  We introduce pretty $k$-clean monomial ideals and $k$-decomposable
multicomplexes, respectively, as the extensions of the notions of $k$-clean
monomial ideals and $k$-decomposable simplicial complexes. We show that a
multicomplex $\Gamma$ is $k$-decomposable if and only if its associated
monomial ideal $I(\Gamma)$ is pretty $k$-clean. Also, we prove that an
arbitrary monomial ideal $I$ is pretty $k$-clean if and only if its
polarization $I^p$ is $k$-clean. Our results extend and generalize some results
due to Herzog-Popescu, Soleyman Jahan and the current author.
