  The aim of this paper is to transfer the restrictedness theory to Hom-Lie
algebras. The concept of restricted Hom-Lie algebras which is introduced in
\cite{BM2} will be used in this paper. First, the existence of $p$-structures
on a Hom-Lie algebra is studied and the direct sum of restricted Hom-Lie
algebras is analyzed. Then, the definition of a restrictable Hom-Lie algebra is
given and the equivalence relation between restrictable Hom-Lie algebras and
restricted Hom-Lie algebras is constructed. Finally, the $p$-envelopes of a
Hom-Lie algebra are defined and studied.

Rings and Algebras