Median algebra: Difference between revisions
Jump to navigation
Jump to search
imported>Richard Pinch (New article, my own wording from Wikipedia) |
mNo edit summary |
||
(2 intermediate revisions by 2 users not shown) | |||
Line 1: | Line 1: | ||
{{subpages}} | |||
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]]. | ||
Line 13: | Line 14: | ||
also suffice. | also suffice. | ||
In a [[ | 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 < 0,x,1 > = x is a [[distributive lattice]]. | Birkhoff and Kiss showed that a median algebra with elements 0 and 1 satisfying < 0,x,1 > = x is a [[distributive lattice]]. | ||
Line 23: | Line 24: | ||
==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: | |||
Latest revision as of 11:00, 17 September 2024
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
- < x,y,y > = y
- < x,y,z > = < z,x,y >
- < x,y,z > = < x,z,y >
- < < 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
- Birkhoff, Garrett (1947). "A ternary operation in distributive lattices". Bull. Amer. Math. Soc. 53: 749-752.
- Isbell, John R. (August 1980). "Median algebra". Trans. Amer. Math. Soc. 260 (2): 319-362.
- Knuth, Donald E. (2008). Introduction to combinatorial algorithms and Boolean functions, 64-74. ISBN 0-321-53496-4.