Median algebra

From Citizendium, the Citizens' Compendium
Jump to: navigation, search
This article is developing and not approved.
Main Article
Talk
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and not meant to be cited; by editing it you can help to improve it towards a future approved, citable version. These unapproved articles are subject to a disclaimer.

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 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

External links