Affiliations: Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan | Deanship of Preparatory Years, Umm al Qurrra University, Makkah, Saudi Arabia
Note: [] Corresponding author. Saleem Abdullah, Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan. E-mail: saleemabdullah81@yahoo.com
Abstract: Using the notions of belonging (∈) and quasi-k-coincidence (qk) of a fuzzy point with a fuzzy set, we define the concepts of $(\overline{\in}, \overline{\in} \vee \overline{q_{k}})$-fuzzy normal subgroups and $(\overline{\in }, \overline{\in } \vee \overline{q_{k}})$-fuzzy cosets which is a generalization of fuzzy normal subgroups, fuzzy coset, $(\overline{\in}, \overline{\in} \vee \overline{q})$-fuzzy normal subgroups and $(\overline{\in}, \overline{\in} \vee \overline{q})$-fuzzy cosets. We give characterizations of an $(\overline{\in}, \overline{\in} \vee \overline{q_{k}})$-fuzzy normal subgroup and $(\overline{\in}, \overline{\in} \vee \overline{q_{k}})$-fuzzy coset, and deal with several related properties. The important achievement of the study with an $(\overline{\in}, \overline{\in} \vee \overline{q_{k}})$-fuzzy normal subgroup and $(\overline{\in}, \overline{\in} \vee \overline{q_{k}})$-fuzzy cosets is the generalization of that the notions of fuzzy normal subgroups, fuzzy coset, $(\overline{\in} ,\overline{\in} \vee \overline{q})$-fuzzy normal subgroups and $(\overline{\in}, \overline{\in} \vee \overline{q})$-fuzzy cosets. We prove that the set of all $(\overline{\in}, \overline{\in} \vee \overline{q_{k}})$-fuzzy cosets of G is a group, where the multiplication is defined by $\overleftarrow{f_{x}}\cdot \overleftarrow{f_{y}} = \overleftarrow{f_{xy}}$ for all $x,y\in G.$ If $\widetilde{f}:F \rightarrow [0,1]$ is defined by $\widetilde{f}(\overleftarrow{f_{x}}) = f(x) $ for all $x\in G.$ Then $\widetilde{f}$ is a fuzzy normal subgroup of F.
Keywords: Group, fuzzy normal subgroup, fuzzy coset, $(\overline{\in}, \overline{\in} \vee \overline{q_{k}})$-fuzzy normal subgroup, $(\overline{\in}, \overline{\in} \vee \overline{q_{k}})$-fuzzy coset
