MODULAR INTERVAL SPACES

Modular interval spaces represent a common generalization of Banach spaces of type L1(mu) 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 and lattices) can be characterized by forbidden subspaces.