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

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