Injectivity and weak ternary separation

A collection H of nonempty subsets of a finite set E determines a weak ternary separation relation r(H) on the set E as follows: (a,b,c)∈r(H) if and only if for all clusters A belonging to H, |A∩{a,b,c}|=2 implies that c∉A. We determine conditions on a collection W of set systems of E such that the mapping r that sends each H belonging to W to the weak ternary separation relation r(H) is injective.