Distributivity and pseudocomplementation of lattices of generalized \(L\)-subgroups

Abstract

The main goal of this paper is to study the lattice of \((0,\mu)\)-\(L\)-subgroups of a group. We characterize abelian groups by the lattice of \((0,\mu)\)-\(L\)-subgroups. Also, we show that a group $G$ is locally cyclic if and only if the lattice of \((0,\mu)\)-\(L\)-subgroups is distributive. As consequence, we obtain that the lattices of all \((\in,\in\vee q)\)-fuzzy subgroups and all fuzzy subgroups of a finite cyclic group are distributive. Finally, we study groups which of the lattice of \((\lambda,\mu)\)-\(L\)-subgroups is pseudocomplemented lattice.