title: Median algebra
text: # 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
  
  * $\langle x,y,y\rangle =y$
  * $\langle x,y,z\rangle =\langle z,x,y\rangle$
  * $\langle x,y,z\rangle =\langle x,z,y\rangle$
  * $\langle \langle x,w,y\rangle ,w,z\rangle =\langle x,w,\langle y,w,z\rangle \rangle$
  
  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
  
  * $\langle x,y,y\rangle =y$
  * $\langle u,v,\langle u,w,x\rangle \rangle =\langle u,x,\langle w,u,v\rangle \rangle$
  
  also suffice.
  
  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 $\langle 0,x,1\rangle =x$ 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.
  
  