mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, especially

order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...

, the covering relation of a

partially ordered set In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...

is the

binary relation In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...

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

\le

. As usual, let

<

be the relation on

X

such that

x if and only if x\le y and x\neq y .

Let x and y be elements of X .

Then y covers x, written x\lessdot y,
if x and there is no element z such that x . Equivalently, y covers x if the interval,y /math> is the two-element set \ .

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

* In a finite linearly ordered set

\

i+1

covers

i

for all

i

between

1

and

n-1

, and there are no other covering relations. * In the

Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...

of the

power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...

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 A skeleton is the structural frame that supports the body of most animals. There are several types of skeletons, including the exoskeleton, which is a rigid outer shell that holds up an organism's shape; the endoskeleton, a rigid internal fra ...

of an associahedron. * The covering relation of any finite

distributive lattice In mathematics, a distributive lattice is a lattice (order), lattice in which the operations of join and meet distributivity, distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice o ...

forms a median graph. * On the

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s with the usual total order ≤, 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 In mathematics, a partial order or total order < on a

set

X

is said to be dense if, for all

x

, such as the

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...

s with the standard order, no element covers another.

References

* . * . * . {{Order theory Binary relations Order theory