Let be a set with a partial order .
As usual, let be the relation on such that if and only if and .
Let and be elements of .
Then covers, written ,
if and there is no element such that . Equivalently, covers if the
interval is the two-element set .
When , it is said that is a cover of . Some authors also use the term cover to denote any such pair 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.
On the
real numbers with the usual total order ≤, the cover set is empty: no number covers another.
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.