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We introduce pretty $k$-clean monomial ideals and $k$-decomposablemulticomplexes, respectively, as the extensions of the notions of $k$-cleanmonomial ideals and $k$-decomposable simplicial complexes. We show that amulticomplex $\Gamma$ is $k$-decomposable if and only if its associatedmonomial ideal $I(\Gamma)$ is pretty $k$-clean. Also, we prove that anarbitrary monomial ideal $I$ is pretty $k$-clean if and only if itspolarization $I^p$ is $k$-clean. Our results extend and generalize some resultsdue to Herzog-Popescu, Soleyman Jahan and the current author.

Pretty $k$-clean monomial ideals and $k$-decomposable multicomplexes