On split regular Hom-Lie superalgebras

We introduce the class of split regular Hom-Lie superalgebras as the natural extension of the one of split Hom-Lie algebras and Lie superalgebras, and study its structure by showing that an arbitrary split regular Hom-Lie superalgebra $\mathfrak{L}$ is of the form $\mathfrak{L}=U+{\sum }_j{I}_j$ with $U$ a linear subspace of a maximal abelian graded subalgebra $H$ and any $I_j$ a well described (split) ideal of $\mathfrak{L}$ satisfying $[I_j,I_k]=0$ if $j\mathrm{\ne }k$. Under certain conditions, the simplicity of $\mathfrak{L}$ is characterized and it is shown that $\mathfrak{L}$ is the direct sum of the family of its simple ideals.