Covering relation

In mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements that are immediate neighbours. The covering relation is commonly used to graphically express the partial order by means of the Hasse diagram.

Definition

Let $X$ be a set with a partial order $\leq$ . As usual, let $<$ be the relation on $X$ such that $x < y$ if and only if $x \leq y$ and $x \neq y$ .

Let $x$ and $y$ be elements of $X$ .

Then $y$ covers $x$ , written $x ⋖ y$ , if $x < y$ and there is no element $z$ such that $x < z < y$ . Equivalently, $y$ covers $x$ if the interval $[x, y]$ is the two-element set ${x, y}$ . In more intuitive words, $x ⋖ y$ if $y$ immediately supersedes or succeeds $x$ in terms of their respective poset's order relation.

When $x ⋖ y$ , it is said that $y$ is a cover of $x$ . Some authors also use the term cover to denote any such pair $(x, y)$ in the covering relation.

Examples

In a finite linearly ordered set ${1, 2, ..., n}$ , $i + 1$ 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 finite distributive lattice forms a median graph.
On the real numbers with the usual total order ≤, no number covers another. Any such partial order in which the covering relation is empty is called "dense."

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.

References

Knuth, Donald E. (2006), The Art of Computer Programming, Volume 4, Fascicle 4, Addison-Wesley, ISBN 0-321-33570-8.
Stanley, Richard P. (1997), Enumerative Combinatorics, vol. 1 (2nd ed.), Cambridge University Press, ISBN 0-521-55309-1.
Brian A. Davey; Hilary Ann Priestley (2002), Introduction to Lattices and Order (2nd ed.), Cambridge University Press, ISBN 0-521-78451-4, LCCN 2001043910.

v t e Order theory
Topics Glossary Category
Key concepts	Binary relation Boolean algebra Cyclic order Lattice Partially ordered set Preorder Total order Weak ordering
Results	Boolean prime ideal theorem Cantor–Bernstein theorem Cantor's isomorphism theorem Dilworth's theorem Dushnik–Miller theorem Hausdorff maximal principle Knaster–Tarski theorem Kruskal's tree theorem Laver's theorem Mirsky's theorem Szpilrajn extension theorem Zorn's lemma
Properties & Types (list)	Antisymmetric Asymmetric Boolean algebra topics Completeness Connected Covering Dense Directed (Partial) Equivalence Foundational Heyting algebra Homogeneous Idempotent Lattice Bounded Complemented Complete Distributive Join and meet Partially ordered set Chain-complete Eulerian Graded Locally finite Strict Prefix order Preorder Total Reflexive Semilattice Semiorder Symmetric Tolerance Total Transitive Well-founded Well-quasi-ordering (Better) (Pre) Well-order
Constructions	Composition Converse/Transpose Lexicographic order Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure
Topology & Orders	Alexandrov topology & Specialization preorder Ordered topological vector space Normal cone Order topology Order topology Topological vector lattice Banach Fréchet Locally convex Normed
Related	Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding Isomorphism Order type Ordered field Positive cone of an ordered field Ordered vector space Partially ordered Positive cone of an ordered vector space Riesz space Partially ordered group Positive cone of a partially ordered group Upper set Young's lattice