  When a monomial ideal has linear quotients with respect to an admissible
order of increasing support-degree, we provide two proofs of different flavors
to show that it is componentwise support-linear. We also introduce the variable
decomposable monomial ideals. In squarefree case, these ideals correspond to
the vertex decomposable simplicial complexes. We study the relationships of the
variable decomposable ideals with weakly polymatroidal ideals, weakly stable
ideals and ideals with linear quotients. We also investigate the componentwise
properties of all these ideals.
