Dendrons and their products admit a natural, continuous median operator. We prove that there exists a two‐dimensional metric continuum with a continuous median operator, for which there is no median‐preserving embedding in a product of finitely many dendrons. Our method involves ideas and results concerning graph colouring and abstract convexity. The main result answers a question in [16] negatively, and is sharply contrasting with a result of Stralka [15] on embeddings of compact lattices.
A convex structure is binary if every finite family of pairwise intersecting convex sets has a non‐empty intersection. Distributive lattices with the convexity of all order‐convex sublattices are a prominent type of examples, because they correspond exactly to the intervals of a binary convex structure which has a certain separation properly. In one direction, this result relies on a study of so‐called base‐point orders induced by a convex structure. Thesis ordering are used to construct an ‘intrinsic’ topology. For binary convexities, certain basic questions are answered with the aid of some results on completely distributive lattices. Several applications are given. Dimension problems are studied in a subsequent paper.
Modular interval spaces represent a common generalization of Banach spaces of type LI(p) or B(X), of hyperconvex metric spaces, modular lattices, modular graphs, and median algebras. It turns out that several types of structures are susceptible for a notion capturing essential features of modularity in lattices, e.g., semilattices, multilattices, metric spaces, ternary algebras, and graphs. There is no perfect correspondence between modular structures of various types unless the existence of a neutral point is imposed. Modular structures with neutral points embed in modular lattices. Particular modular interval spaces (e.g., median spaces, or more generally, modular spaces in which intervals are lattices) can be characterized by forbidden subspaces.
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