On interrelations between strongly, weakly and chord separated set-systems (a geometric approach)

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.