Grupların esnek simetrik fark tümleyen-gama çarpımı

Yıl 2025, Cilt: 8 Sayı: 2, 74 - 85, 31.12.2025

İbrahim Durak , Aslıhan Sezgin

https://doi.org/10.55117/bufbd.1751678

Öz

Matematiksel olarak tutarlı ve cebirsel açıdan güçlü bir biçimsel sistem olan esnek küme teorisi, karmaşık karar verme ve bilgi sistemlerinde var olan belirsizlik, belirsiz tanımlılık ve parametreye bağlı değişkenliği yakalamaya çok uygundur. Bu teorik çerçeve içinde, bu çalışma, parametre uzayları grup teorisine dayalı özelliklerle yapılandırılmış esnek kümeler üzerinde tanımlanan yeni bir ikili işlem olan esnek simetrik fark tümleyen–gamma çarpımını tanıtmaktadır. Sıkı aksiyomatik temeller üzerine formüle edilen bu işlem, genelleştirilmiş esnek altküme ilişkisi ve esnek eşitlikle tam uyumlu olduğu gösterilerek, esnek küme teorisinin daha geniş cebirsel yapısı içinde yapısal entegrasyonunu sağlamaktadır. İşlemin kapalılık, birleşme, değişme ve idempotentlik gibi temel cebirsel özelliklerini tanımlamak ve doğrulamak amacıyla kapsamlı bir cebirsel inceleme yapılmıştır. Ayrıca, işlemin grup parametreli alanların dayattığı kısıtlamalar çerçevesinde birim ve yutan elemanlar ile boş ve mutlak esnek kümelerle olan etkileşimleri titizlikle analiz edilmiştir. Elde edilen bulgular, esnek simetrik fark tümleyen–gamma çarpımının grup teorisi temelli parametre yapıları tarafından dikte edilen tüm cebirsel koşulları sağladığını ve böylece esnek kümeler evreni üzerinde tutarlı ve sağlam bir cebirsel çerçeve oluşturduğunu doğrulamaktadır. Temel değeri yanında, bu işlem esnek küme teorisinin yapısal çeşitliliğini önemli ölçüde zenginleştirmekte ve grup temelli parametrelerle indekslenen esnek kümelerin klasik cebirsel işlemleri taklit ettiği genelleştirilmiş bir esnek grup teorisi inşasına yönelik biçimsel bir yol sunmaktadır. Genelleştirilmiş esnek eşitlik kavramlarıyla olan tutarlılığı ve katmanlı esnek içerme hiyerarşilerine sorunsuz entegrasyonu, işlemin cebirsel derinliğini ve uygulanabilirliğini daha da vurgulamaktadır. Buna bağlı olarak, bu çalışma yalnızca önemli bir cebirsel yenilik sunmakla kalmayıp, aynı zamanda belirsizlik yönetimi, soyut cebirsel modelleme ve çok kriterli karar verme sistemleri gerektiren alanlara esnek küme teorisinin genişletilmesi için prensip temelli bir platform sağlamaktadır.

Anahtar Kelimeler

Esnek kümeler , Esnek alt kümeler , Esnek eşitlikler , esnek simetrik fark tümleyen-gama çarpımı

Soft symmetric difference complement-gamma product of groups

Öz

As a mathematically consistent and algebraically potent formal system, soft set theory is well-suited for capturing the inherent uncertainty, vagueness, and parameter-based variability in complex decision-making and information systems. Within this theoretical framework, the present study introduces the soft symmetric difference complement–gamma product, a novel binary operation defined on soft sets whose parameter spaces are intrinsically structured by group-theoretic properties. Formulated through a rigorous axiomatic foundation, the operation is shown to maintain full compatibility with generalized soft subsethood and soft equality, thereby ensuring its structural integration within the broader algebraic fabric of soft set theory. A comprehensive algebraic examination is conducted to identify and verify the operation’s key structural properties, including closure, associativity, commutativity, and idempotency. The operation’s interactions with identity and absorbing elements, as well as with the null and absolute soft sets, are rigorously analyzed within the constraints imposed by group-parameterized domains. The findings confirm that the soft symmetric difference complement–gamma product satisfies all required algebraic conditions dictated by group-theoretic parameter structures, resulting in a coherent and robust algebraic framework over the universe of soft sets. In addition to its foundational value, the operation substantially enriches the structural landscape of soft set theory and offers a formal pathway toward the construction of a generalized soft group theory, where soft sets indexed by group-based parameters emulate classical algebraic operations. Its consistency with generalized notions of soft equality and its seamless integration into layered soft inclusion hierarchies further emphasize its algebraic depth and applicability. Accordingly, this work not only delivers a significant algebraic innovation but also provides a principled platform for extending soft set theory to fields that demand formal mechanisms for handling uncertainty, abstract algebraic modeling, and multi-criteria decision-making frameworks.

Anahtar Kelimeler

Soft sets , Soft subsets , Soft equalities , Soft symmetric difference complement-gamma

Kaynakça

• L. A. Zadeh, “Fuzzy sets”. Inf Control, vol. 8, no. 3, pp. 338–353, 1965.
• D. Molodtsov, “Soft set theory”. Comput Math Appl, vol. 37, no. 1, pp. 19–31, 1999.
• P. K. Maji, R. Biswas, A. R. Roy, “Soft set theory”. Comput Math Appl, vol. 45, no. 1, pp. 555–562, 2003.
• D. Pei, D. Miao, “From soft sets to information systems”. Granular Computing, IEEE, pp. 617–621, 2005.
• M. I. Ali, F. Feng, X. Liu, W. K. Min, M. Shabir, “On some new operations in soft set theory”. Comput Math Appl, vol. 57, no. 9, pp. 1547–1553, 2009.
• C. F. Yang, “A note on: soft set theory”. Comput Math Appl, vol. 56, no. 7, pp. 1899–1900, 2008.
• F. Feng, Y. M. Li, B. Davvaz, M. I. Ali, “Soft sets combined with fuzzy sets and rough sets: a tentative approach”. Soft Comput, vol. 14, pp. 899–911, 2010.
• Y. Jiang, Y. Tang, Q. Chen, J. Wang, S. Tang, “Extending soft sets with description logics”. Comput Math Appl, vol. 59, no. 6, pp. 2087–2096, 2010.
• M. I. Ali, M. Shabir, M. Naz, “Algebraic structures of soft sets associated with new operations”. Comput Math Appl, vol. 61, no. 9, pp. 2647–2654, 2011.
• I. J. Neog, D. K. Sut, “A new approach to the theory of softset”. Int J Comput Appl, vol. 32, no. 2, pp. 1–6, 2011.
• L. Fu, “Notes on soft set operations”. ARPN J Syst Softw, vol. 1, pp. 205–208, 2011.
• X. Ge, S. Yang, “Investigations on some operations of soft sets”. World Acad Sci Eng Technol, vol. 75, pp. 1113–1116, 2011.
• D. Singh, I. A. Onyeozili, “Notes on soft matrices operations”. ARPN J Sci Technol, vol. 2, no. 9, pp. 861– 869, 2012.
• D. Singh, I. A. Onyeozili, “On some new properties on soft set operations”. Int J Comput Appl, vol. 59, no. 4, pp. 39–44, 2012.
• D. Singh, I. A. Onyeozili, “Some results on distributive and absorption properties on soft operations”. IOSR J Math, vol. 4, no. 2, pp. 18–30, 2012.
• D. Singh, I. A. Onyeozili, “Some conceptual misunderstanding of the fundamentals of soft set theory”. ARPN J Syst Softw, vol. 2, no. 9, pp. 251–254, 2012.
• P. Zhu, Q. Wen, “Operations on soft sets revisited”. J Appl Math, vol. 2013, Article ID 105752, 7 pages, 2013.
• I. A. Onyeozili, T. M. Gwary, “A study of the fundamentals of soft set theory”. Int J Sci Technol Res, vol. 3, no. 4, pp. 132–143, 2014.
• J. Sen, “On algebraic structure of soft sets”. Ann Fuzzy Math Inform, vol. 7, no. 6, pp. 1013–1020, 2014.
• Ö. F. Eren, H. Çalışıcı, “On some operations of soft sets”. Proc Int Conf Comput Math Eng Sci, 2019.
• N. S. Stojanovic, “A new operation on soft sets: Extended symmetric difference of soft sets”. Military Technical Courier, vol. 69, no. 4, pp. 779–791, 2021.
• A. Sezgin, F. Aybek, A. O. Atagün, “A new soft set operation: complementary soft binary piecewise intersection operation”. Black Sea J Eng Sci, vol. 6, no. 4, pp. 330–346, 2023.
• A. Sezgin, F. Aybek, N. B. Güngör, “A new soft set operation: complementary soft binary piecewise union operation”. Acta Inform Malays, vol. 7, no. 1, pp. 38–53, 2023.
• A. Sezgin, F. N. Aybek, “A new soft set operation: complementary soft binary piecewise gamma operation”. Matrix Science Mathematic (MSMK), vol. 7, no. 1, pp. 27-45, 2023.
• A. Sezgin, N. Çağman, “An extensive study on restricted and extended symmetric difference operations of soft sets”. Utilitas Math, in press, 2025.
• A. Sezgin, H. Çalışıcı, “A comprehensive study on soft binary piecewise difference operation”. Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B – Teorik Bilimler, vol. 12, no. 1, pp. 1–23, 2024.
• A. Sezgin, K. Dagtoros, “Complementary soft binary piecewise symmetric difference operation: a novel soft set operation”. Sci J Mehmet Akif Ersoy Univ, vol. 6, no. 2, pp. 31–45, 2023.
• 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, 2024. https://doi.org/10.1007/s43994-024-00187-1
• A. Sezgin, M. Sarıalioğlu, “A new soft set operation: complementary soft binary piecewise theta operation”. J Kadirli Fac Appl Sci, vol. 4, no. 2, pp. 325–357, 2024.
• A. Sezgin, M. Sarıalioğlu, “Complementary extended gamma operation: a new soft set operation”. Nat Appl Sci J, vol. 7, no. 1, pp. 15–44, 2024.
• A. Sezgin, E. Şenyiğit, “A new product for soft sets with its decision-making: soft star-product”. Big Data Comput Visions, vol. 5, no. 1, pp. 52–73, 2025.
• A. Sezgin, E. Yavuz, “A new soft set operation: soft binary piecewise symmetric difference operation”. Necmettin Erbakan Univ J Sci Eng, vol. 5, no. 2, pp. 150–168, 2023.
• A. Sezgin, E. Yavuz, “A new soft set operation: complementary soft binary piecewise lambda operation”. Sinop Univ J Nat Sci, vol. 8, no. 2, pp. 101–133, 2023.
• A. Sezgin, E. Yavuz, “Soft binary piecewise plus operation: a new type of operation for soft sets”. Uncertainty Discourse Appl, vol. 1, no. 1, pp. 79–100, 2024.
• F. Feng, Y. B. Jun, X. Zhao, “Soft semirings”. Comput Math Appl, vol. 56, no. 10, pp. 2621–2628, 2008.
• K. Qin, Z. Hong, “On soft equality”. J Comput Appl Math, vol. 234, no. 5, pp. 1347–1355, 2010.
• Y. B. Jun, X. Yang, “A note on the paper combination of interval-valued fuzzy set and soft set”. Comput Math Appl, vol. 61, no. 5, pp. 1468–1470, 2011.
• X. Liu, F. Feng, Y. B. Jun, “A note on generalized soft equal relations”. Comput Math Appl, vol. 64, no. 4, pp. 572–578, 2012.
• F. Feng, Y. Li, “Soft subsets and soft product operations”. Inf Sci, vol. 232, no. 20, pp. 44–57, 2013.
• M. Abbas, B. Ali, S. Romaguera, “On generalized soft equality and soft lattice structure”. Filomat, vol. 28, no. 6, pp. 1191–1203, 2014
• M. Abbas, M. I. Ali, S. Romaguera, “Generalized operations in soft set theory via relaxed conditions on parameters”. Filomat, vol. 31, no. 19, pp. 5955–5964, 2017.
• T. M. Al-Shami, “Investigation and corrigendum to some results related to g-soft equality and gf-soft equality relations”. Filomat, vol. 33, no. 11, pp. 3375–3383, 2019.
• T. Alshami, M. El-Shafei, “T-soft equality relation”. Turk J Math, vol. 44, no. 4, pp. 1427–1441, 2020.
• N. Çağman, S. Enginoğlu, “Soft set theory and uni-int decision making”. Eur J Oper Res, vol. 207, no. 2, pp. 848–855, 2010.
• S. Sezer, “A new view to ring theory via soft union rings, ideals and bi-ideals”. Knowl Based Syst, vol. 36, pp. 300–314, 2012.
• A. Sezgin, “A new approach to semigroup theory I: soft union semigroups, ideals and bi-ideals”. Algebra Lett, vol. 3, pp. 1–46, 2016.
• K. Kaygisiz, “On soft int-groups”. Ann Fuzzy Math Inform, vol. 4, no. 2, pp. 363–375, 2012.
• E. Muştuoğlu, A. Sezgin, Z. K. Türk, “Some characterizations on soft uni-groups and normal soft unigroups”. Int J Comput Appl, vol. 155, no. 10, pp. 1–8, 2016.
• 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, vol. 29, no. 5, pp. 917–946, 2015.
• A. Sezgin, N. Çağman, A. O. Atagün, “A completely new view to soft intersection rings via soft uni-int product”. Appl Soft Comput, vol. 54, pp. 366–392, 2017.
• A. Sezgin, İ. Durak, Z. Ay, “Some new classifications of soft subsets and soft equalities with soft symmetric difference-difference product of groups”. Amesia, vol. 6, no. 1, 16-32, 2025.
• A. Sezgin, N. Çağman, A. O. Atagün, F. N. Aybek, “Complemental binary operations of sets and their application to group theory”. Matrix Science Mathematic, vol. 7, no. 2, pp. 114–121, 2023.
• N. Çağman, F. Çitak, H. Aktaş, “Soft int-group and its applications to group theory”. Neural Comput Appl, vol. 2, pp. 151–158, 2012.
• A. Sezgin, M. Orbay, “Analysis of semigroups with soft intersection ideals”. Acta Univ Sapientiae Math, vol. 14, no. 1, pp. 166–210, 2022.
• A. O. Atagün, A. Sezgin, “Int-soft substructures of groups and semirings with applications. Applied Mathematics & Information Sciences, vol. 11, no. (1), pp. 105-113, 2017.
• Jana, M. Pal, F. Karaaslan, A. Sezgin, “(α, β)-soft intersectional rings and ideals with their applications”. New Math Nat Comput, vol. 15, no. 2, pp. 333–350, 2019.
• A. S. Sezer, N. Çağman, A. O. Atagün, “Uni-soft substructures of groups”. Ann Fuzzy Math Inform, vol. 9, no. 2, pp. 235–246, 2015.
• A. S. Sezer, “A new approach to LA-semigroup theory via the soft sets”. J Intell Fuzzy Syst, vol. 26, no. 5, pp. 2483-2495, 2014.
• A. O. Atagün, A. S. Sezer, “Soft sets, soft semimodules and soft substructures of semimodules”. Mathematical Sciences Letters, vol. 4, no. 3, pp. 235-242, 2015.
• A. O. Atagün, A. Sezgin, “Soft subnear-rings, soft ideals and soft n-subgroups of near-rings”, Math Sci Letters, vol. 7, no. 1, pp. 37–42, 2018.
• A. Sezgin, N. Çağman, F. Çıtak, “α-inclusions applied to group theory via soft set and logic”, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 1, pp. 334-352, 2019.
• A. O. Atagün, A. Sezgin, “More on prime, maximal and principal soft ideals of soft rings”, New Math Nat Comput, vol. 18, no. 1, pp. 195–207, 2022.
• A. Sezgin, A. İlgin, A. O. Atagün, “Soft intersection almost tri-bi-ideals of semigroups”, J Science & Technology Asia, vol. 29, no. 4, pp. 1–23, 2024.
• M. Gulistan, F. Feng, M. Khan, A. Sezgin, “Characterizations of right weakly regular semigroups in terms of generalized cubic soft sets”, Mathematics, vol. 6, no. 293, pp. 1–15, 2018.
• F. Karaaslan, “Some properties of AG*-groupoids and AG-bands under SI-product operation”, J Intell Fuzzy Syst, vol. 36, no. 1, pp. 231–239, 2019.
• A. Sezgin, A. İlgin, “Soft intersection almost ideals of semigroups.”, Journal of Innovative Engineering and Natural Science, vol. 4, no. 2, pp. 466-481, 2024.
• A. Khan, I. Izhar, A. Sezgin, “Characterizations of Abel Grassmann's groupoids by the properties of their double-framed soft ideals”, Int J Anal Appl, vol. 15, no. 1, pp. 62–74, 2017.
• T. Mahmood, A. Waqas, M. A. Rana, “Soft intersectional ideals in ternary semiring”, Sci Int, vol. 27, no. 5, pp. 3929–3934, 2015.
• T. Manikantan, P. Ramasany, A. Sezgin, “Soft quasi-ideals of soft near-rings”, Sigma J Eng Nat Sci, vol. 41, no. 3, pp. 565–574, 2023.
• S. Memiş, “Another view on picture fuzzy soft sets and their product operations with soft decision-making”, J New Theory, no. 38, pp. 1–13, 2022.
• 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”. CMES Comput Model Eng Sci, vol. 136, no. 2, pp. 1759–1783, 2023.
• A. O. Atagün, H. Kamacı, İ. Tastekin, A. S. Sezer, “P-properties in near-rings.” Journal of Mathematical and Fundamental Sciences, vol. 51, no. 2, pp. 152–167, 2019.
• A. Sezer, A. O. Atagün, N. Çağman, “N-group SI-action and its applications to N-group theory”. Fasciculi Math, vol. 52, pp. 139–153, 2017.
• A. Sezer, A. O. Atagün, N. Çağman, “A new view to N-group theory: soft N-groups”. Fasciculi Math, vol. 51, pp. 123–140, 2013.
• A. Sezgin, A. İlgin, “Soft intersection almost subsemigroups of semigroups”. Int J Math Phys, vol. 15, no. 1, pp. 13–20, 2024.
• A. Sezgin, A. O. Atagün, N. Çağman, “A complete study on and-product of soft sets”. Sigma J Eng Nat Sci, vol. 43, no. 1, pp. 1–14, 2025.
• A. O. Atagün, A. Sezgin, “A new view to near-ring theory: soft near-rings”. South East Asian Journal of Mathematics & Mathematical Sciences, vol. 14, no. 3, pp. 19–32, 2018.
• A. Ullah, F. Karaaslan, I. Ahmad, “Soft uni-abel-grassmann's groups”. Eur J Pure Appl Math, vol. 11, no. 2, pp. 517–536, 2018.
• Z. Ay, A. Sezgin, “Soft symmetric difference-lambda product of groups”, Doğu Fen Bilimleri Dergisi, In press.