In this paper, we mainly introduce the concept of lattice-valued betweenness relations and study its relationships with L-fuzzifying convex structures in depth. Such an extended betweenness relation is defined to be a map from X × X × X to a complete lattice provided that fulfills a set of axioms. We also discuss the relationships among them and other algebraic and geometric structures. Moreover, a category of approach is provided to present lattice-valued betweenness relations. It is shown that the category of lattice-valued betweenness spaces and the category of lattice-valued geometric interval spaces are isomorphic.
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