A New Soft Set Operation: Complementary Soft Binary Piecewise Star (*) Operation

Abstract

Soft set theory, introduced by Molodtsov, is as an important mathematical tool to deal with uncertainty and it has been conveyed to many fields both as theoretical and application aspect. Since its inception, different kinds of soft set operations are defined and used in various types. In this paper, we define a new kind of soft set operation called, complementary soft binary piecewise star operation and we investigate its basic algebraic properties. Moreover, it is aimed to contribute to the soft set literature by obtaining the relationships between this new soft set operation and some other types of soft set operations by examing the distribution of complementary soft binary piecewise star operation over extended soft set operations, complementary extended soft set operations, soft binary piecewise operations, complementary soft binary piecewise operations and restricted soft set operations

Keywords

• soft set
• soft set operations
• conditional complements

References

1. Akbulut, E. (2023) New type of extended operations of soft set: Complementary extended lambda and difference operations. MSc. Thesis, Amasya University, The Graduate School of Natural and Applied Sciences Master of Science in Mathematics Department, Amasya.
2. Ali, M.I., Feng, F., Liu, X., Min, W.K., Shabir, M. (2009) On some new operations in soft set theory. Computers and Mathematics with Applications, 57(9): 1547-1553.
3. Ali, M.I., Shabir, M., Naz, M., (2011) Algebraic structures of soft sets associated with new operations, Computers and Mathematics with Applications 61 2647–2654.
4. Aybek, F. (2023) New restricted and extended soft set operations. MSc. thesis, Amasya University, The Graduate School of Natural and Applied Sciences Master of Science in Mathematics Department, Amasya.
5. Çağman N. (2021) Conditional complements of sets and their application to group theory. Journal of New Results in Science, 10 (3): 67-74.
6. Çağman N., Enginoğlu, S. (2010) Soft set theory and uni-int decision making, Eurepean Journal of Operational Research, 207: 848-855.
7. Dalkılıç O. (2021) A novel approach to soft set theory in decision-making under uncertainty, International Journal of Computer Mathematics, 98:10, 1935-1945.
8. Dalkılıç O. (2021) Determining the (non-)membership degrees in the range (0,1) independently of the decision-makers for bipolar soft sets, Journal of Taibah University for Science, 15:1, 609-618.

9. Dalkılıç O. (2022) Determining the membership degrees in the range (0, 1) for hypersoft sets independently of the decision-maker, International Journal of Systems Science, 53:8, 1733-1743.
10. Dalkılıç O. (2022) Approaches that take into account interactions between parameters: pure (fuzzy) soft sets, International Journal of Computer Mathematics, 99:7, 1428 1437.
11. Demirci A.M. (2023) New type of extended operations of soft set: Complementary extended plus, union and theta operations. MSc. Thesis, Amasya University, The Graduate School of Natural and Applied Sciences Master of Science in Mathematics Department, Amasya.
12. Eren Ö.F. and Çalışıcı H. (2019) On some operations of soft sets, The Fourth International Conference on Computational Mathematics and Engineering Sciences (CMES-2019), Antalya.
13. Feng, F., Li, Y.M., Davvaz, B., Ali, M. I., (2010) Soft Sets Combined With Fuzzy Sets and Rough Sets: A Tantative Approach, Soft Computing, 14 899–911.
14. Fu, L., Notes on soft set operations, (2011) ARPN Journal of systems and softwares, 1, 205-208.
15. Ge, X., Yang S., (2011) Investigations on some operations of soft sets, World academy of Science, Engineering and Technology, 75 1113-1116.
16. Husain, S., Shamsham, Km., (2018) A study of properties of soft set and its applications, International Research Journal of Engineering and Technology, 5 (1) 363-372.
17. Jayanta, S., (2014) On algebraic structure of soft sets, Annals of Fuzzy Mathematics and Informatics, Volume 7 (6) 1013-1020.
18. Jiang, J., Tang, Y., Chen, Q., Wang, J., Tang, S., (2010) Extending soft sets with description logics, Computers and Mathematics with Applications, 59 2087–2096.
19. Neog, I.J, Sut, D.K., (2011) A new Approach to the Theory of Soft Set, International Journal of Computer Applications, 32 (2) 1-6.
20. Maji, P.K, Bismas. R., Roy, A.R. (2003) Soft set theory. Computers and Mathematics with Applications, 45 (1): 555-562.
21. Molodtsov, D. (1999) Soft set theory-first results. Computers and Mathematics with Applications, 37 (1): 19-31
22. Onyeozili, L.A., T.A., Gwary, (2014) A study of the fundamentals of soft set theory, Internatıonal Journal Of Scıentıfıc & Technology Research, 3 (4) 132-143.
23. Özlü, Ş., (2022a) Interval valued q- rung orthopair hesitant fuzzy choquet aggregating operators in multi-criteria decision making problems. Gazi University Journal of Science Part C: Design and Technology, 10 (4). 1006-1025.
24. Özlü Ş., (2022b) Interval valued bipolar fuzzy prioritized weighted dombi averaging operator based on multi-criteria decision making problems. Gazi University Journal of Science Part C: Design and Technology, 10 (4). 841-857.
25. Özlü Ş. and Sezgin A., (2021) Soft covered ideals in semigroups. Acta Universitatis Sapientiae Mathematica, 12 (2). 317-346.
26. Pei, D. and Miao, D. (2005) From Soft Sets to Information Systems. In: Proceedings of Granular Computing. IEEE, 2: 617-621.
27. Ping, Z., Qiaoyan, W.,(2013) Operations on Soft Sets Revisited, Journal of Applied Mathematics, Volume 2013 Article ID 105752 7 pages.
28. Sarıalioğlu, M. (2023) New type of extended operations of soft set: Complementary extended gamma, intersection and star operations. MSc. Thesis, Amasya University, The Graduate School of Natural and Applied Sciences Master of Science in Mathematics Department, Amasya.
29. Sezgin, A. and Atagün, A.O. (2011) On operations of soft sets. Computers and Mathematics with Applications, 61(5):1457-1467.
30. Sezgin A. and Atagün, A.O., (2023) New soft set operation: Complementary soft binary piecewise plus operation. accepted in Matrix Science Mathematic.
31. Sezgin A. and Aybek, F. (2023) New soft set operation: Complementary soft binary piecewise gamma operation. Matrix Science Mathematics (7) 1, 27-45.
32. Sezgin A., Aybek F., Güngör Bilgili N. (2023a) New soft set operation: Complementary soft binary piecewise union operation. Acta Informatica Malaysia 7(1), 38-53.
33. Sezgin A., Aybek F., Atagün A.O. (2023b) New soft set operation: Complementary soft binary piecewise intersection operation. accepted in Black Sea Journal of Engineering and Science.
34. Sezgin A. and Çağman N. (2023) New soft set operation: Complementary soft binary piecewise difference operation. accepted in Osmaniye Korkut Ata University Journal of the Institute of Science and Technology.
35. Sezgin A., Çağman N., Atagün A.O., Aybek F. (2023c) Complemental binary operations of sets and their application to group theory, accepted in Information Management and Computer Science.
36. Sezgin A. and Sarıalioğlu, M. (2023) New soft set operation: Complementary soft binary piecewise theta operation. accepted in Journal of Kadirli Faculty of Applied Sciences.
37. Sezgin, A., Shahzad A. and Mehmood, A. (2019) New Operation on Soft Sets: Extended Difference of Soft Sets. Journal of New Theory, (27): 33-42.
38. Singh, D., Onyeozili, L.A., (2012) Some conceptual misunderstanding of the fundamentals of soft set theory, ARPN Journal of systems and softwares, 2 (9) 251-254.
39. Singh, D., Onyeozili, L.A., (2012) Some results on Distributive and absorption properties on soft operations, IOSR Journal of mathematics, 4 (2) 18-30.
40. Singh, D., Onyeozili, L.A., (2012) On some new properties on soft set operations, International Journal of Computer Applications, 59 (4) 39-44.
41. Singh, D., Onyeozili, L.A., (2012) Notes on soft matrices operations, ARPN Journal of science and technology, 2(9) 861-869.
42. Stojanovic, N.S. (2021) A new operation on soft sets: extended symmetric difference of soft sets. Military Technical Courier, 69(4): 779-791.
43. Yang, C.F., (2008) “A note on: “Soft set theory” [Computers & Mathematics with Applications 45 (2003), no. 4-5, 555–562],” Computers & Mathematics with Applications, 56 (7) 1899–1900.
44. Yavuz E., (2023) Soft binary piecewise operations and their properties, MSc. Thesis, Amasya University, The Graduate School of Natural and Applied Sciences Master of Science in Mathematics Department, Amasya.