Soft union-star product of groups

Year 2026, Volume: 7 Issue: 1, 1 - 8, 30.01.2026

İbrahim Durak , Aslıhan Sezgin

Abstract

Soft set theory constitutes a logically rigorous and algebraically expressive formalism for representing systems permeated by ambiguity, epistemic uncertainty, and parameter-dependent variability. In this context, the present study introduces the soft union–star product, a novel binary operation defined on soft sets whose parameter do-mains are endowed with an intrinsic group-theoretic structure. Formulated within a strictly axiomatic framework, the operation is proven to exhibit full compatibility with generalized formulations of soft subsethood and soft equality. A comprehensive algebraic analysis is conducted to establish its core structural invariants, including closure, associativity, commutativity, and idempotency. Moreover, the operation’s behavior is rigorously characterized in relation to the identity and absorbing elements, as well as its interaction with the null and absolute soft sets. The findings confirm that the soft union–star product satisfies all algebraic conditions imposed by group-parameterized domains, thereby generating a robust and internally coherent algebraic structure over the universe of soft sets. Beyond its foundational significance, the operation meaningfully enriches the operational landscape of soft set theory and provides a formal platform for advancing a generalized soft group theory. Its structural compatibility with key relational constructs—particularly generalized soft equalities and inclusion hierarchies—underscores its potential utility in diverse application domains, including algebraic abstraction, uncertainty-sensitive classification, and multi-criteria decision-making. As such, this work contributes not only a substantive theoretical advancement but also a mathematically principled pathway for practical deployment in uncertainty-aware systems.

Keywords

Soft sets , Soft subsets , Soft equalities , Soft union-star product

References

• L. A. Zadeh, "Fuzzy sets", Information and Control, 8(3), pp. 338–353, 1965. https://doi.org/10.1016/S0019-9958(65)90241-X
• D. Molodtsov, “Soft set theory-first results”, Computers and Mathematics with Applications, 37 (1), pp. 19-31, 1999. https://doi.org/10.1016/S0898-1221(99)00056-5
• P.K. Maji, R. Bismas, A.R. Roy, “Soft set theory”, Computers and Mathematics with Applications, 45 (1), pp. 555-562, 2003. https://doi.org/10.1016/S0898-1221(02)00216-X
• D. Pei and D. Miao, “From Soft Sets to Information Systems”, In: Proceedings of Granular Computing IEEE, 2, pp. 617-621, 2005. DOI: 10.1109/GRC.2005.1547365
• M. I. Ali, F. Feng, X. Liu, W. K. Min., M. Shabir, “On some new operations in soft set theory”, Computers and Mathematics with Applications, 57(9), pp. 1547-1553, 2009. https://doi.org/10.1016/j.camwa.2008.11.009
• C. F. Yang, "A note on: soft set theory", Computers and Mathematics with Applications, 56(7), pp. 1899–1900, 2008. https://doi.org/10.1016/j.camwa.2008.03.019
• F. Feng, Y. M. Li, B. Davvaz, M. I. Ali, "Soft sets combined with fuzzy sets and rough sets: a tentative approach", Soft Computing, 14, pp. 899–911, 2010. https://doi.org/10.1007/s00500-009-0465-6
• Y. Jiang, Y. Tang, Q. Chen, J. Wang, S. Tang, "Extending soft sets with description logics", Computers and Mathematics with Applications, 59(6), pp. 2087–2096, 2010. https://doi.org/10.1016/j.camwa.2009.12.014
• M. I. Ali, M. Shabir, M. Naz, "Algebraic structures of soft sets associated with new operations", Computers and Mathematics with Applications, 61(9), pp. 2647–2654, 2011. https://doi.org/10.1016/j.camwa.2011.03.011
• I. J. Neog, D. K. Sut, "A new approach to the theory of soft set", International Journal of Computer Applications, 32(2), pp. 1–6, 2011. DOI=10.5120/3874-5415
• L. Fu, "Notes on soft set operations", ARPN Journal of Systems and Software, 1, pp. 205–208, 2011.
• X. Ge, S. Yang, "Investigations on some operations of soft sets", World Academy of Science Engineering and Technology, 75, pp. 1113–1116, 2011.
• D. Singh, I. A. Onyeozili, "Notes on soft matrices operations", ARPN Journal of Science and Technology, 2(9), pp. 861–869, 2012.
• D. Singh, I. A. Onyeozili, "On some new properties on soft set operations", International Journal of Computer Applications, 59(4), pp. 39–44, 2012.
• D. Singh, I. A. Onyeozili, "Some results on distributive and absorption properties on soft operations", IOSR Journal of Mathematics, 4(2), pp. 18–30, 2012.
• D. Singh, I. A. Onyeozili, "Some conceptual misunderstanding of the fundamentals of soft set theory", ARPN Journal of Systems and Software, 2(9), pp. 251–254, 2012.
• P. Zhu, Q. Wen, "Operations on soft sets revisited", Journal of Applied Mathematics, 2013, Article ID 105752, 7 pages. https://doi.org/10.1155/2013/105752
• I. A. Onyeozili, T. M. Gwary, "A study of the fundamentals of soft set theory", International Journal of Science and Technology Research, 3(4), pp. 132–143, 2014.
• J. Sen, "On algebraic structure of soft sets", Annals of Fuzzy Mathematics and Informatics, 7(6), pp. 1013–1020, 2014.
• Ö. F. Eren, H. Çalışıcı, “On some operations of soft sets”, The Fourth International Conference on Computational Mathematics and Engineering Sciences, Antalya, 2019.
• N.S. Stojanovic, “A new operation on soft sets: extended symmetric difference of soft sets”, Military Technical Courier, 69(4), pp.779-791, 2021. https://doi.org/10.5937/vojtehg69-33655
• A. Sezgin, F. N. Aybek, “A new soft set operation: Complementary soft binary piecewise gamma operation”, Matrix Science Mathematic, 7 (1), pp. 27-45, 2023. http://doi.org/10.26480/msmk.01.2023.27.45
• A. Sezgin, H. Çalışıcı, “A comprehensive study on soft binary piecewise difference operation”, Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B -Teorik Bilimler, 12 (1), pp. 32-54, 2024. DOI:10.20290/estubtdb.1356881
• A. Sezgin, N. Çağman, A.O. Atagün, F.N Aybek, “Complemental binary operations of sets and their application to group theory,” Matrix Science Mathematic, 7(2), pp. 114-121, 2023. http://doi.org/10.26480/msmk.02.2023.114.121
• A. Sezgin, F.N. Aybek, A.O. Atagün, “A new soft set operation: Complementary soft binary piecewise intersection operation”, Black Sea Journal of Engineering and Science, 6 (4), pp.330-346, 2023. https://doi.org/10.34248/bsengineering.1319873
• A. Sezgin, E. Yavuz, Ş. Özlü, “Insight into soft binary piecewise lambda operation: a new operation for soft sets”, Journal of Umm al-Qura University for Applied Sciences, pp. 1-15, 2025. https://doi.org/10.1007/s43994-024-00187-1
• A. Sezgin, A.M. Demirci, “A new soft set operation: complementary soft binary piecewise star operation”, Ikonion Journal of Mathematics, 5 (2), pp.24-52, 2023. https://doi.org/10.54286/ikjm.1304566
• A. Sezgin, E. Yavuz, “A new soft set operation: Complementary soft binary piecewise lambda operation”, Sinop University Journal of Natural Sciences, 8 (2), pp. 101-133, 2023. https://doi.org/10.33484/sinopfbd.1320420
• A. Sezgin, N. Çağman, “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, 2024. https://doi.org/10.47495/okufbed.1308379
• A. Sezgin, K. Dagtoros, “Complementary soft binary piecewise symmetric difference operation: A novel soft set operation,” Scientific Journal of Mehmet Akif Ersoy University, 6 (2), pp. 31–45, 2023.
• A. Sezgin, E. Yavuz, “A new soft set operation: Soft binary piecewise symmetric difference operation”, Necmettin Erbakan University Journal of Science and Engineering, 5 (2), pp. 189-208, 2023. DOI: 10.47112/neufmbd.2023.18
• A. Sezgin, N. Çağman, "An extensive study on restricted and extended symmetric difference operations of soft sets", Utilitas Mathematica, in press, 2025.
• A. Sezgin, M. Sarıalioğlu, "A new soft set operation: complementary soft binary piecewise theta operation", Journal of Kadirli Faculty of Applied Sciences, 4(2), pp. 325–357, 2024.
• A. Sezgin, M. Sarıalioğlu, "Complementary extended gamma operation: a new soft set operation", Natural and Applied Sciences Journal, 7(1), pp. 15–44, 2024. https://doi.org/10.38061/idunas.1482044
• A. Sezgin, E. Şenyiğit, "A new product for soft sets with its decision-making: soft star-product", Big Data Computing Visions, 5(1), pp. 52–73, 2025. https://doi.org/10.22105/bdcv.2024.492834.1221
• A. Sezgin, F. N. Aybek, " New restricted and extended soft set operations: restricted gamma and extended gamma operations", Big Data and Computing Vision, 4(4), pp. 272–306, 2024.  https://doi.org/10.22105/bdcv.2024.478983.1199
• A. Sezgin, E. Yavuz, "Soft binary piecewise plus operation: a new type of operation for soft sets", Uncertainty Discourse and Applications, 1(1), pp. 79–100, 2024. https://doi.org/10.48313/uda.v1i1.26
• F. Feng, Y. B. Jun, X. Zhao, "Soft semirings", Computers and Mathematics with Applications, 56(10), pp. 2621–2628, 2008. https://doi.org/10.1016/j.camwa.2008.05.011
• K. Qin, Z. Hong, "On soft equality", Journal of Computational and Applied Mathematics, 234(5), pp. 1347–1355, 2010. https://doi.org/10.1016/j.cam.2010.02.028
• Y. B. Jun, X. Yang, "A note on the paper combination of interval-valued fuzzy set and soft set", Computers and Mathematics with Applications, 61(5), pp. 1468–1470, 2011. https://doi.org/10.1016/j.camwa.2010.12.077
• X. Liu, F. Feng, Y. B. Jun, "A note on generalized soft equal relations", Computers and Mathematics with Applications, 64(4), pp. 572–578, 2012. https://doi.org/10.1016/j.camwa.2011.12.052
• F. Feng, Y. Li, "Soft subsets and soft product operations", Information Sciences, 232(20), pp. 44–57, 2013. https://doi.org/10.1016/j.ins.2013.01.001
• M. Abbas, B. Ali, S. Romaguera, "On generalized soft equality and soft lattice structure", Filomat, 28(6), pp. 1191–1203, 2014. https://doi.org/10.2298/FIL1406191A
• M. Abbas, M. I. Ali, S. Romaguera, "Generalized operations in soft set theory via relaxed conditions on parameters", Filomat, 31(19), pp. 5955–5964, 2017. https://doi.org/10.2298/FIL1719955A
• T. M. Al-shami, "Investigation and corrigendum to some results related to g-soft equality and gf-soft equality relations", Filomat, 33(11), pp. 3375–3383, 2019. https://doi.org/10.2298/FIL1911375A
• T. M. Al-shami, M. El-Shafei, "T-soft equality relation", Turkish Journal of Mathematics, 44(4), pp. 1427–1441, 2020. https://doi.org/10.3906/mat-2005-117
• N. Çağman, S. Enginoğlu, "Soft set theory and uni-int decision making", European Journal of Operational Research, 207(2), pp. 848–855, 2010. https://doi.org/10.1016/j.ejor.2010.05.004
• A. S. Sezer, "A new view to ring theory via soft union rings, ideals and bi-ideals", Knowledge-Based Systems, 36, pp. 300–314, 2012. https://doi.org/10.1016/j.knosys.2012.04.031
• A. Sezgin, "A new approach to semigroup theory I: soft union semigroups, ideals and bi-ideals", Algebra Letters, 3, pp. 1–46, 2016.
• K. Kaygisiz, "On soft int-groups", Annals of Fuzzy Mathematics and Informatics, 4(2), pp. 363–375, 2012.
• E. Muştuoğlu, A. Sezgin, Z. K. Türk, "Some characterizations on soft uni-groups and normal soft uni-groups", International Journal of Computer Applications, 155(10), pp. 1–8, 2016. 10.5120/ijca2016912412
• A. S. Sezer, N. Çağman, A. O. Atagün, M. I. Ali, E. Türkmen, "Soft intersection semigroups, ideals and bi-ideals; a new application on semigroup theory I", Filomat, 29(5), pp. 917–946, 2015. https://dx.doi.org/10.2298/FIL1505917S
• A. Sezgin, N. Çağman, A. O. Atagün, "A completely new view to soft intersection rings via soft uni-int product", Applied Soft Computing, 54, pp. 366–392, 2017. https://doi.org/10.1016/j.asoc.2016.10.004
• A. Sezgin, İ. Durak, Z. Ay, "Some new classifications of soft subsets and soft equalities with soft symmetric difference-difference product of groups", Amesia , 6(1), pp. 16-32, 2025. https://doi.org/10.54559/amesia.1730014
• A. Sezgin, N. Çağman, F. Çitak, "α-inclusions applied to group theory via soft set and logic", Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(1), pp. 334-352, 2019. https://doi.org/10.31801/cfsuasmas.420457
• A. Sezgin, M. Orbay, "Analysis of semigroups with soft intersection ideals", Acta Universitatis Sapientiae Mathematics, 14(1), pp. 166–210, 2022.  https://doi.org/10.2478/ausm-2022-0012
• A. Sezgin, A. İlgin, "Soft intersection almost ideals of semigroups", Journal of Innovative Engineering and Natural Science, 4(2), pp. 466-481, 2024. https://doi.org/10.61112/jiens.1464344
• C. Jana, M. Pal, F. Karaaslan, A. Sezgin, "(α, β)-soft intersectional rings and ideals with their applications", New Mathematics and Natural Computation, 15(2), pp. 333–350, 2019. DOI: 10.1142/S179300571950018
• A.O. Atagün, A.S. Sezer, "Soft sets, soft semimodules and soft substructures of semimodules", Mathematical Sciences Letters, 43 (3), pp. 235–242, 2015. http://dx.doi.org/10.12785/msl/040303
• A. S. Sezer, "A new approach to LA-semigroup theory via the soft sets”, Journal of Intelligent and Fuzzy Systems, 26 (5), pp. 2483-2495, 2014. DOI: 10.3233/IFS-130918
• A. Sezgin, A. İlgin, A. O. Atagün, "Soft intersection almost tri-bi-ideals of semigroups", Science & Technology Asia, 29(4), pp. 1–13, 2024. doi: 10.14456/scitechasia.2024.66
• A. O. Atagün, A. Sezgin, "Soft subnear-rings, soft ideals and soft n-subgroups of near-rings", Mathematical Sciences Letters, 7(1), pp. 37–42, 2018. DOI: 10.18576/msl/070106
• A. Sezgin, "A new view on AG-groupoid theory via soft sets for uncertainty modeling", Filomat, 32(8), pp. 2995–3030, 2018. https://doi.org/10.2298/FIL1808995S
• A. O. Atagün, A. Sezgin, "More on prime, maximal and principal soft ideals of soft rings", New Mathematics and Natural Computation, 18(1), pp. 195-207, 2022. https://doi.org/10.1142/S1793005722500119
• A. O. Atagün. A. Sezgin, "Int-soft substructures of groups and semirings with applications", Applied Mathematics & Information Sciences, 11(1), pp. 105–113, 2017. http://dx.doi.org/10.18576/amis/110113
• M. Gulistan, F. Feng, M. Khan, A. Sezgin, "Characterizations of right weakly regular semigroups in terms of generalized cubic soft sets", Mathematics, 6, 293, 2018. https://doi.org/10.3390/math6120293
• F. Karaaslan, "Some properties of AG*-groupoids and AG-bands under SI-product operation", Journal of Intelligent & Fuzzy Systems, 36(1), pp. 231–239, 2019. https://doi.org/10.3233/JIFS-181208
• A. Sezgin, A. İlgin, "Soft intersection almost bi-quasi ideals of semigroups", Soft computing fusion with applications, 1(1), pp. 28–43, 2024. https://doi.org/10.22105/scfa.v1i1.26
• A. Khan, I. Izhar, A. Sezgin, "Characterizations of Abel Grassmann's groupoids by the properties of their double-framed soft ideals", International Journal of Analysis and Applications, 15(1), pp. 62–74, 2017. https://hdl.handle.net/20.500.12450/1095
• A. O. Atagün, H. Kamacı, İ. Taştekin, A. Sezgin, " P-properties in near-rings", Journal of Mathematical and Fundamental Science, 51(2), pp. 152–167, 2019. https://doi.org/10.5614/j.math.fund.sci.2019.51.2.5
• T. Manikantan, P. Ramasany, A. Sezgin, "Soft quasi-ideals of soft near-rings", Sigma Journal of Engineering and Natural Sciences, 41(3), pp. 565–574, 2023.  DOI: 10.14744/sigma.2023.00062
• S. Memiş, "Another view on picture fuzzy soft sets and their product operations with soft decision-making", Journal of New Theory, 38, pp. 1–13, 2022. https://doi.org/10.53570/jnt.1037280
• M. Riaz, M. R. Hashmi, F. Karaaslan, A. Sezgin, M. M. A. A. Shamiri, M. M. Khalaf, "Emerging trends in social networking systems and generation gap with neutrosophic crisp soft mapping", Computational Modeling in Engineering & Sciences, 136(2), pp. 1759–1783, 2023. https://doi.org/10.32604/cmes.2023.023327
• A. S. Sezer, A. O. Atagün, "A new kind of vector space: soft vector space", Southeast Asian Bulletin of Mathematics, 40(5), pp. 753–770, 2016.
• A. Sezer, A. O. Atagün, N. Çağman, "N-group SI-action and its applications to N-group theory", Fasciculi Mathematici, 52, pp. 139–153, 2017.
• A. Sezer, A. O. Atagün, N. Çağman, "A new view to N-group theory: soft N-groups", Fasciculi Mathematici, 51, pp. 123–140, 2013.
• A. Sezgin, A. İlgin, "Soft intersection almost subsemigroups of semigroups", International Journal of Mathematics and Physics, 15(1), pp. 13–20, 2024. https://doi.org/10.26577/ijmph.2024v15i1a2
• A. Sezgin, A. O. Atagün, N. Çağman, "A complete study on and-product of soft sets", Sigma Journal of Engineering and Natural Sciences, 43(1), pp. 1–14, 2025. DOI: 10.14744/sigma.2025.00002
• M. Tunçay, A. Sezgin, "Soft union ring and its applications to ring theory", International Journal of Computer Applications, 151(9), pp. 7–13, 2016. 10.5120/ijca2016911867
• A. O. Atagün, A. Sezgin, "A new view to near-ring theory: Soft near-rings", South East Asian Journal of Mathematics & Mathematical Sciences, 14(3), pp. 1-14, 2018.