mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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 partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...

is the

binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...

which holds between

comparable Comparable may refer to: * Comparability In mathematics, two elements ''x'' and ''y'' of a set ''P'' are said to be comparable with respect to a binary relation ≤ if at least one of ''x'' ≤ ''y'' or ''y'' ≤ ''x'' is true. They are ca ...

elements that are immediate neighbours. The covering relation is commonly used to graphically express the partial order by means of the

Hasse diagram In order theory, a Hasse diagram (; ) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ''(S, ≤)'' one represents ...

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 variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...

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 post ...

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 In mathematics, Young's lattice is a lattice that is formed by all integer partitions. It is named after Alfred Young, who, in a series of papers ''On quantitative substitutional analysis,'' developed the representation theory of the symmetric ...

, formed by the

partitions Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of ...

of all nonnegative integers, a partition ''λ'' covers a partition ''μ'' if and only if the

Young diagram In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups ...

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 an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...

of an

associahedron In mathematics, an associahedron is an -dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a string of letters, and the edges correspond to single application of ...

. * The covering relation of any finite

distributive lattice In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set ...

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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s 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 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 (e.g. ). The set of all ra ...

s with the standard order, no element covers another.

References

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