Median algebra

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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 notion of median or majority function, 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, 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 < 0,x,1 > = x is a distributive lattice.

Relation to median graphs

A median graph is an undirected graph in which for every three vertices x, y, and z there is a unique vertex < x,y,z > that belongs to shortest paths between any two of x, y, and z. If this is the case, then the operation < x,y,z > defines a median algebra having the vertices of the graph as its elements.

Conversely, in any median algebra, one may define an interval [x, z] to be the set of elements y such that < x,y,z > = y. One may define a graph from a median algebra by creating a vertex for each algebra element and an edge for each pair (x, z) such that the interval [x, z] contains no other elements. If the algebra has the property that every interval is finite, then this graph is a median graph, and it accurately represents the algebra in that the median operation defined by shortest paths on the graph coincides with the algebra's original median operation.


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