Partially ordered set

Set ordered by a transitive, antisymmetric, and reflexive binary relation

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 relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." 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. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable.

A partial order defines a notion of comparison. Two elements x and y 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.

A set with a partial order is called a partially ordered set (also called a poset). 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. In particular, totally ordered sets can also be referred to as "ordered sets", especially in areas where these structures are more common than posets.

A poset can be visualized through its Hasse diagram, which depicts the ordering relation.[1]

A partial order relation is a homogeneous relation that is transitive and antisymmetric.[2] There are two common sub-definitions for a partial order relation, for reflexive and irreflexive partial order relations, also called "non-strict" and "strict" respectively. The two definitions 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. The term partial order typically refers to a non-strict partial order relation.

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

Irreflexivity and transitivity together imply asymmetry. Also, asymmetry implies irreflexivity. In other words, a transitive relation is asymmetric if and only if it is irreflexive.[4] So the definition is the same if it omits either irreflexivity or asymmetry (but not both).

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.

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.3-5):

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.

Another way to combine two (disjoint) posets is the ordinal sum[9] (or linear sum),[10] Z = XY, defined on the union of the underlying sets X and Y by the order aZ b if and only if:

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.

Sequence 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, ... (sequence in the OEIS) is obtained.

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

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.

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 xzy, then z is also in I. (This definition generalizes the interval definition for real numbers.)

For ab, the closed interval [a, b] is the set of elements x satisfying axb (that is, ax and xb). It contains at least the elements a and b.

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). An open interval may be empty even if a < b. For example, the open interval (1, 2) on the integers is empty since there are no integers I such that 1 < I < 2.

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

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.