  We show that all monomial ideals in the polynomial ring in at most 3
variables are pretty clean and that an arbitrary monomial ideal $I$ is pretty
clean if and only if its polarization $I^p$ is clean. This yields a new
characterization of pretty clean monomial ideals in terms of the arithmetic
degree, and it also implies that a multicomplex is shellable if and only the
simplicial complex corresponding to its polarization is (non-pure) shellable.
We also discuss Stanley decompositions in relation to prime filtrations.
