Covering relation

Inmathematics,especiallyorder theory,thecovering relationof apartially ordered setis thebinary relationwhich holds betweencomparableelements that are immediate neighbours. The covering relation is commonly used to graphically express the partial order by means of theHasse diagram.

Definition[edit]

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 $x\lessdot y$ , if $x<y$ and there is no element $z$ such that $x<z<y$ .Equivalently, $y$ covers $x$ if theinterval $[x,y]$ is the two-element set $\{x,y\}$ .

When $x\lessdot y$ ,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[edit]

In a finitelinearly orderedset {1, 2,...,n},i+ 1 coversifor allibetween 1 andn− 1 (and there are no other covering relations).
In theBoolean algebraof thepower setof a setS,a subsetBofScovers a subsetAofSif and only ifBis obtained fromAby adding one element not inA.
InYoung's lattice,formed by thepartitionsof all nonnegative integers, a partitionλcovers a partitionμif and only if theYoung diagramofλis obtained from the Young diagram ofμby adding an extra cell.
The Hasse diagram depicting the covering relation of aTamari latticeis theskeletonof anassociahedron.
The covering relation of any finitedistributive latticeforms amedian graph.
On thereal numberswith the usual total order ≤, the cover set is empty: no number covers another.

Properties[edit]

If a partially ordered set is finite, its covering relation is thetransitive reductionof the partial order relation. Such partially ordered sets are therefore completely described by their Hasse diagrams. On the other hand, in adense order,such as therational numberswith the standard order, no element covers another.

References[edit]

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 Partial order 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 Reflexive Partial order Chain-complete Graded Eulerian Strict Prefix order Preorder Total Semilattice Semiorder Symmetric Total Tolerance 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