Median algebra: Difference between revisions

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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]].
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]].


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also suffice.
also suffice.


In a [[Boolean algebra (introduction)|Boolean algebra]] the median function <math>\langle x,y,z \rangle = (x \vee y) \wedge (y \vee z) \wedge (z \vee x)</math> satisfies these axioms, so that every Boolean algebra is a median algebra.
In a [[Boolean algebra]] the median function <math>\langle x,y,z \rangle = (x \vee y) \wedge (y \vee z) \wedge (z \vee x)</math> 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 &lt; 0,x,1 &gt; = x is a [[distributive lattice]].
Birkhoff and Kiss showed that a median algebra with elements 0 and 1 satisfying &lt; 0,x,1 &gt; = x is a [[distributive lattice]].
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==External links==
==External links==
* [http://www.cs.unm.edu/~veroff/MEDIAN_ALGEBRA/ Median Algebra Proof]
* [http://www.cs.unm.edu/~veroff/MEDIAN_ALGEBRA/ Median Algebra Proof][[Category:Suggestion Bot Tag]]
 
[[Category:Algebra]]
 
{{algebra-stub}}

Latest revision as of 11:00, 17 September 2024

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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

  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