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In mathematics, especially order theory, a partial order on a 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 needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable.

Formally, a partial order is a homogeneous binary relation that is reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair (called the ground set itself is sometimes called a poset.

Partial order relations

The term partial order usually refers to the reflexive partial order relations, referred to in this article as non-strict partial orders. However some authors use the term for the other common type of partial order relations, the irreflexive partial order relations, also called strict partial orders. Strict and non-strict partial orders can be put into a one-to-one correspondence, so for every strict partial order there is a unique corresponding non-strict partial order, and vice versa.

Partial orders

A reflexive, weak, or , commonly referred to simply as a partial order, is a homogeneous relation ≤ on a set that is reflexive, antisymmetric, and transitive it must satisfy:

Reflexivity, i.e. every element is related to itself.
Antisymmetry, i.e. no two distinct elements precede each other.
Transitivity.

A non-strict partial order is also known as an antisymmetric preorder.

Strict partial orders

An irreflexive, strong, or that is irreflexive, asymmetric and transitive

Irreflexivity, i.e. no element is related to itself (also called anti-reflexive).
Asymmetry.
Transitivity.

A transitive relation is asymmetric if and only if it is irreflexive. So the definition is the same if it omits either irreflexivity or asymmetry (but not both).

A strict partial order is also known as a strict preorder.

Correspondence of strict and non-strict partial order relations

is the identity relation denotes set subtraction is a non-strict partial order, then the corresponding strict partial order < is the irreflexive kernel is the reflexive closure

Dual orders

The dual (or opposite be the converse relation. The dual of a non-strict partial order is a non-strict partial order, and the dual of a strict partial order is a strict partial order. The dual of a dual of a relation is the original relation.

Notation

(the irreflexive kernel. Strictly speaking, the term partially ordered set.

The term ordered set is sometimes used as a shorthand for partially ordered set, as long as it is clear from the context that no other kind of order is meant. In particular, totally ordered sets to distinguish partial orders from total orders.

should not be taken as the complement if, and only if is a total order.

Alternative definitions

Another way of defining a partial order, found in computer science, is via a notion of comparison as defined previously, it can be observed that two elements x and y may stand in any of four mutually exclusive relationships to each other: either , or , or , or x and y are incomparable that returns one of four codes when given two elements. This definition is equivalent to a partial order on a setoid, where equality is taken to be a defined equivalence relation rather than set equality.

Wallis defines a more general notion of a partial order relation as any homogeneous relation that is transitive and antisymmetric. This includes both reflexive and irreflexive partial orders as subtypes.

A finite poset can be visualized through its Hasse diagram., a directed acyclic graph to be an edge. The transitive reduction of this DAG is then the Hasse diagram. Similarly this process can be reversed to construct strict partial orders from certain DAGs. In contrast, the graph associated to a non-strict partial order has self-loops at every node and therefore is not a DAG; when a non-strict order is said to be depicted by a Hasse diagram, actually the corresponding strict order is shown.

Examples

Standard examples of posets arising in mathematics include:

The real numbers, or in general any totally ordered set, ordered by the standard less-than-or-equal relation ≤, is a partial order.
, the usual less than relation < is a strict partial order. The same is also true of the usual greater than.
By definition, every strict weak order is a strict partial order.
The set of subsets of a given set (its power set) ordered by inclusion (see Fig.1). Similarly, the set of sequences ordered by subsequence, and the set of stringss ordered by substring.
The set of natural numbers equipped with the relation of divisibility. (see Fig.3 and Fig.6)
The vertex set of a directed acyclic graph ordered by reachability.
The set of subspaces of a vector space ordered by inclusion.
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; that is, a componentwise order.
For a set X and a partially ordered set P, the function space containing all functions from X to P, where if and only if
A fence, a partially ordered set defined by an alternating sequence of order relations
The set of events in special relativity and, in most cases, general relativity, where for two events X and Y, if and only if Y is in the future light cone of X. An event Y can be causally affected by X only if .

One familiar example of a partially ordered set is a collection of people ordered by genealogical descendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs of people are incomparable, with neither being a descendant of the other.

Orders on the Cartesian product of partially ordered sets

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 Fig.4):

the lexicographical order: if or ( and );
the product order: (a, b) ≤ (c, d) if a ≤ c and b ≤ d;
the reflexive closure of the direct product of the corresponding strict orders: if ( and ) or ( and ).

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

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

Sums of partially ordered sets

Another way to combine two (disjoint) posets is the ordinal sum (or linear sum), , defined on the union of the underlying sets X and Y by the order if and only if:

a, b ∈ X with a ≤X b, or
a, b ∈ Y with a ≤Y b, or
a ∈ X and b ∈ Y.

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

Series-parallel partial orders are formed from the ordinal sum operation (in this context called series composition) and another operation called parallel composition. Parallel composition is the disjoint union of two partially ordered sets, with no order relation between elements of one set and elements of the other set.

Derived notions

consisting of the set of all subsets ordered by set inclusion (see Fig.1).

a is related to b when a ≤ b. This does not imply that b is also related to a, because the relation need not be symmetric but not the reverse.
a and b are comparable if or . Otherwise they are incomparable are not.
A total order or linear order is a partial order under which every pair of elements is comparable, i.e. trichotomy holds. For example, the natural numbers with their standard order.
A chain is a chain.
An antichain is a subset of a poset in which no two distinct elements are comparable. For example, the set of singletons
An element a is said to be strictly less than an element b, if a ≤ b
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 are true, and a ≤ c ≤ b is false for each c Using the strict order <, the relation a ⋖ b can be equivalently rephrased as " but not for any c

Extrema

notably:

Greatest element is a is a is the least.
Maximal elements are the maximal and minimal elements. Removing these, there are 3 maximal elements and 3 minimal elements (see Fig.5).
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 is an

As another 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. 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. 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 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. If the number 0 is included, this will be the greatest element, since this is a multiple of every integer (see Fig.6).

Mappings between partially ordered sets

Given two partially ordered sets and is called order-preserving, or monotone, or isotone implies . If are order-preserving, their composition is called order-reflecting If is both order-preserving and order-reflecting, then it is called an order-embedding of into . In the latter case, is necessarily injective If an order-embedding between two posets S and T exists, one says that S can be embedded into T is bijective, it is called an order isomorphism, and the partial orders and are said to be isomorphic. Isomorphic orders have structurally similar Hasse diagrams (see Fig. yields the identity function on S and T, respectively, then S and T are order-isomorphic.

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. It is order-preserving: if divides , then each prime divisor of is also a prime divisor of ) nor order-reflecting (since 12 does not divide 6). Taking instead each number to the set of its prime power), but it can be made one by restricting its codomain Fig. and its isomorphic image under . 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.

Number of partial orders

Sequence [ A001035] in OEIS gives the number of partial orders on a set of n labeled elements:

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

If the count is made only up to isomorphism, the sequence 1, 1, 2, 5, 16, 63, 318, ... is obtained.

Subposets

is called a subposet is a subset.

induced.

Linear extension

is called an extension A linear extension is an extension that is also a linear (that is, total) order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order. Every partial order can be extended to a total order (order-extension principle).

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

In category theory

Every poset (and every preordered set) may be considered as a category there is at most one morphism More explicitly, let if (and otherwise the empty set Such categories are sometimes called thin.

Posets are equivalent to one another if and only if they are isomorphic. In a poset, the smallest element, if it exists, is an initial object, and the largest element, if it exists, is a terminal object. Also, every preordered set is equivalent to a poset. Finally, every subcategory of a poset is isomorphism-closed.

Partial orders in topological spaces

is a partially ordered set that has also been given the structure of a topological space is a closed subset of the topological product space Under this assumption partial order relations are well behaved at limits

Intervals

A convex set in a poset P is a subset of P with the property that, for any x and y in and any z in P, if x ≤ z ≤ y, then z is also in . This definition generalizes the definition of intervalss of real numbers. When there is possible confusion with convex sets of geometry, one uses order-convex instead of "convex".

A convex sublattice of a lattice L is a sublattice of L that is also a convex set of L. Every nonempty convex sublattice can be uniquely represented as the intersection of a filter and an ideal of L.

An interval in a poset P is a subset that can be defined with interval notation:

For a ≤ b, the closed interval is the set of elements x satisfying (that is, and ). It contains at least the elements a and b.
Using the corresponding strict relation "<", the open interval is the set of elements x satisfying (i.e. and ). An open interval may be empty even if . For example, the open interval on the integers is empty since there is no integer such that .
The half-open intervals and are defined similarly.

Whenever does not hold, all these intervals are empty. Every interval is a convex set, but the converse does not hold; for example, in the poset of divisors of 120, ordered by divisibility (see Fig.7b), the set is convex, but not an interval.

An interval such that . Every interval that can be represented in interval notation is obviously bounded, but the converse is not true. For example, let as a subposet of the real numbers. The subset is a bounded interval, but it has no infimum or supremum inP, so it cannot be written in interval notation using elements ofP.

A poset is called locally finite is not locally finite, since . Using the interval notation, the property "a is covered by b

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.