Hierarchies and ternary separation

If HH is a hierarchy on some finite set SS, then HH determines a ternary relation s(H)s(H) as follows: (a,b,c)(a,b,c) belongs to s(H)s(H) if and only if there exists a cluster AA in HH such that a,b∈Aa,b∈A and c∉Ac∉A. A well known and useful fact is that the function ss, which maps hierarchies on SS to ternary separation relations on SS, is injective. We consider ternary separation from a new point of view by showing that ss satisfies three natural algebraic properties and that these three properties are only satisfied by functions that are closely connected to ss.