  In this paper we show that the Stanley depth, as well as the usual depth, are
essentially determined by the lcm-lattice. More precisely, we show that for
quotients $I/J$ of monomial ideals $J\subset I$, both invariants behave
monotonic with respect to certain maps defined on their lcm-lattice. This
allows simple and uniform proofs of many new and known results on the Stanley
depth. In particular, we obtain a generalization of our result on polarization
presented in the reference [IKMF14]. We also obtain a useful description of the
class of all monomial ideals with a given lcm-lattice, which is independent
from our applications to the Stanley depth.
