# Median algebra

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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

- < x,y,y > = y
- < x,y,z > = < z,x,y >
- < x,y,z > = < x,z,y >
- < < 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 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.