In this paper we study the regular Hom-Lie algebra L graded by an arbitrary set S (set grading). We show that L decomposes as L=U⊕∑[λ]∈(ΛS∖{0})/∼L[λ], where U is a linear complement of ∑[λ]∈(ΛS∖{0})/∼L0,[λ] in L0 and any L[λ] a well-described graded ideals of L, satisfying [L[λ],L[μ]]=0 if [λ]≠[μ]. Under certain conditions, the simplicity of L is characterized and it is shown that the above decomposition is actually the direct sum of the family of its minimal graded ideals, each one being a simple set-graded regular Hom-Lie algebras.
