Partially ordered set

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 is an ordered pair consisting of a set and a partial order on. When the meaning is clear from context and there is no ambiguity about the partial order, the 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. That is, for all it must satisfy:

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

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

Strict partial orders

An irreflexive, strong, or is a homogeneous relation < on a set that is irreflexive, asymmetric and transitive; that is, it satisfies the following conditions for all

Irreflexivity: , i.e. no element is related to itself.
Asymmetry: if then not.
Transitivity: if and then.

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

Correspondence of strict and non-strict partial order relations

Strict and non-strict partial orders on a set are closely related. A non-strict partial order may be converted to a strict partial order by removing all relationships of the form that is, the strict partial order is the set where is the identity relation on and denotes set subtraction. Conversely, a strict partial order < on may be converted to a non-strict partial order by adjoining all relationships of that form; that is, is a non-strict partial order. Thus, if is a non-strict partial order, then the corresponding strict partial order < is the irreflexive kernel given by
Conversely, if < is a strict partial order, then the corresponding non-strict partial order is the reflexive closure given by:

Dual orders

The dual of a partial order relation is defined by letting be the converse relation of, i.e. if and only if. 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

Given a set and a partial order relation, typically the non-strict partial order, we may uniquely extend our notation to define four partial order relations and, where is a non-strict partial order relation on, is the associated strict partial order relation on , is the dual of, and is the dual of. Strictly speaking, the term partially ordered set refers to a set with all of these relations defined appropriately. But practically, one need only consider a single relation, or, or, in rare instances, the non-strict and strict relations together,.
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 can also be referred to as "ordered sets", especially in areas where these structures are more common than posets. Some authors use different symbols than such as or to distinguish partial orders from total orders.
When referring to partial orders, should not be taken as the complement of. The relation is the converse of the irreflexive kernel of, which is always a subset of the complement of, but is equal to the complement of 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. Specifically, given 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. This can be represented by a function 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. Specifically, taking a strict partial order relation, a directed acyclic graph may be constructed by taking each element of to be a node and each element of 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.
On the real numbers, the usual less than relation < is a strict partial order. The same is also true of the usual greater than relation > on.
By definition, every strict weak order is a strict partial order.
The set of subsets of a given set ordered by inclusion. Similarly, the set of sequences ordered by subsequence, and the set of strings ordered by substring.
The set of natural numbers equipped with the relation of divisibility.
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. Formally, if and only if for all ; 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 for all
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 :

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

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

Sums of partially ordered sets

Another way to combine two posets is the ordinal 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 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

The examples use the poset consisting of the set of all subsets of a three-element set ordered by set inclusion.

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. For example, is related to but not the reverse.
a and b are comparable if or. Otherwise they are incomparable. For example, and are comparable, while and 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 subset of a poset that is a totally ordered set. For example, 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 and For example, is strictly less than
An element a is said to be covered by another element b, written a ⋖ b, if a is strictly less than b and no third element c fits between them; formally: if both a ≤ b and are true, and a ≤ c ≤ b is false for each c with Using the strict order <, the relation a ⋖ b can be equivalently rephrased as " but not for any c". For example, is covered by but is not covered by