Median algebra

2020 Mathematics Subject Classification: Primary: 08-XX [MSN][ZBL]

A set with a ternary operation $\langle x,y,z \rangle$ satisfying a set of axioms which generalise the notion of median, or majority vote, as a Boolean 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 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 $\langle 0,x,1 \rangle = x$ is a distributive lattice.

References

• Birkhoff, Garrett; Kiss, ; A ternary operation in distributive lattices, Bull. Amer. Math. Soc., 53 (1947) pp. 749-752
• Isbell, John R.; Median algebra, Trans. Amer. Math. Soc., 260 (1980) pp. 319-362
• Knuth, Donald E.; Introduction to combinatorial algorithms and Boolean functions, ser. The Art of Computer Programming 4.0 (2008) pp. 64-74 ISBN: 0-321-53496-4