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 Γ is kdecomposable if and only if its associated monomial ideal I(Γ) is pretty k-clean. Also, we prove that an arbitrary monomial ideal I is pretty k-clean if and only if its polarization Ip is k-clean. Our results extend and generalize some results due to Herzog-Popescu, Soleyman Jahan and the first author
