View source for Median algebra

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{{MSC|08}}

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 
# $\langle x,y,y \rangle = y$                   
# $\langle x,y,z \rangle = \langle z,x,y \rangle$            
# $\langle x,y,z \rangle = \langle x,z,y \rangle$            
# $\langle \langle x,w,y \rangle ,w,z \rangle = \langle x,w, \langle y,w,z \rangle \rangle$

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==
* Garrett Birkhoff, S.A. Kiss, "A ternary operation in distributive lattices", ''Bull. Amer. Math. Soc.'' '''53''' (1947) pp. 749-752  {{ZBL|0031.25002}}
* John R. Isbell, "Median algebra", ''Trans. Amer. Math. Soc.'' '''260''' (1980) pp. 319-362 {{ZBL|0446.06007}}
* Donald E. Knuth, "Introduction to combinatorial algorithms and Boolean functions", ser. The Art of Computer Programming 4.0 (2008) pp. 64-74 ISBN 0-321-53496-4  {{ZBL|1178.68372}}