  We consider three types of set-systems that have interesting applications in
algebraic combinatorics and representation theory: maximal collections of the
so-called strongly separated, weakly separated, and chord separated subsets of
a set $[n]=\{1,2,\ldots,n\}$. These collections are known to admit nice
geometric interpretations; namely, they are bijective, respectively, to rhombus
tilings on the zonogon $Z(n,2)$, combined tilings on $Z(n,2)$, and fine
zonotopal tilings (or `cubillages') on the 3-dimensional zonotope $Z(n,3)$. We
describe interrelations between these three types of set-systems in $2^{[n]}$,
by studying interrelations between their geometric models. In particular, we
completely characterize the sets of rhombus and combined tilings properly
embeddable in a fixed cubillage, explain that they form distributive lattices,
give efficient methods of extending a given rhombus or combined tiling to a
cubillage, and etc.
