Median algebra

In mathematics, a median algebra is a set with a ternary operation < x,y,z > satisfying a set of axioms which generalise the notion of median, or majority vote, as a Boolean function.

The axioms are

1. < x,y,y > = y
2. < x,y,z > = < z,x,y >
3. < x,y,z > = < x,z,y >
4. < < x,w,y > ,w,z > = < x,w, < y,w,z > >

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

• < x,y,y > = y
• < u,v, < u,w,x > > = < u,x, < w,u,v > >

also suffice.

In a Boolean algebra 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 is a median algebra.

Birkhoff and Kiss showed that a median algebra with elements 0 and 1 satisfying < 0,x,1 > = x is a distributive lattice.

References

• Birkhoff, Garrett (1947). "A ternary operation in distributive lattices". Bull. Amer. Math. Soc. 53: 749-752.

• Isbell, John R. (August 1980). "Median algebra". Trans. Amer. Math. Soc. 260 (2): 319-362.

• Knuth, Donald E. (2008). Introduction to combinatorial algorithms and Boolean functions, 64-74. ISBN 0-321-53496-4. 

External links

• Median Algebra Proof