RESTRICTED AND EXTENDED THETA OPERATIONS OF SOFT SETS: NEW RESTRICTED AND EXTENDED SOFT SET OPERATIONS
Abstract
Since its introduction by Molodtsov in 1999, soft set theory has gained widespread recognition as a method for addressing uncertainty-related issues and modeling uncertainty. It has been used to solve several theoretical and practical issues. Since its introduction, the central idea of the theory-soft set operations-has captured the attention of scholars. Numerous limited and expanded businesses have been identified, and their attributes have been scrutinized thus far. We present a detailed analysis of the fundamental algebraic properties of our proposed restricted theta and extended theta operations, which are unique restricted and extended soft set operations. We also investigate these operations’ distributions over various kinds of soft set operations. We demonstrate that, when coupled with other types of soft set operations, the extended theta operation forms numerous significant algebraic structures, such as semirings in the collection of soft sets over the universe, by taking into account the algebraic properties of the extended theta operation and its distribution rules. This theoretical subject is very important from both a theoretical and practical perspective since soft sets' operations form the foundation for numerous applications, including cryptology and decision-making procedures.
References
Akbulut, E. 2024. New Type of Extended Operations of Soft Sets: Complementary Extended Lambda and Difference Operation. (Master Thesis), Amasya University the Graduate School of Natural and Applied Sciences, Amasya.
Ali, M. I., Feng, F., Liu, X., Min, W.K. & Shabir, M. 2009. On some new operations in soft set theory, Computers Mathematics with Applications, 57(9), pp. 1547-1553.
Ali, M. I., Shabir, M. & Naz M. 2011. Algebraic structures of soft sets associated with new operations, Computers and Mathematics with Applications, 61(9), pp. 2647-2654.
Aybek, F. 2024. New restricted and extended soft set operations. (Master Thesis), Amasya University The Graduate School of Natural and Applied Sciences, Amasya.
Birkhoff, G. 1967. Lattice theory, American Mathematical Society, Providence, Rhode.
Çağman, N. 2021. Conditional complements of sets and their application to group theory, Journal of New Results in Science, 10(3), pp. 67-74.
Çağman, N., Çıtak, F. & Aktaş, H. 2012. Soft int-group and its applications to group theory, Neural Computing and Applications, 2, pp. 151-158.
Clifford, A. H. 1954. Bands of semigroups, Proceedings of the American Mathematical Society, 5(3), pp. 499-504.
Eren, Ö.F. & Çalışıcı, H. 2019. On some operations of soft sets, The Fourth International Conference on Computational Mathematics and Engineering Sciences (CMES 2019), Antalya, Türkiye.
Ge, X. & Yang, S. 2011. Investigations on some operations of soft sets, World Academy of Science, Engineering and Technology, 75, pp. 1113-1116.
Hoorn, W. G. V. & Rootselaar, V. B. 1967. Fundamental notions in the theory of seminearrings, Compositio Mathematica, 18(1-2), pp. 65-78.
Husain, S. & Shivani K. M. 2018. A study of propertıes of soft set and its applications, International Research Journal of Engineering and Technology, 5(1), pp. 363-372.
Jabir, N. J., Abdulhasan, A. M. R. & Hassan, A. F. 2024. Algebraic properties of operations on n ary relation soft set, Applied Geomatics, 16(1), pp. 41-45.
Jana, C., Pal, M., Karaaslan, F. & Sezgin, A. 2019. (α, β)-soft intersectional rings and ideals with their applications, New Mathematics and Natural Computation, 15(2), pp. 333-350.
Kilp, M., Knauer, U. & Mikhalev, A. V. 2001. Monoids, acts and categories (Second edition). Berlin: De Gruyter Expositions in Mathematics, (29).
Lawrence S. & Manoharan R. 2023. New approach to De Morgan’s laws for soft sets via soft ideals, Applied Nanoscience, 13, pp. 2585-2592.
Liu, X. Y., Feng F. & Jun, Y. B. 2012. A note on generalized soft equal relations, Computers and Mathematics with Applications, 64(4), pp. 572-578.
Mahmood, T., Rehman, Z. U. & Sezgin, A. 2018. Lattice ordered soft near rings, Korean Journal of Mathematics, 26(3), pp. 503-517.
Maji, P. K, Biswas, R. & Roy, A. R. 2003. Soft set theory, Computers and Mathematics with Applications, 45(1), pp. 55-562.
Molodtsov, D. 1999. Soft set theory-first results, Computers and Mathematics with Applications, 37(4-5), pp. 19-31.
Muştuoğlu, E., Sezgin, A. & Türk, Z. K. 2016. Some characterizations on soft uni-groups and normal soft uni-groups, International Journal of Computer Applications, 155(10), pp. 1-8.
Neog,T. J. & Sut D. K. 2011. A new approach to the theory of soft set, International Journal of Computer Applications, 32(2), pp. 1-6.
Sezgin, A. 2016. A new approach to semigroup theory I: Soft union semigroups, ideals and bi-ideals, Algebra Letters, 2016(3), pp. 1-46.
Pant, S., Dagtoros, K., Kholil, M. I. & Vivas A. 2024. Matrices: Peculiar determinant property, Optimum Science Journal, 1, pp. 1-7.
Pei, D. & Miao, D. 2005. From soft sets to information systems, IEEE International Conference on Granular Computing, 2, pp. 617-621.
Sezer, A. S. 2014. Certain Characterizations of LA-semigroups by soft sets, Journal of Intelligent and Fuzzy Systems, 27(2), pp. 1035-1046.
Sezer, A. S., Çağman, N. & Atagün, A. O. 2015. Uni-soft substructures of groups, Annals of Fuzzy Mathematics and Informatics, 9(2), pp. 235-246.
Sezgin, A. 2018. A new view on AG-groupoid theory via soft sets for uncertainty modeling, Filomat, 32(8), 2995-3030.
Sezgin, A., Ahmad S., Mehmood A. 2019. A new operation on soft sets: Extended difference of soft sets, Journal of New Theory, 27, pp. 33-42.
Sezgin, A. & Aybek, F.N. 2023. A new soft set operation: Complementary soft binary piecewise gamma operation, Matrix Science Mathematic (1), pp. 27-45.
Sezgin, A., Aybek, F. N. & tagün A. O. 2023a. A new soft set operation: Complementary soft binary piecewise intersection operation, Black Sea Journal of Engineering and Science, 6(4), pp. 330-346.
Sezgin, A., Aybek, F. N. & Güngör, N. B. 2023b. A new soft set operation: Complementary soft binary piecewise union operation, Acta Informatica Malaysia, (7)1, pp. 38-53.
Sezgin, A., Atagün, A. O., Çağman, N. & Demir H. 2022. On near-rings with soft union ideals and applications, New Mathematics and Natural Computation, 18(2), pp. 495-511.
Sezgin, A. & Atagün, A. O. 2011. On operations of soft sets. Computers and Mathematics with Applications, 61(5), pp. 1457-1467.
Sezgin, A. & Atagün, A.O. 2023. A new soft set operation: Complementary soft binary piecewise plus operation, Matrix Science Mathematic, 7(2), pp. 125-142.
Sezgin, A. & 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), pp. 58-94.
Sezgin, A., Çağman, N. & Atagün, A. O. 2017. A completely new view to soft intersection rings via soft uni-int product, Applied Soft Computing, 54, pp. 366-392.
Sezgin, A., Çağman, N., Atagün, A. O. & Aybek, F. 2023c. Complemental binary operations of sets and their pplication to group theory. Matrix Science Mathematic, 7(2), pp. 114-121.
Sezgin, A & Çalışıcı, H. 2024. A comprehensive study on soft binary piecewise difference operation, Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler, 12(1), pp. 32-54.
Sezgin, A. & Dagtoros, K. 2023. Complementary soft binary piecewise symmetric difference operation: A novel soft set operation, Scientific Journal of Mehmet Akif Ersoy University, 6(2), pp. 31-45.
Sezgin, A. & Demirci, A.M. 2023. A new soft set operation: complementary soft binary piecewise star operation, Ikonion Journal of Mathematics, 5(2), pp. 24-52.
Sezgin, A. & Sarıalioğlu M. 2024. A new soft set operation complementary soft binary piecewise theta operation, Journal of Kadirli Faculty of Applied Sciences, 4(1), pp. 1-33.
Sezgin, A. & Yavuz, E. 2023a. A new soft set operation: complementary soft binary piecewise lambda operation, Sinop University Journal of Natural Sciences, 8(2), pp. 101-133.
Sezgin, A. & Yavuz, E. 2023b. A new soft set operation: Soft binary piecewise symmetric difference operation, Necmettin Erbakan University Journal of Science and Engineering, 5(2), pp. 189-208.
Singh, D. & Onyeozili, I. A. 2012a. Notes on soft matrices operations, ARPN Journal of Science and Technology, 2(9), pp. 861-869.
Singh, D. & Onyeozili, I. A. 2012b. On some new properties on soft set operations, International Journal of Computer Applications, 59(4), pp. 39-44.
Singh, D. & Onyeozili, I. A. 2012c. Some conceptual misunderstanding of the fundamentals of soft set theory, ARPN Journal of Systems and Software, 2(9), pp. 251-254.
Singh, D. & Onyeozili, I. A. 2012d. Some Results on Distributive and Absorption Properties on Soft Operations. IOSR Journal of Mathematics, 4(2), pp. 18-30.
Stojanovic, N. S. 2021. A new operation on soft sets: Extended symmetric difference of soft sets, Military Technical Courier, 69(4), pp. 779-791.
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), pp. 914-920.
Yang, C. F. 2008. “A note on: “Soft set theory” [Computers and Mathematics with Applications, 45(4-5) (2003) 555–562],” Computers and Mathematics with Applications, 56(7), pp. 1899-1900.
Yavuz E., 2024. Soft binary piecewise operations and their properties, (Master Thesis), Amasya University the Graduate School of Natural and Applied Sciences, Amasya.
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