The axioms are
The second and third axioms imply commutativity: it is possible (but not easy) to show that in the presence of the other three, axiom (3) is redundant. The fourth axiom implies associativity. There are other possible axiom systems: for example the two
Birkhoff and Kiss showed that a median algebra with elements and satisfying is a distributive lattice.
Relation to median graphs
A median graph is an undirected graph in which for every three vertices , , and there is a unique vertex that belongs to shortest paths between any two of , , and . If this is the case, then the operation defines a median algebra having the vertices of the graph as its elements.
Conversely, in any median algebra, one may define an interval to be the set of elements such that . One may define a graph from a median algebra by creating a vertex for each algebra element and an edge for each pair such that the interval contains no other elements. If the algebra has the property that every interval is finite, then this graph is a median graph, and it accurately represents the algebra in that the median operation defined by shortest paths on the graph coincides with the algebra's original median operation.
- Birkhoff, Garrett; Kiss, S.A. (1947). "A ternary operation in distributive lattices". Bull. Amer. Math. Soc. 53 (8): 749–752. doi:10.1090/S0002-9904-1947-08864-9.
- Isbell, John R. (August 1980). "Median algebra". Trans. Amer. Math. Soc. American Mathematical Society. 260 (2): 319–362. doi:10.2307/1998007. JSTOR 1998007.
- Knuth, Donald E. (2008). Introduction to combinatorial algorithms and Boolean functions. The Art of Computer Programming. 4.0. Upper Saddle River, NJ: Addison-Wesley. pp. 64–74. ISBN 0-321-53496-4.