| Title of paper:
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Approximations of crisp set and intuitionistic fuzzy set based on intuitionistic fuzzy normal subgroup
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| Author(s):
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| Prasenjit Mandal
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| Bhalukdungri Jr High School, Raigara, Purulia (W.B.), 723153, India
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| prasenjitmandal08@yahoo.com
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| A. S. Ranadive
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| Department of Pure and Applied Mathematics, Guru Ghasidas University, Bilaspur (C.G.), India
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| asranadive04@yahoo.co.in
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| Published in:
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"Notes on IFS", Volume 23, 2017, Number 4, pages 91—105
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| Download:
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 PDF (157 Kb Kb, File info)
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| Abstract:
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We consider a group G, with identity element e, as a universal set and assume that the knowledge about objects is restricted by an intuitionistic fuzzy normal subgroup [math]\displaystyle{ A=(\mu_A,\nu_A) }[/math] of G. For each α, β ∈ [0, 1] such that α + β ≤ 1, the set [math]\displaystyle{ U (A, \alpha, \beta) = U (\mu_A , \alpha) \cap U (\nu_A, \beta) }[/math] is a congruence relation on G, where [math]\displaystyle{ U (\mu_A , \alpha) = \{ (x,y) \in G \times G: \mu_A(xy^{-1}) \geq \alpha \} }[/math] and [math]\displaystyle{ U (\nu_A , \beta) = \{ (x,y) \in G \times G: \nu_A(xy^{-1}) \leq \beta \} }[/math]. In this paper, the notion of U (A, α, β)-lower and U (A, α, β)-upper approximation of a non-empty subset of G and an intuitionistic fuzzy set of G are introduced and some important properties of the above approximations are presented.
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| Keywords:
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Rough set, Fuzzy set, Intuitionistic fuzzy set, Intuitionistic fuzzy normal subgroup, Congruence relation, Lower and Upper approximations of crisp and fuzzy sets.
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| AMS Classification:
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03E72, 06F35, 08A72, 03E99.
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| References:
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- Atanassov, K. T. (1986). Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 87–96.
- Atanassov, K. T. (1994). New operations defined over the intuitionistic fuzzy sets, Fuzzy Sets and Systems, 61, 137–142.
- Atanassova, L. (2007). On intuitionistic fuzzy versions of L. Zadeh's extension principle, Notes on Intuitionistic Fuzzy Sets, 13(3), 33–36.
- Banerjee, B., & Basnet Kr. D. (2003). Intuitionistic fuzzy subrings and ideals, The Journal of fuzzy mathematics, 11(1), 139–155.
- Biswas, R., Nanda, S. (1994). Rough groups and rough subgroups, Bull. Polish Acad. Sci. Math., 42, 251–254.
- Davvaz, B. (2004). Roughness in rings, Information Sciences, 164, 147–163.
- Davvaz, B., Dudak, W. A., & Jun, Y. B. (2006). Intuitionistic fuzzy HV-submodules, Information Sciences, 176, 285–300.
- Davvaz, B., (2006). Roughness based on fuzzy ideals, Information Sciences, 176, 2417–2437.
- Davvaz, B., & Mahdavipour, M. (2006). Roughness in modules, Information Sciences, 176,3658–3674.
- Dubois, D., & Prade, H. (1990). Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems, 17(2 3), 191–209.
- Estaji, A. A., Khodaii, S., & Bahrami, S. (2011). On rough set and fuzzy sublattice, Information Sciences, 181, 3981–3994
- Hur, K., Kang, H. W., & Song, H. K. (2003). Intuitionistic fuzzy subgroups and subrings, Honam Mathematical Journal, 25, 19–41.
- Kuroki, N., & Wang, P. P. (1996). The lower and upper approximations in a fuzzy group, Information Sciences, 90, 203–220.
- Kuroki, N., & Mordeson, J. N. (1997). Structure of rough sets and rough groups, The Journal of fuzzy mathematics, 5(1), 183–191.
- Kuroki, N. (1997). Rough ideals in semigroups, Information Sciences, 100, 139–163.
- Pawlak, Z. (1982). Rough sets, International Journal of Computer and Information Sciences, 11, 341–356.
- Xiao, Q. M., & Zhang, Z. L. (2008). Rough prime ideals and rough fuzzy prime ideals in semigroups, Information Sciences, 178, 425–438.
- Zadeh, L. A. (1965). Fuzzy sets, Information and Control, 8 338–353.
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