Fuzzy structures induced by fuzzy betweenness relations

Abstract

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.