Fuzzy structures induced by fuzzy betweenness relations

In the setting of complete residuated lattices, we explore the relationships between the recently introduced fuzzy betweenness relations and three important mathematical notions: fuzzy interval operators, fuzzy partial orders and fuzzy Peano–Pasch spaces. After recalling the concept of a fuzzy betweenness relation w.r.t. a fuzzy equivalence relation, we prove that the resulting category is isomorphic to that of geometric fuzzy interval spaces w.r.t. the same fuzzy equivalence relation. Next, we construct a fuzzy partial order via a fuzzy betweenness relation w.r.t. a fuzzy equivalence relation and analyze their relationships in depth. Finally, taking a field as underlying set, we introduce the concept of a fuzzy betweenness field. Furthermore, in the setting of completely distributive lattices, we provide an interesting example showing that a vector space over a fuzzy betweenness field can yield a fuzzy Peano–Pasch space.