Abstract
The main objective of the proposed work in this paper is to introduce a generalized form of rough fuzzy subsemigroups, which is rough fuzzy ternary subsemigroups (RFTSs) combining the notions of fuzziness and roughness in ternary semigroups. In RFTSs, we deal with vague and incomplete information in decision-making problems. RFTSs are characterized by lower and upper approximations using fuzzy ideals. In this research, we propose the three-dimensional k-level relation and proved that this relation is a congruence relation on a ternary semigroup. Furthermore, comparing it with the previous literature, we conclude that our proposed technique is better and effective because it deals with vague problems and there are many structures which are not handled using binary multiplication such as all the sets of negative numbers. In addition, we have proved by counterexamples that converses of many parts of many results do not hold which have negated the results proved in Q. Wang’s paper.
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References
Biswas R, Nanda S (1994) Rough groups and rough subgroups. Bull Polish Acad Sci Math 42:251–254
Bonikowaski Z (1995) Algebraic structures of rough sets. In: Ziako WP (ed) Rough sets fuzzy sets, and knowledge discovery. Springer-Verlag, Berlin
Davvaz B (2004) Roughness in rings. Inf Sci 164:147–163
Davvaz B (2006) Roughness based on fuzzy ideals. Inf Sci 176:2417–2437
Davvaz B, Dudek WA (2009) Fuzzy n-ary groups as a generalization of Rosenfeld’s fuzzy groups. J. Multi-Valued Logic Soft Comput. 15:451–469
Dixit VN, Dewan S (1995) A note on quasi and bi-ideals in ternary semigroups. Int J Math Math Sci 18:501–508
Dudek WA (2000) Fuzzification of n-ary groupoids. Quasigroups Relat Syst 7:45–66
Iwinski T (1987) Algebraic approach to rough sets. Bull Polish Acad Sci Math 35:673–683
Jun YB (2003) Roughness of gamma-subsemigroups/ideals in gamma-subsemigroups. Bull Korean Math Soc 40:531–536
Kar S, Maity BK (2007) Congruences on ternary semigroups. J Chugcheong Math Soc 20:191–200
Kar S, Sarkar P (2012) Fuzzy ideals of ternary semigroup. Fuzzy Inf. Eng 2:181–193
Kuroki N (1997) Rough ideals in semigroups. Inf Sci 100:139–163
Lehmer DH (1932) A ternary analogue of Abelian groups. Am J Math 59:329–338
Los J (1955) On the extending of model I. Fund Math 42(38–54):571
Pawlak Z (1982) Rough sets. Int J Inf Comput Sci 11:341–356
Petchkhaew P, Chinram R (2009) Fuzzy, rough and rough fuzzy ideals in ternary semigroups. Int J Pure Appl Math 56:21–36
Pomykala J, Pomykala JA (1988) The stone algebra of rough sets. Bull Polish Acad Sci Math 36:495–508
Rainich GY (1952) Ternary relations in geometry and algebra. Mich Math J 1:97–111
Rameez M, Ali MI, Ejaz A (2017) Generalized roughness in (ϵ, ϵ V q)-fuzzy ideals of hemirings. Kuwait J Sci 44(3):34–43
Sioson FM (1965) Ideal theory in ternary semigroups. Math Jpn 10:63–84
Wang Q, Zhan J (2016) Rough semigroups and rough fuzzy semigroups based on fuzzy ideals. Open Math 14:1114–1121
Xiao QM, Zhang ZL (2006) Rough prime ideals and rough fuzzy prime ideals in semigroups. Inform Sci 176:725–733
Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353
Zhan J, Dudek WA (2007) Fuzzy h-ideals of hemirings. Inf Sci 177:876–886
Zhan J, Yin Y (2012) A new view of fuzzy k-ideals of hemirings. J Intell Fuzzy Syst 23:169–176
Zhan J, Zhou X, Xiang D (2016) Roughness in n-ary semigroups based on fuzzy ideals. J Intel Fuzzy Syst 30:2833–2841
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Conceptualization: SB and MS; methodology: RM; software: HA; validation: HA, SB; formal analysis: RM; investigation: MS; resources: SB; data curation, MS; writing—original draft preparation: HA; writing—review and editing: SB and RM; visualization: HA; supervision: SB; project administration: HA.
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Communicated by Marcos Eduardo Valle.
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Bashir, S., Abbas, H., Mazhar, R. et al. Rough fuzzy ternary subsemigroups based on fuzzy ideals with three-dimensional congruence relation. Comp. Appl. Math. 39, 90 (2020). https://doi.org/10.1007/s40314-020-1079-y
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DOI: https://doi.org/10.1007/s40314-020-1079-y