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

• \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 \langle0,x,1 \rangle = x is a distributive lattice.