Partially ordered set

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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. Partially ordered set_sentence_0

A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. Partially ordered set_sentence_1

The relation itself is called a "partial order." Partially ordered set_sentence_2

The word partial in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. Partially ordered set_sentence_3

That is, there may be pairs of elements for which neither element precedes the other in the poset. Partially ordered set_sentence_4

Partial orders thus generalize total orders, in which every pair is comparable. Partially ordered set_sentence_5

Formally, a partial order is any binary relation that is reflexive (each element is comparable to itself), antisymmetric (no two different elements precede each other), and transitive (the start of a chain of precedence relations must precede the end of the chain). Partially ordered set_sentence_6

One familiar example of a partially ordered set is a collection of people ordered by genealogical descendancy. Partially ordered set_sentence_7

Some pairs of people bear the descendant-ancestor relationship, but other pairs of people are incomparable, with neither being a descendant of the other. Partially ordered set_sentence_8

A poset can be visualized through its Hasse diagram, which depicts the ordering relation. Partially ordered set_sentence_9

Formal definition Partially ordered set_section_0

A (non-strict) partial order is a homogeneous binary relation ≤ over a set P satisfying particular axioms which are discussed below. Partially ordered set_sentence_10

When a ≤ b, we say that a is related to b. Partially ordered set_sentence_11

(This does not imply that b is also related to a, because the relation need not be symmetric.) Partially ordered set_sentence_12

The axioms for a non-strict partial order state that the relation ≤ is reflexive, antisymmetric, and transitive. Partially ordered set_sentence_13

That is, for all a, b, and c in P, it must satisfy: Partially ordered set_sentence_14

Partially ordered set_ordered_list_0

  1. a ≤ a (reflexivity: every element is related to itself).Partially ordered set_item_0_0
  2. if a ≤ b and b ≤ a, then a = b (antisymmetry: two distinct elements cannot be related in both directions).Partially ordered set_item_0_1
  3. if a ≤ b and b ≤ c, then a ≤ c (transitivity: if a first element is related to a second element, and, in turn, that element is related to a third element, then the first element is related to the third element).Partially ordered set_item_0_2

In other words, a partial order is an antisymmetric preorder. Partially ordered set_sentence_15

A set with a partial order is called a partially ordered set (also called a poset). Partially ordered set_sentence_16

The term ordered set is sometimes also used, as long as it is clear from the context that no other kind of order is meant. Partially ordered set_sentence_17

In particular, totally ordered sets can also be referred to as "ordered sets", especially in areas where these structures are more common than posets. Partially ordered set_sentence_18

For a, b, elements of a partially ordered set P, if a ≤ b or b ≤ a, then a and b are comparable. Partially ordered set_sentence_19

Otherwise they are incomparable. Partially ordered set_sentence_20

In the figure on top-right, e.g. {x} and {x, y, z} are comparable, while {x} and {y} are not. Partially ordered set_sentence_21

A partial order under which every pair of elements is comparable is called a total order or linear order; a totally ordered set is also called a chain (e.g., the natural numbers with their standard order). Partially ordered set_sentence_22

A subset of a poset in which no two distinct elements are comparable is called an antichain (e.g. the set of singletons {{x}, {y}, {z}} in the top-right figure). Partially ordered set_sentence_23

An element a is said to be strictly less than an element b, if a ≤ b and a ≠ b. Partially ordered set_sentence_24

An element a is said to be covered by another element b, written a ⋖ b (or a <: b), if a is strictly less than b and no third element c fits between them; formally: if both a ≤ b and a ≠ b are true, and a ≤ c ≤ b is false for each c with a ≠ c ≠ b. Partially ordered set_sentence_25

A more concise definition will be given below using the strict order corresponding to "≤". Partially ordered set_sentence_26

For example, {x} is covered by {x, z} in the top-right figure, but not by {x, y, z}. Partially ordered set_sentence_27

Examples Partially ordered set_section_1

Standard examples of posets arising in mathematics include: Partially ordered set_sentence_28

Partially ordered set_unordered_list_1

  • The real numbers ordered by the standard less-than-or-equal relation ≤ (a totally ordered set as well).Partially ordered set_item_1_3
  • The set of subsets of a given set (its power set) ordered by inclusion (see the figure on top-right). Similarly, the set of sequences ordered by subsequence, and the set of strings ordered by substring.Partially ordered set_item_1_4
  • The set of natural numbers equipped with the relation of divisibility.Partially ordered set_item_1_5
  • The vertex set of a directed acyclic graph ordered by reachability.Partially ordered set_item_1_6
  • The set of subspaces of a vector space ordered by inclusion.Partially ordered set_item_1_7
  • For a partially ordered set P, the sequence space containing all sequences of elements from P, where sequence a precedes sequence b if every item in a precedes the corresponding item in b. Formally, (an)n∈ℕ ≤ (bn)n∈ℕ if and only if an ≤ bn for all n in ℕ, i.e. a componentwise order.Partially ordered set_item_1_8
  • For a set X and a partially ordered set P, the function space containing all functions from X to P, where f ≤ g if and only if f(x) ≤ g(x) for all x in X.Partially ordered set_item_1_9
  • A fence, a partially ordered set defined by an alternating sequence of order relations a < b > c < d ...Partially ordered set_item_1_10
  • The set of events in special relativity and, in most cases, general relativity, where for two events X and Y, X ≤ Y if and only if Y is in the future light cone of X. An event Y can only be causally affected by X if X ≤ Y.Partially ordered set_item_1_11

Extrema Partially ordered set_section_2

There are several notions of "greatest" and "least" element in a poset P, notably: Partially ordered set_sentence_29

Partially ordered set_unordered_list_2

  • Greatest element and least element: An element g in P is a greatest element if for every element a in P, a ≤ g. An element m in P is a least element if for every element a in P, a ≥ m. A poset can only have one greatest or least element.Partially ordered set_item_2_12
  • Maximal elements and minimal elements: An element g in P is a maximal element if there is no element a in P such that a > g. Similarly, an element m in P is a minimal element if there is no element a in P such that a < m. If a poset has a greatest element, it must be the unique maximal element, but otherwise there can be more than one maximal element, and similarly for least elements and minimal elements.Partially ordered set_item_2_13
  • Upper and lower bounds: For a subset A of P, an element x in P is an upper bound of A if a ≤ x, for each element a in A. In particular, x need not be in A to be an upper bound of A. Similarly, an element x in P is a lower bound of A if a ≥ x, for each element a in A. A greatest element of P is an upper bound of P itself, and a least element is a lower bound of P.Partially ordered set_item_2_14

For example, consider the positive integers, ordered by divisibility: 1 is a least element, as it divides all other elements; on the other hand this poset does not have a greatest element (although if one would include 0 in the poset, which is a multiple of any integer, that would be a greatest element; see figure). Partially ordered set_sentence_30

This partially ordered set does not even have any maximal elements, since any g divides for instance 2g, which is distinct from it, so g is not maximal. Partially ordered set_sentence_31

If the number 1 is excluded, while keeping divisibility as ordering on the elements greater than 1, then the resulting poset does not have a least element, but any prime number is a minimal element for it. Partially ordered set_sentence_32

In this poset, 60 is an upper bound (though not a least upper bound) of the subset {2, 3, 5, 10}, which does not have any lower bound (since 1 is not in the poset); on the other hand 2 is a lower bound of the subset of powers of 2, which does not have any upper bound. Partially ordered set_sentence_33

Orders on the Cartesian product of partially ordered sets Partially ordered set_section_3

In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the Cartesian product of two partially ordered sets are (see figures): Partially ordered set_sentence_34

Partially ordered set_unordered_list_3

  • the lexicographical order:   (a,b) ≤ (c,d) if a < c or (a = c and b ≤ d);Partially ordered set_item_3_15
  • the product order:   (a,b) ≤ (c,d) if a ≤ c and b ≤ d;Partially ordered set_item_3_16
  • the reflexive closure of the direct product of the corresponding strict orders:   (a,b) ≤ (c,d) if (a < c and b < d) or (a = c and b = d).Partially ordered set_item_3_17

All three can similarly be defined for the Cartesian product of more than two sets. Partially ordered set_sentence_35

Applied to ordered vector spaces over the same field, the result is in each case also an ordered vector space. Partially ordered set_sentence_36

See also orders on the Cartesian product of totally ordered sets. Partially ordered set_sentence_37

Sums of partially ordered sets Partially ordered set_section_4

Another way to combine two posets is the ordinal sum (or linear sum), Z = X ⊕ Y, defined on the union of the underlying sets X and Y by the order a ≤Z b if and only if: Partially ordered set_sentence_38

Partially ordered set_unordered_list_4

  • a, b ∈ X with a ≤X b, orPartially ordered set_item_4_18
  • a, b ∈ Y with a ≤Y b, orPartially ordered set_item_4_19
  • a ∈ X and b ∈ Y.Partially ordered set_item_4_20

If two posets are well-ordered, then so is their ordinal sum. Partially ordered set_sentence_39

The ordinal sum operation is one of two operations used to form series-parallel partial orders, and in this context is called series composition. Partially ordered set_sentence_40

The other operation used to form these orders, the disjoint union of two partially ordered sets (with no order relation between elements of one set and elements of the other set) is called in this context parallel composition. Partially ordered set_sentence_41

Strict and non-strict partial orders Partially ordered set_section_5

In some contexts, the partial order defined above is called a non-strict (or reflexive) partial order. Partially ordered set_sentence_42

In these contexts, a strict (or irreflexive) partial order "<" is a binary relation that is irreflexive, transitive and asymmetric, that is, which satisfies the following relations for all a, b, and c in P: Partially ordered set_sentence_43

Partially ordered set_unordered_list_5

  • not a < a (irreflexivity),Partially ordered set_item_5_21
  • if a < b and b < c then a < c (transitivity), andPartially ordered set_item_5_22
  • if a < b then not b < a (asymmetry; implies irreflexivity; implied by irreflexivity and transitivity).Partially ordered set_item_5_23

Strict and non-strict partial orders are closely related. Partially ordered set_sentence_44

A non-strict partial order may be converted to a strict partial order by removing all relationships of the form a ≤ a. Conversely, a strict partial order may be converted to a non-strict partial order by adjoining all relationships of that form. Partially ordered set_sentence_45

Thus, if "≤" is a non-strict partial order, then the corresponding strict partial order "<" is the irreflexive kernel given by Partially ordered set_sentence_46

Partially ordered set_description_list_6

  • a < b if a ≤ b and a ≠ b.Partially ordered set_item_6_24

Conversely, if "<" is a strict partial order, then the corresponding non-strict partial order "≤" is the reflexive closure given by: Partially ordered set_sentence_47

Partially ordered set_description_list_7

  • a ≤ b if a < b or a = b.Partially ordered set_item_7_25

This is the reason for using the notation "≤". Partially ordered set_sentence_48

Using the strict order "<", the relation "a is covered by b" can be equivalently rephrased as "a < b, but not a < c < b for any c". Partially ordered set_sentence_49

Strict partial orders are useful because they correspond more directly to directed acyclic graphs (dags): every strict partial order is a dag, and the transitive closure of a dag is both a strict partial order and also a dag itself. Partially ordered set_sentence_50

Inverse and order dual Partially ordered set_section_6

The inverse (or converse) of a partial order relation ≤ is the converse of ≤. Partially ordered set_sentence_51

Typically denoted ≥, it is the relation that satisfies x ≥ y if and only if y ≤ x. Partially ordered set_sentence_52

The inverse of a partial order relation is reflexive, transitive, and antisymmetric, and hence itself a partial order relation. Partially ordered set_sentence_53

The order dual of a partially ordered set is the same set with the partial order relation replaced by its inverse. Partially ordered set_sentence_54

The irreflexive relation > is to ≥ as < is to ≤. Partially ordered set_sentence_55

Any one of the four relations ≤, <, ≥, and > on a given set uniquely determines the other three. Partially ordered set_sentence_56

In general two elements x and y of a partial order may stand in any of four mutually exclusive relationships to each other: either x < y, or x = y, or x > y, or x and y are incomparable (none of the other three). Partially ordered set_sentence_57

A totally ordered set is one that rules out this fourth possibility: all pairs of elements are comparable and we then say that trichotomy holds. Partially ordered set_sentence_58

The natural numbers, the integers, the rationals, and the reals are all totally ordered by their algebraic (signed) magnitude whereas the complex numbers are not. Partially ordered set_sentence_59

This is not to say that the complex numbers cannot be totally ordered; we could for example order them lexicographically via x + iy < u + iv if and only if x < u or (x = u and y < v), but this is not ordering by magnitude in any reasonable sense as it makes 1 greater than 100i. Partially ordered set_sentence_60

Ordering them by absolute magnitude yields a preorder in which all pairs are comparable, but this is not a partial order since 1 and i have the same absolute magnitude but are not equal, violating antisymmetry. Partially ordered set_sentence_61

Mappings between partially ordered sets Partially ordered set_section_7

Given two partially ordered sets (S, ≤) and (T, ≤), a function f: S → T is called order-preserving, or monotone, or isotone, if for all x and y in S, x ≤ y implies f(x) ≤ f(y). Partially ordered set_sentence_62

If (U, ≤) is also a partially ordered set, and both f: S → T and g: T → U are order-preserving, their composition g∘f : S → U is order-preserving, too. Partially ordered set_sentence_63

A function f: S → T is called order-reflecting if for all x and y in S, f(x) ≤ f(y) implies x ≤ y. Partially ordered set_sentence_64

If f is both order-preserving and order-reflecting, then it is called an order-embedding of (S, ≤) into (T, ≤). Partially ordered set_sentence_65

In the latter case, f is necessarily injective, since f(x) = f(y) implies x ≤ y and y ≤ x. Partially ordered set_sentence_66

If an order-embedding between two posets S and T exists, one says that S can be embedded into T. If an order-embedding f: S → T is bijective, it is called an order isomorphism, and the partial orders (S, ≤) and (T, ≤) are said to be isomorphic. Partially ordered set_sentence_67

Isomorphic orders have structurally similar Hasse diagrams (cf. Partially ordered set_sentence_68

right picture). Partially ordered set_sentence_69

It can be shown that if order-preserving maps f: S → T and g: T → S exist such that g∘f and f∘g yields the identity function on S and T, respectively, then S and T are order-isomorphic. Partially ordered set_sentence_70

For example, a mapping f: ℕ → ℙ(ℕ) from the set of natural numbers (ordered by divisibility) to the power set of natural numbers (ordered by set inclusion) can be defined by taking each number to the set of its prime divisors. Partially ordered set_sentence_71

It is order-preserving: if x divides y, then each prime divisor of x is also a prime divisor of y. Partially ordered set_sentence_72

However, it is neither injective (since it maps both 12 and 6 to {2, 3}) nor order-reflecting (since 12 doesn't divide 6). Partially ordered set_sentence_73

Taking instead each number to the set of its prime power divisors defines a map g: ℕ → ℙ(ℕ) that is order-preserving, order-reflecting, and hence an order-embedding. Partially ordered set_sentence_74

It is not an order-isomorphism (since it e.g. doesn't map any number to the set {4}), but it can be made one by restricting its codomain to g(ℕ). Partially ordered set_sentence_75

The right picture shows a subset of ℕ and its isomorphic image under g. The construction of such an order-isomorphism into a power set can be generalized to a wide class of partial orders, called distributive lattices, see "Birkhoff's representation theorem". Partially ordered set_sentence_76

Number of partial orders Partially ordered set_section_8

Sequence in OEIS gives the number of partial orders on a set of n labeled elements: Partially ordered set_sentence_77

Partially ordered set_table_general_0

Number of n-element binary relations of different typesPartially ordered set_table_caption_0
Elem­entsPartially ordered set_header_cell_0_0_0 AnyPartially ordered set_header_cell_0_0_1 TransitivePartially ordered set_header_cell_0_0_2 ReflexivePartially ordered set_header_cell_0_0_3 PreorderPartially ordered set_header_cell_0_0_4 Partial orderPartially ordered set_header_cell_0_0_5 Total preorderPartially ordered set_header_cell_0_0_6 Total orderPartially ordered set_header_cell_0_0_7 Equivalence relationPartially ordered set_header_cell_0_0_8
0Partially ordered set_cell_0_1_0 1Partially ordered set_cell_0_1_1 1Partially ordered set_cell_0_1_2 1Partially ordered set_cell_0_1_3 1Partially ordered set_cell_0_1_4 1Partially ordered set_cell_0_1_5 1Partially ordered set_cell_0_1_6 1Partially ordered set_cell_0_1_7 1Partially ordered set_cell_0_1_8
1Partially ordered set_cell_0_2_0 2Partially ordered set_cell_0_2_1 2Partially ordered set_cell_0_2_2 1Partially ordered set_cell_0_2_3 1Partially ordered set_cell_0_2_4 1Partially ordered set_cell_0_2_5 1Partially ordered set_cell_0_2_6 1Partially ordered set_cell_0_2_7 1Partially ordered set_cell_0_2_8
2Partially ordered set_cell_0_3_0 16Partially ordered set_cell_0_3_1 13Partially ordered set_cell_0_3_2 4Partially ordered set_cell_0_3_3 4Partially ordered set_cell_0_3_4 3Partially ordered set_cell_0_3_5 3Partially ordered set_cell_0_3_6 2Partially ordered set_cell_0_3_7 2Partially ordered set_cell_0_3_8
3Partially ordered set_cell_0_4_0 512Partially ordered set_cell_0_4_1 171Partially ordered set_cell_0_4_2 64Partially ordered set_cell_0_4_3 29Partially ordered set_cell_0_4_4 19Partially ordered set_cell_0_4_5 13Partially ordered set_cell_0_4_6 6Partially ordered set_cell_0_4_7 5Partially ordered set_cell_0_4_8
4Partially ordered set_cell_0_5_0 65,536Partially ordered set_cell_0_5_1 3,994Partially ordered set_cell_0_5_2 4,096Partially ordered set_cell_0_5_3 355Partially ordered set_cell_0_5_4 219Partially ordered set_cell_0_5_5 75Partially ordered set_cell_0_5_6 24Partially ordered set_cell_0_5_7 15Partially ordered set_cell_0_5_8
nPartially ordered set_cell_0_6_0 2Partially ordered set_cell_0_6_1 Partially ordered set_cell_0_6_2 2Partially ordered set_cell_0_6_3 Partially ordered set_cell_0_6_4 Partially ordered set_cell_0_6_5 ∑n

k=0  k! S(n, k)Partially ordered set_cell_0_6_6

n!Partially ordered set_cell_0_6_7 ∑n

k=0  S(n, k)Partially ordered set_cell_0_6_8

OEISPartially ordered set_header_cell_0_7_0 Partially ordered set_header_cell_0_7_1 Partially ordered set_header_cell_0_7_2 Partially ordered set_header_cell_0_7_3 Partially ordered set_header_cell_0_7_4 Partially ordered set_header_cell_0_7_5 Partially ordered set_header_cell_0_7_6 Partially ordered set_header_cell_0_7_7 Partially ordered set_header_cell_0_7_8

The number of strict partial orders is the same as that of partial orders. Partially ordered set_sentence_78

If the count is made only up to isomorphism, the sequence 1, 1, 2, 5, 16, 63, 318, … (sequence in the OEIS) is obtained. Partially ordered set_sentence_79

Linear extension Partially ordered set_section_9

A partial order ≤ on a set X is an extension of another partial order ≤ on X provided that for all elements x and y of X, whenever x ≤ y, it is also the case that x ≤ y. Partially ordered set_sentence_80

A linear extension is an extension that is also a linear (i.e., total) order. Partially ordered set_sentence_81

Every partial order can be extended to a total order (order-extension principle). Partially ordered set_sentence_82

In computer science, algorithms for finding linear extensions of partial orders (represented as the reachability orders of directed acyclic graphs) are called topological sorting. Partially ordered set_sentence_83

In category theory Partially ordered set_section_10

Every poset (and every preordered set) may be considered as a category where, for objects x and y, there is at most one morphism from x to y. Partially ordered set_sentence_84

More explicitly, let hom(x, y) = {(x, y)} if x ≤ y (and otherwise the empty set) and (y, z)∘(x, y) = (x, z). Partially ordered set_sentence_85

Such categories are sometimes called posetal. Partially ordered set_sentence_86

Posets are equivalent to one another if and only if they are isomorphic. Partially ordered set_sentence_87

In a poset, the smallest element, if it exists, is an initial object, and the largest element, if it exists, is a terminal object. Partially ordered set_sentence_88

Also, every preordered set is equivalent to a poset. Partially ordered set_sentence_89

Finally, every subcategory of a poset is isomorphism-closed. Partially ordered set_sentence_90

Partial orders in topological spaces Partially ordered set_section_11

Main article: Partially ordered space Partially ordered set_sentence_91

Intervals Partially ordered set_section_12

An interval in a poset P is a subset I of P with the property that, for any x and y in I and any z in P, if x ≤ z ≤ y, then z is also in I. Partially ordered set_sentence_92

(This definition generalizes the interval definition for real numbers.) Partially ordered set_sentence_93

For a ≤ b, the closed interval [a, b] is the set of elements x satisfying a ≤ x ≤ b (i.e. a ≤ x and x ≤ b). Partially ordered set_sentence_94

It contains at least the elements a and b. Partially ordered set_sentence_95

Using the corresponding strict relation "<", the open interval (a, b) is the set of elements x satisfying a < x < b (i.e. a < x and x < b). Partially ordered set_sentence_96

An open interval may be empty even if a < b. Partially ordered set_sentence_97

For example, the open interval (1, 2) on the integers is empty since there are no integers I such that 1 < I < 2. Partially ordered set_sentence_98

The half-open intervals [a, b) and (a, b] are defined similarly. Partially ordered set_sentence_99

Sometimes the definitions are extended to allow a > b, in which case the interval is empty. Partially ordered set_sentence_100

An interval I is bounded if there exist elements a and b of P such that I ⊆ [a, b]. Partially ordered set_sentence_101

Every interval that can be represented in interval notation is obviously bounded, but the converse is not true. Partially ordered set_sentence_102

For example, let P = (0, 1) ∪ (1, 2) ∪ (2, 3) as a subposet of the real numbers. Partially ordered set_sentence_103

The subset (1, 2) is a bounded interval, but it has no infimum or supremum in P, so it cannot be written in interval notation using elements of P. Partially ordered set_sentence_104

A poset is called locally finite if every bounded interval is finite. Partially ordered set_sentence_105

For example, the integers are locally finite under their natural ordering. Partially ordered set_sentence_106

The lexicographical order on the cartesian product ℕ×ℕ is not locally finite, since (1, 2) ≤ (1, 3) ≤ (1, 4) ≤ (1, 5) ≤ ... ≤ (2, 1). Partially ordered set_sentence_107

Using the interval notation, the property "a is covered by b" can be rephrased equivalently as [a, b] = {a, b}. Partially ordered set_sentence_108

This concept of an interval in a partial order should not be confused with the particular class of partial orders known as the interval orders. Partially ordered set_sentence_109

See also Partially ordered set_section_13

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: ordered set.