  In multipartite entanglement theory, the partial separability properties have
an elegant, yet complicated structure, which boils down in the case when
multipartite correlations are considered. In this work, we elaborate this, by
giving necessary and sufficient conditions for the existence and uniqueness of
the class of a given class-label, by the use of which we work out the structure
of the classification for some important particular cases, namely, for the
finest classification, for the classification based on k-partitionability and
k-producibility, and for the classification based on the atoms of the
correlation properties.
