Median algebra

Summary

A median algebra is a set with a ternary operation $\langle x, y, z \rangle$ that behaves like the median of three numbers. Every distributive lattice is a median algebra with $\langle x, y, z \rangle = (x \wedge y) \vee (y \wedge z) \vee (z \wedge x)$. The vertex sets of median graphs, such as trees and hypercubes, are median algebras. There $\langle x, y, z \rangle$ is the unique vertex lying on shortest paths between each pair of the three vertices. Median algebras give a common language for distributive lattices, median graphs and some structures of geometric group theory.