Soft union-intersection product and soft symmetric difference-union product of groups
DOI:
https://doi.org/10.31181/jdaic10012122025sKeywords:
soft subsets, soft equalities, soft union-intersection product, soft symmetric difference-union productAbstract
In settings where parametric variability is present, soft set theory has evolved into a robust and versatile mathematical framework for modeling and analyzing uncertainty. Central to this framework are the operations and product constructions on soft sets, which together provide a powerful algebraic infrastructure for addressing complex parameter-dependent problems. Formally, it is shown that under the union operation, the collection of all soft sets defined over a fixed parameter set forms a bounded semilattice, thereby supplying essential algebraic structure and coherence. Through careful analysis, it is further demonstrated that the algebraic system consisting of all soft sets over a fixed parameter set endowed with a group structure, equipped with the union operation and the proposed product, satisfies the axiomatic framework of a hemiring. This structural characterization yields two major theoretical implications: it strengthens the algebraic foundations of soft set theory and lays the groundwork for constructing a soft group theory analogous to its classical counterpart. A new product on soft sets, termed the soft symmetric difference–union product, is then introduced for the case in which the parameter set carries a group structure. This operation is examined in detail from both axiomatic and structural perspectives, with particular attention to its compatibility with soft equality and soft subsethood. The approach presented here makes a substantial contribution to the ongoing algebraic refinement and theoretical advancement of soft set theory, as the formal development of soft algebraic systems fundamentally relies on rigorously defined operations and product constructions.
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References
Abbas, M., Ali, B., & Romaguera, S. (2014). On generalized soft equality and soft lattice structure. Filomat, 28(6), 1191-1203.
Abbas, M., Ali, M. I., & Romaguera, S. (2017). Generalized operations in soft set theory via relaxed conditions on parameters. Filomat, 31(19), 5955-5964.
Aktas, H., & Çağman, N. (2007). Soft sets and soft groups. Information Science, 177(13), 2726-2735.
Alcantud, J. C. R., Khameneh, A. Z., Santos-García, G., & Akram, M. (2024). A systematic literature review of soft set theory. Neural Computing and Applications, 36, 8951–8975.
Ali, B., Saleem, N., Sundus, N., Khaleeq, S., Saeed, M., & George, R. (2022). A contribution to the theory of soft sets via generalized relaxed operations. Mathematics, 10(15), 26-36.
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.
Ali, M. I., Mahmood, M., Rehman, M.U., & Aslam, M. F. (2015). On lattice ordered soft sets. Applied Soft Computing, 36, 499-505.
Ali, M. I., Shabir, M., & Naz, M. (2011). Algebraic structures of soft sets associated with new operations. Computers and Mathematics with Applications, 61(9), 2647-2654.
Al-shami, T. M. (2019). Investigation and corrigendum to some results related to g-soft equality and gf -soft equality relations. Filomat, 33(11), 3375-3383.
Al-shami, T. M., & El-Shafei, M. (2020). T-soft equality relation. Turkish Journal of Mathematics, 44(4), 1427-1441.
Atagün, A. O., & Sezer, A. S. (2015). Soft sets, soft semimodules and soft substructures of semimodules. Mathematical Sciences Letters, 4(3), 235-242.
Atagün, A. O., & Sezgin, A. (2018). A new view to near-ring theory: Soft near-rings. South East Asian Journal of Mathematics & Mathematical Sciences, 14(3), 1-14.
Atagün, A.O., & Sezgin, A. (2017). Int-soft substructures of groups and semirings with applications. Applied Mathematics & Information Sciences, 11(1), 105-113.
Atagün, A.O., Kamacı, H., Taştekin, İ., & Sezer, A.S. (2019). P-properties in near-rings. Journal of Mathematical and Fundamental Sciences, 51(2), 152-167.
Çağman, N., & Enginoğlu, S. (2010). Soft set theory and uni-int decision making. European Journal of Operational Research, 207(2), 848-855.
Clifford, A. H. (1954). Bands of semigroups. Proceedings of the American Mathematical Society, 5(3), 499–504.
Feng, F., & Li, Y. (2013). Soft subsets and soft product operations. Information Sciences, 232(20), 44-57.
Feng, F., Jun, Y. B., & Zhao, X. (2008). Soft semirings. Computers and Mathematics with Applications, 56(10), 2621-2628.
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.
Fu, L. (2011). Notes on soft set operations. ARPN Journal of Systems and Software, 1(6), 205-208.
Ge, X., & Yang, S. (2011). Investigations on some operations of soft sets. World Academy of Science, Engineering and Technology, 75, 1113-1116.
Gulistan, M., Feng, F., Khan, M., & Sezgin, A. (2018). Characterizations of right weakly regular semigroups in terms of generalized cubic soft sets. Mathematics, 6(12), 293.
Gulistan, M., Shahzad, M. (2014). On soft KU-algebras. Journal of Algebra, Number Theory: Advances and Applications, 11(1), 1-20.
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), 333–350.
Jiang, Y., Tang, Y., Chen, Q., Wang, J., & Tang, S. (2010). Extending soft sets with description logics. Computers and Mathematics with Applications, 59(6), 2087-2096.
Jun, Y. B., & Yang, X. (2011). A note on the paper “Combination of interval-valued fuzzy set and soft set” [Comput. Math. Appl. 58 (2009) 521–527]. Computers and Mathematics with Applications, 61(5), 1468-1470.
Karaaslan, F. (2019). Some properties of AG*-groupoids and AG-bands under SI-product Operation. Journal of Intelligent and Fuzzy Systems, 36(1), 231-239.
Kaygisiz, K. (2012). On soft int-groups. Annals of Fuzzy Mathematics and Informatics, 4(2), 363–375.
Khan, M., Ilyas, F., Gulistan, M., & Anis, S. (2015). A study of soft AG-groupoids, Annals of Fuzzy Mathematics and Informatics, 9(4), 621–638.
Liu, X., Feng, F., & Jun, Y. B. (2012). A note on generalized soft equal relations. Computers and Mathematics with Applications, 64(4), 572-578.
Mahmood, T., Waqas, A., & Rana, M. A. (2015). Soft intersectional ideals in ternary semiring. Science International, 27(5), 3929-3934.
Maji, P. K., Biswas, R., & Roy, A. R. (2003). Soft set theory. Computers and Mathematics with Application, 45(1), 555-562.
Memiş, S.(2022). Another view on picture fuzzy soft sets and their product operations with soft decision-making. Journal of New Theory, 38, 1-13.
Molodtsov, D. (1999). Soft set theory. Computers and Mathematics with Applications, 37(1), 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), 1-8.
Neog, I. J & Sut, D. K. (2011). A new approach to the theory of softset. International Journal of Computer Applications, 32(2), 1-6.
Onyeozili, I. A., & Gwary T. M. (2014). A study of the fundamentals of soft set theory. International Journal of Scientific & Technology Research, 3(4), 132-143.
Pei, D., & Miao, D. (2005). From soft sets to information systems. In Hu, X., Liu, Q., Skowron, A., Lin, T. Y., Yager, R. R., & Zhang, B (Eds.), 2005 IEEE International Conference on Granular Computing, vol. 2 (pp. 617-621). Beijing: IEEE.
Qin, K., & Hong, Z. (2010). On soft equality. Journal of Computational and Applied Mathematics, 234(5), 1347-1355.
Riaz, M., Hashmi, M. R., Karaaslan, F., Sezgin, A., Shamiri, M. M. A. A., & Khalaf, M. M. (2023). Emerging trends in social networking systems and generation gap with neutrosophic crisp soft mapping. CMES-computer modeling in engineering and sciences, 136(2), 1759-1783.
Sen, J. (2014).On algebraic structure of soft sets. Annals of Fuzzy Mathematics and Informatics, 7(6), 1013-1020.
Sezer, A. S. (2012). A new view to ring theory via soft union rings, ideals and bi-ideals. Knowledge-Based Systems, 36, 300–314.
Sezer, A. S., & Atagün, A. O. (2016). A new kind of vector space: soft vector space. Southeast Asian Bulletin of Mathematics, 40(5), 753-770.
Sezer, A. S., Çağman, N., & Atagün, A. O. (2014). Soft intersection interior ideals, quasi-ideals and generalized bi-ideals; A new approach to semigroup theory II. Journal of Multiple-Valued Logic and Soft Computing, 23(1-2), 161-207.
Sezer, A. S., Çağman, N., Atagün, A. O., Ali, M. I., & Türkmen, E. (2015). Soft intersection semigroups, ideals and bi-Ideals; A New application on semigroup theory I. Filomat, 29(5), 917-946.
Sezer, A., Atagün, A. O., & Çağman, N. (2013). A new view to N-group theory: soft N-groups. Fasciculi Mathematici, 51, 123-140.
Sezer, A., Atagün, A. O., & Çağman, N. (2017). N-group SI-action and its applications to N-group theory. Fasciculi Mathematici, 52, 139-153.
Sezgin, A. (2016). A new approach to semigroup theory I: Soft union semigroups, ideals and bi-ideals. Algebra Letters, 2016(3), 1-46.
Sezgin, A., & İlgin, A. (2024). Soft intersection almost subsemigroups of semigroups. International Journal of Mathematics and Physics, 15(1), 13-20.
Sezgin, A., Atagün, A. O. & Çağman N. (2025a). A complete study on and-product of soft sets. Sigma Journal of Engineering and Natural Sciences, 43(1), 1−14.
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, 366-392.
Sezgin, A., Durak, İ., & Ay, Z. (2025b). Some new classifications of soft subsets and soft equalities with soft symmetric difference-difference product of groups. Amesia, 6(1), 16-32.
Singh, D., & Onyeozili, I. A. (2012a). Notes on soft matrices operations. ARPN Journal of Science and Technology, 2(9), 861-869.
Singh, D., & Onyeozili, I. A. (2012c). Some results on distributive and absorption properties on soft operations. IOSR Journal of Mathematics (IOSR-JM), 4(2), 18-30.
Singh, D., & Onyeozili, I. A. (2012d). Some conceptual misunderstanding of the fundamentals of soft set theory. ARPN Journal of Systems and Software, 2(9), 251-254.
Singh, D., & Onyeozili, I. A.(2012b). On some new properties on soft set operations. International Journal of Computer Applications, 59(4), 39-44.
Stojanovic, N. S. (2021). A new operation on soft sets: Extended symmetric difference of soft sets. Military Technical Courier/Vojnotehnički glasnik, 69(4), 779-791.
Ullah, A., Karaaslan, F., & Ahmad, I. (2018). Soft uni-abel-grassmann's groups. European Journal of Pure and Applied Mathematics, 11(2), 517-536.
Vandiver, H. S. (1934). Note on a simple type of algebra in which the cancellation law of addition does not hold. Bulletin (New Series) of the American Mathematical Society, 40(12), 914–920.
Yang, C. F. (2008). A note on: Soft set theory. Computers and Mathematics with Applications, 56(7), 1899-1900.
Zadeh, L. A. (1965). Fuzzy sets. Information Control, 8(3), 338-353.
Zhu, P., & Wen, Q. (2013). Operations on soft sets revisited. Journal of Applied Mathematics, 2013, 105752.
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