Issue:Approximations of crisp set and intuitionistic fuzzy set based on intuitionistic fuzzy normal subgroup

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http://ifigenia.org/wiki/issue:nifs/23/4/91-105

Title of paper: Approximations of crisp set and intuitionistic fuzzy set based on intuitionistic fuzzy normal subgroup Author(s): Prasenjit Mandal Bhalukdungri Jr High School, Raigara, Purulia (W.B.), 723153, India prasenjitmandal08@yahoo.com A. S. Ranadive Department of Pure and Applied Mathematics, Guru Ghasidas University, Bilaspur (C.G.), India asranadive04@yahoo.co.in Published in: "Notes on Intuitionistic Fuzzy Sets", Volume 23, 2017, Number 4, pages 91—105 Download:  PDF (157 Kb  Kb, File info) Abstract: 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. Keywords: Rough set, Fuzzy set, Intuitionistic fuzzy set, Intuitionistic fuzzy normal subgroup, Congruence relation, Lower and Upper approximations of crisp and fuzzy sets. AMS Classification: 03E72, 06F35, 08A72, 03E99. References: 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. Citations: The list of publications, citing this article may be empty or incomplete. If you can provide relevant data, please, write on the talk page.