A New Type of Interval Valued Fuzzy Normal Subgroups of Groups

Saleem Abdullah; Muhammad Naeem

A New Type of Interval Valued Fuzzy Normal Subgroups of Groups

Year 2015, Volume: 3 Issue: 1, 62 - 77, 22.12.2014

Abstract

In this paper we are using the notions of not belonging (∈) and non quasi-k-coincidenceqk ( ) of an interval valued fuzzypoint with an interval valued fuzzy set, we define the concepts of interval valued (∈,∈ ∨ qk)-fuzzy normal subgroups and interval valued(∈,∈ ∨ qk)-fuzzy cosets which is a generalization of fuzzy normal subgroups, fuzzy coset, interval valued fuzzy normal subgroups,interval valued fuzzy coset, interval valued (∈,∈ ∨ q)-fuzzy normal subgroups and interval valued (∈,∈ ∨ q)-fuzzy cosets. We givesome characterizations of an interval valued (∈,∈ ∨ qk)-fuzzy normal subgroup and interval valued (∈,∈ ∨ qk)-fuzzy coset, and dealwith several related properties. The important achievement of the study with an interval valued (∈,∈ ∨ qk)-fuzzy normal subgroupand interval valued (∈,∈ ∨ qk)-fuzzy cosets is the generalization of that the notions of fuzzy normal subgroups, fuzzy coset, intervalvalued fuzzy normal subgroups, interval valued fuzzy coset, interval valued (∈,∈ ∨ q)-fuzzy normal subgroups and interval valued(∈,∈ ∨ q)-fuzzy cosets. We prove that the set of all interval valued (∈,∈ ∨ qk)-fuzzy cosets of G is a group, where the multiplicationis defined byλx·λy=λxyfor all x, y∈ G. Ifµ : F → D[0,1] is defined by µ←−←−(←−)λx =λ (x) for all x ∈ G. Then µ is an interval valued( )-fuzzy normal subgroup of F

Keywords

Group, normal subgroup, coset, interval valued (∈, ∈ ∨ qk)-fuzzy normal subgroup, interval valued (∈, ∈ ∨ qk)-fuzzy coset

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