Galoitica: Journal of Mathematical Structures and Applications
Journal DOI
2834-5568ISSN (Online)
Volume 12 , Issue 2 , PP: 01-23, 2025 | Cite this article as | XML | Html | PDF | Full Length Article
Takaaki Fujita 1 * , Arif Mehmood 2
Doi: https://doi.org/10.54216/GJMSA.120201
Numerous mathematical frameworks have been developed to handle uncertainty, including Fuzzy Sets,1 Intuitionistic Fuzzy Sets,2 Hyperfuzzy Sets,3 Picture Fuzzy Sets,4 Hesitant Fuzzy Sets,5, 6 Neutrosophic Sets,7 Plithogenic Sets,8 and Soft Sets,9 and research in this area continues to evolve rapidly. Rough set theory provides a foundational method for approximating subsets using lower and upper bounds based on equivalence relations, offering an effective approach to modeling uncertainty in classification and data analysis.10, 11 Building upon these foundations, extended models such as HyperRough Sets and SuperHyperRough Sets have been proposed.12 In this paper, we present novel definitions that further generalize Graded Rough Sets and Linguistic Rough Sets—specifically, the Graded HyperRough Set and the Linguistic HyperRough Set. These new frameworks are expected to contribute to the advancement of research in fields such as decision-making, language theory, and artificial intelligence.
Rough set , Hyperrough Set , Linguistic Rough Set , SuperHyperRough set , Graded Rough Set
[1] Lotfi A Zadeh. Fuzzy sets. Information and control, 8(3):338–353, 1965.
[2] Krassimir T Atanassov. Circular intuitionistic fuzzy sets. Journal of Intelligent & Fuzzy Systems, 39(5):5981–5986, 2020.
[3] Seok-Zun Song, Seon Jeong Kim, and Young Bae Jun. Hyperfuzzy ideals in bck/bci-algebras. Mathematics, 5(4):81, 2017.
[4] Bui Cong Cuong and Vladik Kreinovich. Picture fuzzy sets-a new concept for computational intelligence problems. In 2013 third world congress on information and communication technologies (WICT 2013), pages 1–6. IEEE, 2013.
[5] Vicenc¸ Torra and Yasuo Narukawa. On hesitant fuzzy sets and decision. In 2009 IEEE international conference on fuzzy systems, pages 1378–1382. IEEE, 2009.
[6] Zeshui Xu. Hesitant fuzzy sets theory, volume 314. Springer, 2014.
[7] Said Broumi, Mohamed Talea, Assia Bakali, and Florentin Smarandache. Interval valued neutrosophic graphs. Critical Review, XII, 2016:5–33, 2016.
[8] Florentin Smarandache. Plithogeny, plithogenic set, logic, probability, and statistics. arXiv preprint arXiv:1808.03948, 2018.
[9] Dmitriy Molodtsov. Soft set theory-first results. Computers & mathematics with applications, 37(4-5):19–31, 1999.
[10] Zdzisław Pawlak. Rough sets. International journal of computer & information sciences, 11:341–356, 1982.
[11] Said Broumi, Florentin Smarandache, and Mamoni Dhar. Rough neutrosophic sets. Infinite Study, 32:493–502, 2014.
[12] Takaaki Fujita. Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. Biblio Publishing, 2025.
[13] Xianyong Zhang, Zhiwen Mo, Fang Xiong, and Wei Cheng. Comparative study of variable precision rough set model and graded rough set model. International Journal of Approximate Reasoning, 53(1):104–116, 2012.
[14] Bo Wen Fang and Bao Qing Hu. Probabilistic graded rough set and double relative quantitative decision-theoretic rough set. International Journal of Approximate Reasoning, 74:1–12, 2016.
[15] Weihua Xu, Shihu Liu, Qiaorong Wang, and Wenxiu Zhang. The first type of graded rough set based on rough membership function. In 2010 Seventh International Conference on Fuzzy Systems and Knowledge Discovery, volume 4, pages 1922–1926. IEEE, 2010.
[16] Manish Agarwal and Themis Palpanas. Linguistic rough sets. International Journal of Machine Learning and Cybernetics, 7:953– 966, 2016.
[17] Ze-Ling Wang, Hu chen Liu, Jin-Ye Xu, and Ye-Jia Ping. A new method for quality function deployment using double hierarchy hesitant fuzzy linguistic term sets and axiomatic design approach. Quality Engineering, 33:511 – 522, 2021.
[18] Tianle Yao, Weili Wang, Run Miao, Qiwei Hu, Jun Dong, and Xuefei Yan. Warhead power assessment based on double hierarchy hesitant fuzzy linguistic term sets theory and gained and lost dominance score method. Chinese Journal of Aeronautics, 2021.
[19] Ying-Ying Li, Hongyu Zhang, and Jian-Qiang Wang. Linguistic neutrosophic sets and their application in multicriteria decision- making problems. International Journal for Uncertainty Quantification, 7(2), 2017.
[20] Xingang Wang, Yushui Geng, Peipei Yao, and Mengjie Yang. Multiple attribute group decision making approach based on extended vikor and linguistic neutrosophic set. Journal of Intelligent & Fuzzy Systems, 36(1):149–160, 2019.
[21] Pradip Kumar Maji, Ranjit Biswas, and A Ranjan Roy. Soft set theory. Computers & mathematics with applications, 45(4-5):555– 562, 2003.
[22] Muhammad Haris Saeed, Atiqe Ur Rahman, and Muhammad Arshad. A novel approach to neutrosophic hypersoft graphs with properties. 2021.
[23] Sagvan Y Musa, Ramadhan A Mohammed, and Baravan A Asaad. N-hypersoft sets: An innovative extension of hypersoft sets and their applications. Symmetry, 15(9):1795, 2023.
[24] Young Bae Jun, Kul Hur, and Kyoung Ja Lee. Hyperfuzzy subalgebras of bck/bci-algebras. Annals of Fuzzy Mathematics and Informatics, 2017.
[25] M MAHARIN. An over view on hyper fuzzy subgroups. Scholar: National School of Leadership, 9(1.2), 2020.
[26] Jayanta Ghosh and Tapas Kumar Samanta. Hyperfuzzy sets and hyperfuzzy group. Int. J. Adv. Sci. Technol, 41:27–37, 2012.
[27] Z Nazari and B Mosapour. The entropy of hyperfuzzy sets. Journal of Dynamical Systems and Geometric Theories, 16(2):173–185, 2018.
[28] Alain Bretto. Hypergraph theory. An introduction. Mathematical Engineering. Cham: Springer, 1, 2013.
[29] Yifan Feng, Haoxuan You, Zizhao Zhang, Rongrong Ji, and Yue Gao. Hypergraph neural networks. In Proceedings of the AAAI conference on artificial intelligence, volume 33, pages 3558–3565, 2019.
[30] Mohammad Hamidi and Mohadeseh Taghinezhad. Application of Superhypergraphs-Based Domination Number in Real World. Infinite Study, 2023.
