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