  The main concern of this paper is how to define proper measures of
multipartite entanglement for mixed quantum states. Since the structure of
partial separability and multipartite entanglement is getting complicated if
the number of subsystems exceeds two, one can not expect the existence of an
ultimate scalar entanglement measure, which grasps even a small part of the
rich hierarchical structure of multipartite entanglement, and some higher order
structure characterizing that is needed. In this paper we make some steps
towards this direction.
  First, we reveal the lattice-theoretic structure of the partial separability
classification, introduced earlier [Sz. Szalay and Z. Kokenyesi, Phys. Rev. A
86, 032341 (2012)]. It turns out that, mathematically, the structure of the
entanglement classes is the up-set lattice of the structure of the different
kinds of partial separability, which is the down-set lattice of the lattice of
the partitions of the subsystems. It turns also out that, physically, this
structure is related to the LOCC convertibility: If a state from a class can be
mapped into another one, then that class can be found higher in the hierarchy.
  Second, we introduce the notion of multipartite monotonicity, expressing that
a given set of entanglement monotones, while measuring the different kinds of
entanglement, shows also the same hierarchical structure as the entanglement
classes. Then we construct such hierarchies of entanglement measures, and
propose a physically well-motivated one, being the direct multipartite
generalization of the entanglement of formation based on the entanglement
entropy, motivated by the notion of statistical distinguishability. The
multipartite monotonicity shown by this set of measures motivates us to
consider the measures to be the different manifestations of some "unified"
notion of entanglement.
