Covering relation
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In mathematics, especially order theory, a covering relation is a binary relation which holds between two comparable elements in a partially ordered set if they are immediate neighbours. The covering relation is commonly used to graphically express the partial order by means of the Hasse diagram.
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[edit] Definition
Let x and y be elements of a partially ordered set P. Then y covers x, written x <: y, if x < y, but there is no element z such that x < z < y. Equivalently, y covers x if the interval [x, y] is the two-element set consisting of x and y.
[edit] Examples
- In a finite linearly ordered set 1,… n, i + 1 covers i for all i between 1 and n − 1 (and there are no other covering relations).
- In the Boolean algebra of the power set of a set S, a subset B of S covers a subset A of S if and only if B is obtained from A by adding one element not in A.
- In Young's lattice, formed by the partitions of all nonnegative integers, a partition λ covers a partition μ if and only if the Young diagram of λ is obtained from the Young diagram of μ by adding an extra cell.
- The Hasse diagram depicting the covering relation of a Tamari lattice is the skeleton of an associahedron.
- The covering relation of any distributive lattice forms a median graph.
[edit] Properties
- If a partially ordered set is finite, its covering relation is the transitive reduction of the partial order relation. Such partially ordered sets are therefore completely described by their Hasse diagrams. On the other hand, in a dense order, such as the rational numbers with the standard order, no element covers another.
[edit] References
- Knuth, Donald E. (2006), The Art of Computer Programming, Volume 4, Fascicle 4, Addison-Wesley, ISBN ISBN 0-321-33570-8.
- Stanley, Richard P. (1997), Enumerative Combinatorics, 1 (2nd ed.), Cambridge University Press, ISBN ISBN 0-521-55309-1, http://www-math.mit.edu/~rstan/ec/.
