Complementary Soft Binary Piecewise Symmetric Difference Operation: A Novel Soft Set Operation

Abstract

Since Molodtsov first introduced soft set theory, a useful mathematical tool for solving problems related to uncertainties, many soft set operations have been described and used in decision making problems. In this study, a new soft set operation called complementary soft binary piecewise symmetric difference operation is defined, and its properties are examined in comparison with the basic algebraic properties of the symmetric difference operation. Moreover, it has been shown that the collection of soft sets with a fixed set of parameter together with complementary soft binary symmetric difference and restricted intersection, is a commutative hemiring with identity and also a Boolean ring.

Keywords

• Soft sets
• soft set operations
• Boolean rings
• hemirings

Supporting Institution

YOK

Project Number

YOK

Ethical Statement

Etik kurul kararının alınması gereken bir çalışma değildir.

References

1. Molodtsov, D., (1999). Soft set theory-first results. Computers and Mathematics with Applications, 37 (1) 19-31.
2. Maji, P.K., Bismas, R., Roy, A.R., (2003). Soft set theory, Computers and Mathematics with Applications, 45 (1) 555-562.
3. Pei D. and Miao, D., (2005). From Soft Sets to Information Systems, In: Proceedings of Granular Computing. IEEE, 2 617-62.
4. 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.
5. Sezgin, A., Atagün, A. O., (2011). On operations of soft sets, Computers and Mathematics with Applications, 61(5) 1457-1467.
6. Sezgin, A. Shahzad, A., Mehmood A. (2019). New operation on soft sets: extended difference of soft Sets, Journal of New Theory, (27) 33-42.
7. Stojanovic, N.S., (2021). A new operation on soft sets: extended symmetric difference of soft sets, Military Technical Courier, 69(4) 779-791.
8. Eren, Ö.F., (2019). On some operations of soft sets, Ondokuz Mayıs University, The Graduate School of Natural and Applied Sciences Master of Science in Mathematics Department, Samsun.

9. Yavuz, E., (2024). Soft binary piecewise operations and their properties, Amasya University, The Graduate School of Natural and Applied Sciences Master of Science in Mathematics Department, Amasya.
10. Sezgin, A., Sarıalioğlu M, (2024). New soft set operation: Complementary soft binary piecewise theta operation, Journal of Kadirli Faculty of Applied Sciences, (4)(1), 1-33.
11. Sezgin A., Aybek, F., Atagün, A.O. (2023). New soft set operation: Complementary soft binary piecewise intersection operation, Black Sea Journal of Engineering and Science, 6(4) 330-346.
12. Sezgin, A. Aybek, F., Atagün, A.O. (2023). New soft set operation: Complementary soft binary piecewise union operation, Acta Informatica Malaysia, (7) 1, 27-45.
13. Sezgin, A., Çağman, N. (2024). 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.
14. 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.
15. Feng, F., Li, Y.M., Davvaz, B., Ali, M. I., (2010). Soft sets combined with fuzzy sets and rough Sets: A tentative approach, Soft Computing, 14 899–911.
16. 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.
17. 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.
18. 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.
19. Fu, L., Notes on soft set operations, (2011). ARPN Journal of Systems and Softwares, 1, 205-208.
20. Ge, X., Yang S., (2011). Investigations on some operations of soft sets, World academy of Science, Engineering and Technology, 75 1113-1116.
21. Singh, D., Onyeozili, L.A., (2012a). Some conceptual misunderstanding of the fundamentals of soft set theory, ARPN Journal of systems and softwares, 2 (9) 251-254.
22. Singh, D., Onyeozili, L.A., (2012b). Some results on distributive and absorption properties on soft operations, IOSR Journal of mathematics, 4 (2) 18-30.
23. Singh, D., Onyeozili, L.A., (2012c). On some new properties on soft set operations, International Journal of Computer Applications, 59 (4) 39-44.
24. Singh, D., Onyeozili, L.A., (2012d). Notes on soft matrices operations, ARPN Journal of science and technology, 2(9) 861-869.
25. Ping, Z., Qiaoyan, W., (2013). Operations on soft sets revisited, Journal of Applied Mathematics, Volume 2013 Article ID 105752 7 pages.
26. Jayanta, S., (2014). On algebraic structure of soft sets, Annals of Fuzzy Mathematics and Informatics, Volume 7 (6) 1013-1020.
27. Onyeozili, L.A., T.A., Gwary, (2014). A study of the fundamentals of soft set theory, International Journal of Scientific & Technology Research, 3 (4) 132-143.
28. 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.
29. Vandiver, H.S., (1934). Note on a simple type of algebra in which the cancellation law of addition does not hold, Bulletin of the American Mathematical Society, 40, (12) 914–920.
30. Vasanthi T., and Sulochana, N., (2013). On the additive and multiplicative structure of semirings, Annals of Pure and Applied Mathematics, 3 (1) 78–84.
31. Kaya A. and Satyanarayana, M., (1981). Semirings satisfying properties of distributive type, Proceedings of the American Mathematical Society, 82 (3) 341–346.
32. Karvellas, P.H., (1974). Inversive semirings, Journal of the Australian Mathematical Society, 18 (3), 277–288.
33. Goodearl, K.R., (1979). Von Neumann Regular Rings, Pitman, London.
34. Petrich, M., (1973). Introduction to Semiring, Charles E Merrill Publishing Company, Ohio.
35. Reutenauer C. and Straubing H., (1984). Inversion of matrices over a commutative semiring, Journal of Algebra, 88 (2) 350–360.
36. Glazek, K., (2002). A guide to litrature on semirings and their applications in mathematics and ınformation sciences: with complete bibliography, Kluwer Acad. Publ., Nederland.
37. Kolokoltsov, V.N., Maslov, V.P., (1997). Idempotent Analysis and its Applications, in: Mathematics and its Applications, 401, Kluwer.
38. Aho, A.W, Ullman, J.D., Introduction to automata theory, languages and computation, Addison Wesley, Reading, MA.
39. Beasley, L.B., Pullman, N.G., (1979). Operators that preserves semiring matrix functions, Linear Algebra Appl. 99 (1988) 199–216.
40. Beasley, L.B., Pullman, N.G., (1992). Linear operators strongly preserving idempotent matrices over semirings, Linear Algebra Appl. 160 217–229.
41. Ghosh, S., (1996). Matrices over semirings, Inform. Sci. 90 221–230.
42. Wechler, W., (1978). The concept of fuzziness in automata and language theory, Akademic Verlag, Berlin, 1978.
43. Golan, J.S., (1999) Semirings and their applications, Kluwer Acad. Publication.
44. Hebisch, U., Weinert, H.J., (1998). Semirings: algebraic theory and applications in the computer science, World Scientific.
45. Mordeson, J.N., Malik, D.S., (2002). Fuzzy automata and languages, theory and applications, in: computational mathematics series, Chapman and Hall, CRC, Boca Rato.