-0012.
  105. A. Sezgin, A Complete Study on and-Product of Soft Sets, Sigma J. Eng. Nat. Sci. 43 (2025), 1-14. https://doi.org/10.14744/sigma.2025.00002.
  106. A. Sezgin, İ. Durak, Z. Ay, Some New Classifications of Soft Subsets and Soft Equalities with Soft Symmetric Difference-Difference Product of Groups, Amesia 6 (2025), 16-32. https://doi.org/10.54559/amesia.1730014.
  107. M. Tuncay, A. Sezgin, Soft Union Ring and Its Applications to Ring Theory, Int. J. Comput. Appl. 151 (2016), 7-13. https://doi.org/10.5120/ijca2016911867.
  108. A. Ullah, F. Karaaslan, I. Ahmad, Soft Uni-Abel-Grassmann's Groups, Eur. J. Pure Appl. Math. 11 (2018), 517-536. https://doi.org/10.29020/nybg.ejpam.v11i2.3228.
  109. A. Sezgin, A. Ilgin, F.Z. Kocakaya, Z.H. Bas, B. Onur, et al., A Remarkable Contribution to Soft INT-Group Theory via a Comprehensive View of Soft Cosets, J. Sci. Arts 24 (2024), 905-934. https://doi.org/10.46939/j.sci.arts-24.4-a13.