The lattice of generalized normal L-subgroups

Many studies have investigated the lattice structure of fuzzy substructures of algebraic sets such as group and ring. Some important results about modularity and distributivity have been obtained in these studies. In this paper, we first define the notion of (normal) (λ,μ)-L-subgroups on a group and investigate some of their properties. In particular, we discuss the relationships among ordinary L-subgroups, (∈,∈ ∨q)-, (∈̄, ∈̄ ∨ q̄)-, (∈λ,∈λ ∨qμ)-fuzzy subgroups and (λ,μ)-L-subgroups of a group. Also, we give a characterization of (normal) (λ,μ)-L-subgroup by means of (normal) L-subgroup. Finally, we discuss some properties of the lattices of normal (λ,μ)-L-subgroups and obtain that the lattice of all normal (0, μ)-L-subgroups of a group is modular. As consequence, we obtain that the lattices of all normal (∈,∈ ∨q)-fuzzy subgroups and all normal L-subgroups of a group are modular.