Median algebra

In mathematics, a median algebra is a set with a ternary operation ${\displaystyle \langle x,y,z\rangle }$ 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. ${\displaystyle \langle x,y,y\rangle =y}$
2. ${\displaystyle \langle x,y,z\rangle =\langle z,x,y\rangle }$
3. ${\displaystyle \langle x,y,z\rangle =\langle x,z,y\rangle }$
4. ${\displaystyle \langle \langle x,w,y\rangle ,w,z\rangle =\langle x,w,\langle y,w,z\rangle \rangle }$

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

• ${\displaystyle \langle x,y,y\rangle =y}$
• ${\displaystyle \langle u,v,\langle u,w,x\rangle \rangle =\langle u,x,\langle w,u,v\rangle \rangle }$

also suffice.

In a Boolean algebra, or more generally a distributive lattice, the median function ${\displaystyle \langle x,y,z\rangle =(x\vee y)\wedge (y\vee z)\wedge (z\vee x)}$ 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 ${\displaystyle \langle 0,x,1\rangle =x}$ is a distributive lattice.