Abstract properties of a class of intervals of lattices of closed classes

The lattice $\mathcal L_k$ of closed classes that contain all projections (that is, the lattice of clones) on a $k$-element set is considered. It is proved that for any $k\geq 2$ the countable direct degree of $\mathcal L_k$ is isomorphic to an interval in $\mathcal L_{k+3}$. In particular, hence it follows that the class of all sublattices (intervals) of the lattice of clones is closed under countable direct degrees.