A New Soft Set Operation: Complementary Soft Binary Piecewise Intersection (∩) Operation

Öz

The soft set theory developed by Molodtsov has been applied both theoretically and practically in many fields. It is a useful piece of mathematics for handling uncertainty. Numerous variations of soft set operations have been described and used since its introduction. In this paper, we define a new soft set operation, called complementary soft binary piecewise intersection operation, a unique soft set operation, and examine its basic algebraic properties. Additionally, we aim to contribute to the literature on soft sets by illuminating the relationships between this new soft set operation and other kinds of soft set operations by researching the distribution of this new soft set operation over other soft set operations. Moreover, we prove that the set of all the soft sets with a fixed parameter set together with the complementary soft binary piecewise intersection operation and the soft binary piecewise union operation is a zero-symmetric near-semiring and hemiring.

Anahtar Kelimeler

• Soft sets
• Soft set operations
• Conditional complements
• Near-semiring

Kaynakça

1. Akbulut E. 2023. New type of extended operations of soft set: Complementary extended lambda and difference operations. Master Thesis, Amasya University, The Graduate School of Natural and Applied Sciences, Mathematics Department, Amasya, Türkiye, pp: 105.
2. Ali MI, Feng F, Liu X, Min WK, Shabir M. 2009. On some new operations in soft set theory. Comput Math Appl, 57(9): 1547-1553.
3. Aybek F. 2023. New restricted and extended soft set operations. Master Thesis, Amasya University, The Graduate School of Natural and Applied Sciences Mathematics Department, Amasya, Türkiye, pp: 186.
4. Çağman N. 2021. Conditional complements of sets and their application to group theory. J New Results Sci, 10(3): 67-74.
5. Demirci AM. 2023. New type of extended operations of soft set: Complementary extended plus, union and theta operations. Master Thesis, Amasya University, The Graduate School of Natural and Applied Sciences, Mathematics Department, Amasya, Türkiye, pp: 105.
6. Eren ÖF, Çalışıcı H. 2019. On some operations of soft sets. The Fourth International Conference on Computational Mathematics and Engineering Sciences (CMES-2019), April 20-22, 2019, Antalya, Türkiye.
7. Hoorn WGV, Rootselaar VB. 1967. Fundamental notions in the theory of seminearrings. Composit Math, 18(1-2): 65.
8. Maji PK, Bismas R, Roy AR. 2003. Soft set theory. Comput Math Appl, 45(1): 555-562.

9. Molodtsov D. 1999. Soft set theory-first results. Comput Math Appl, 37(1): 19-31.
10. Özlü Ş. 2022a. Interval Valued q- rung orthopair hesitant fuzzy choquet aggregating operators in multi-criteria decision making problems. Gazi Univ J Sci Part C: Design Tech, 10(4): 1006-1025.
11. Özlü Ş. 2022b. Interval valued bipolar fuzzy prioritized weighted dombi averaging operator based on multi-criteria decision making problems. Gazi Univ J Sci Part C: Design Tech, 10(4): 841-857.
12. Özlü Ş, Sezgin A. 2020. Soft covered ideals in semigroups. Acta Univ Sapientiae Math, 12(2): 317-346.
13. Pei D, Miao D. 2005. From soft sets to information systems. In: Proceedings of Granular Computing IEEE, 2: 617-621.
14. Pilz G. 1977. Near Rings. Amsterdam, Nederland.
15. Sarıalioğlu M. 2023a. New type of extended operations of soft set: Complementary extended gamma, intersection and star operations. Master Thesis, Amasya University, The Graduate School of Natural and Applied Sciences, Mathematics Department, Amasya, Türkiye, pp: 78.
16. Sezgin A, Atagün AO. 2011. On operations of soft sets. Comput Math Appl, 61(5):1457-1467.
17. Sezgin A, Aybek F. 2023. New soft set operation: Complementary soft binary piecewise gamma operation. Matrix Sci Math, 7(1): 27-45.
18. Sezgin A, Aybek F, Güngör Bilgili N. 2023a. New soft set operation: Complementary soft binary piecewise union operation. Acta Inform Malaysia. 7(1): 38-53
19. Sezgin A, Çağman N, Atagün AO, Aybek F. 2023b. Complemental binary operations of sets and their application to group theory. Matrix Sci Math. 7 (2) (in press)
20. Sezgin A, Demirci AM. 2023. New soft set operation: Complementary soft binary piecewise star operation. Ikonion J Math.. 5(2): 24-52.
21. Sezgin A, Shahzad A, Mehmood A. 2019. New operation on soft sets: extended difference of soft sets. J New Theory, (27): 33-42.
22. Sezgin A, Yavuz E. 2023. New soft set operation: Complementary soft binary piecewise lambda operation. Sinopjns, DOİ:10.33484/sinopfbd.1320420
23. Stojanovic NS. 2021. A new operation on soft sets: extended symmetric difference of soft sets. Military Tech Courier, 69(4): 779-791.
24. Taşdemir F, Atagün AO, Altındiş H. 2013. Different prime N-ideals and IFP N-Ideals. Indian J Pure Appl Math, 44(4): 527-542.
25. Taşdemir F, Taştekin İ. 2019. On P-3-prime and P-c-prime ideals in near-rings. J Algebra Number Theory Appl, 42(1): 49-57.
26. Yavuz E. 2023. Soft binary piecewise operations and their properties, Master Thesis, Amasya University, The Graduate School of Natural and Applied Sciences, Mathematics Department, Amasya, Türkiye, pp: 169.