Betweenness Closure of Ternary and Fuzzy Ternary Relations

Metric betweenness and fuzzy betweenness relations are revisited and their relationship with indistinguishability operators is established. Proximity relations (fuzzy reflexive and symmetric binary relations) do not generate betweenness and fuzzy betweenness relations in general but what we call pre-betweenness and fuzzy pre-betweenness rela- tions. How to obtain their betweenness closure, i.e., the smallest betweenness or fuzzy betweenness relation containing them, is studied. Operations between ternary and fuzzy ternary relations similar to the sup-[Formula: see text] products of proximity relations are studied and their relationship with the betweenness closures is established.