Median algebra

In mathematics, a median algebra is a set with a ternary operation $$\langle x,y,z \rangle$$ satisfying a set of axioms which generalise the notions of medians of triples of real numbers and of the Boolean majority function.

The axioms are
 * 1)   $$\langle x,y,y \rangle = y$$
 * 2)   $$\langle x,y,z \rangle = \langle z,x,y \rangle$$
 * 3)   $$\langle x,y,z \rangle = \langle x,z,y \rangle$$
 * 4)   $$\langle \langle x,w,y\rangle ,w,z \rangle = \langle x,w, \langle y,w,z \rangle\rangle$$

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 also suffice.
 * $$\langle x,y,y \rangle = y$$
 * $$\langle u,v, \langle u,w,x \rangle\rangle = \langle u,x, \langle w,u,v \rangle\rangle$$

In a Boolean algebra, or more generally a distributive lattice, the median function $$\langle x,y,z \rangle = (x \vee y) \wedge (y \vee z) \wedge (z \vee x)$$ satisfies these axioms, so that every Boolean algebra and every distributive lattice forms a median algebra.

Birkhoff and Kiss showed that a median algebra with elements 0 and 1 satisfying $$\langle0,x,1 \rangle = x$$ is a distributive lattice.

Relation to median graphs
A median graph is an undirected graph in which for every three vertices $$x$$, $$y$$, and $$z$$ there is a unique vertex $$\langle x,y,z \rangle$$ that belongs to shortest paths between any two of $$x$$, $$y$$, and $$z$$. If this is the case, then the operation $$\langle x,y,z \rangle$$ defines a median algebra having the vertices of the graph as its elements.

Conversely, in any median algebra, one may define an interval $$[x, z]$$ to be the set of elements $$y$$ such that $$\langle x,y,z \rangle = y$$. One may define a graph from a median algebra by creating a vertex for each algebra element and an edge for each pair $$(x, z)$$ such that the interval $$[x, z]$$ 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.