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
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.