Abstract
Fuzzy rough sets are a mathematical concept that combines fuzzy sets and rough sets to deal with uncertainty and incompleteness in data and information. In this study, different from the definition of Dubois and Prade (1990), the fuzzy rough set is defined within the framework of the rough group concept defined by Biswas and Nanda (1994), and some of its algebraic properties are discussed. Then, the concepts of fuzzy rough subgroup and fuzzy rough normal subgroup are introduced in the rough group. In addition, some basic features and examples of these concepts are given.
MSC (2010): Primary: 03E99, 20N99.
References
Zadeh LA. Fuzzy sets. Inf Control1965; 8: 338-53. https://doi.org/10.1016/S0019-9958(65)90241-X
Rosenfeld A. Fuzzy groups. J Math Anal Appl. 971; 35: 512-7. https://doi.org/10.1016/0022-247X(71)90199-5
Das PS. Fuzzy groups and level subgroups. J Math Anal Appl. 1981; 84: 264-9. https://doi.org/10.1016/0022-247X(81)90164-5
Anthony JM, Sherwood H. A characterization of fuzzy subgroups. Fuzzy Sets Syst. 1982; 7: 297-305. https://doi.org/10.1016/0165-0114(82)90057-4
Pawlak Z. Rough sets. Int J Comput Inform Sci. 1982; 11: 341-56. https://doi.org/10.1007/BF01001956
Iwinski T. Algebraic approach to rough sets. Bull Polish Acad Sci Math. 1987; 35: 673-83.
Kuroki N, Wang PP. The lower and upper approximations in a fuzzy group. Inf Sci. 1996; 90: 203-20. https://doi.org/10.1016/0020-0255(95)00282-0
Cheng W, Mo Z-W, Wang J. Notes on “the lower and upper approximations in a fuzzy group” and “rough ideals in semigroups.” Inf Sci. 2007; 177: 5134-40. https://doi.org/10.1016/j.ins.2006.12.006
Li F, Zhang Z. The homomorphisms and operations of rough groups. Sci World J. 2014; 2014: 1-6. https://doi.org/10.1155/2014/507972
Wang Z, Shu L. The lower and upper approximations in a group. Int J Math Comput Sci. 2012; 6: 158-62.
Wang C, Chen D. A short note on some properties of rough groups. Comput Math Appl. 2010; 59: 431-6. https://doi.org/10.1016/j.camwa.2009.06.024
Wang C, Chen D, Hu Q. On rough approximations of groups. Int J Mach Learn Cyber. 2013; 4: 445-9. https://doi.org/10.1007/s13042-012-0108-6
Biswas R, Nanda S. Rough groups and rough subgroups. Bull Polish Acad Sci Math. 1994; 42: 251-4.
Miao D, Han S, Li D, Sun L. Rough Group, Rough Subgroup and Their Properties. In: Ślęzak D, Wang G, Szczuka M, Düntsch I, Yao Y, Eds., Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing. vol. 3641. Berlin, Heidelberg: Springer; 2005. https://doi.org/10.1007/11548669
Bagirmaz N, Ozcan AF. Rough semigroups on approximation spaces. Int J Algebra. 2015; 9: 339-50. https://doi.org/10.12988/ija.2015.5742
Bağırmaz N, İçen İ, Özcan AF. Topological rough groups. Topol Algebra Appl. 2016; 4: 31-8. https://doi.org/10.1515/taa-2016-0004
Li P-Y, Liu W-L, Mou L, Guo Z-F. On separation axioms of topological rough groups. Soft Comput. 2023; 27: 57-61. https://doi.org/10.1007/s00500-022-07521-x
Pei D. A generalized model of fuzzy rough sets. Int J Gen Syst. 2005; 34: 603-13. https://doi.org/10.1080/03081070500096010
Liu G, Zhu W. The algebraic structures of generalized rough set theory. Inf Sci. 2008; 178: 4105-13. https://doi.org/10.1016/j.ins.2008.06.021
Kondo M. On the structure of generalized rough sets. Information Sciences. 2006; 176: 589-600. https://doi.org/10.1016/j.ins.2005.01.001
Dubois D, Prade H. Rough fuzzy sets and fuzzy rough sets. Int J General Syst. 1990; 17: 191-209. https://doi.org/10.1080/03081079008935107
Radzikowska AM, Kerre EE. A comparative study of fuzzy rough sets. Fuzzy Sets Syst. 2002; 126: 137-55. https://doi.org/10.1016/S0165-0114(01)00032-X
Zhan J, Liu Q. Rough fuzzy (fuzzy rough) strong H-ideals of hemirings. Ital J Pure Appl Math. 2015: 483-96.
Pan W, Zhan J. Rough fuzzy groups and rough soft groups. Rough fuzzy (fuzzy rough) strong H-ideals of hemirings. Ital J Pure Appl Math. 2016; 36: 617-28.
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Copyright (c) 2023 Nurettin Bağırmaz