The algebra of metric betweenness II: Geometry and equational characterization of weakly median graphs

We elaborate on the convexity properties of (not necessarily finite) weakly median graphs and their prime constituents in order to establish a number of equations in terms of the weakly median operation. Then the weakly median graphs can be identified with the discrete members of an equational class of ternary algebras satisfying five independent axioms on two to four points. This demonstrates that the median algebras featured by Avann and Sholander half a century ago and, more generally, Isbell’s isotropic media can be generalized much further, without losing the close ties with graphs.