Definitions

# Partially ordered set

In mathematics, especially order theory, a partially ordered set (or poset) formalizes 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 that describes, for certain pairs of elements in the set, the requirement that one of the elements must precede the other. However, a partially ordered set differs from a total order in that some pairs of elements may not be related to each other in this way. A finite poset can be visualized through its Hasse diagram, which depicts the ordering relation between certain pairs of elements and allows one to reconstruct the whole partial order structure.

A familiar real-life example of a partially ordered set is a collection of people ordered by genealogical descendancy. Some pairs of people bear the ancestor-descendant relationship but, in general, other pairs bear no such relationship.

## Formal definition

A partial order is a binary relation "≤" over a set P which is reflexive, antisymmetric, and transitive, i.e., for all a, b, and c in P, we have that:

• a ≤ a (reflexivity);
• if a ≤ b and b ≤ a then a = b (antisymmetry);
• if a ≤ b and b ≤ c then a ≤ c (transitivity).

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

A set with a partial order is called a partially ordered set (also called a poset). The term ordered set is sometimes also used for posets, as long as it is clear from the context that no other kinds of orders are 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.

## Examples

Standard examples of posets arising in mathematics include:

• The real numbers ordered by the standard less-than-or-equal relation ≤ (a totally ordered set as well).
• The set of natural numbers equipped with the relation of divisibility.
• The set of subsets of a given set (its power set) ordered by inclusion (see the figure on top-right).
• 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, $\left(a_n\right)_\left\{ninmathbf\left\{N\right\}\right\} le \left(b_n\right)_\left\{ninmathbf\left\{N\right\}\right\}$ if and only if $a_n le b_n$ for all n in N.
• For a set X and a partially ordered set P, the function space containing all functions from X to P, where fg if and only if f(x)g(x) for all x in X.
• The vertex set of a directed acyclic graph ordered by reachability.

## 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:

• Lexicographical order: (a,b) ≤ (c,d) if and only if a < c or (a = c and bd).
• (a,b) ≤ (c,d) if and only if ac and bd (the product order).
• (a,b) ≤ (c,d) if and only if (a < c and b < d) or (a = c and b = d) (the reflexive closure of the of the corresponding strict total orders).

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.

## Strict and non-strict partial orders

In some contexts, the partial order defined above is called a non-strict (or reflexive, or weak) partial order. In these contexts a strict (or irreflexive) partial order "<" is a binary relation that is irreflexive and transitive, and therefore asymmetric. In other words, asymmetric (hence irreflexive) and transitive.

Thus, for all a, b, and c in P, we have that:

• ¬(a < a) (irreflexivity);
• if a < b then ¬(b < a) (asymmetry); and
• if a < b and b < c then a < c (transitivity).

There is a 1-to-1 correspondence between all non-strict and strict partial orders.

If "≤" is a non-strict partial order, then the corresponding strict partial order "<" is the reflexive reduction given by:

a < b if and only if (ab and ab)

Conversely, if "<" is a strict partial order, then the corresponding non-strict partial order "<" is the "≤" given by:

ab if and only if a < b or a = b.
This is the reason for using the notation "≤".

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.

## Inverse and order dual

The inverse or converse ≥ of a partial order relation ≤ satisfies xy if and only if yx. The inverse of a partial order relation is reflexive, transitive, and antisymmetric, and hence itself a partial order relation. The order dual of a partially ordered set is the same set with the partial order relation replaced by its inverse. The irreflexive relation > is to ≥ as < is to ≤.

Any of these four relations ≤, <, ≥, and > on a given set uniquely determine the other three.

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). 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. 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. 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. 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.

## Number of partial orders

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

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

## Linear extension

A total order T is a linear extension of a partial order P if, whenever xy in P it also holds that xy in T. In computer science, algorithms for finding linear extensions of partial orders are called topological sorting.

## Category theory

When considered as a category where hom(x, y) = {(x, y)} if xy (and otherwise the empty set) and (y, z)o(x, y) = (x, z), posets are equivalent to one another if and only if they are isomorphic. In a poset, the smallest element, if any, is an initial object, and the largest element, if any, a terminal object. Also, every pre-ordered set is equivalent to a poset. Finally, every subcategory of a poset is isomorphism-closed.

## Partial orders in topological spaces

If P is a topological space, then it is customary to assume that R is closed in $Ptimes P$. Under this assumption relations are well behaved in limits; if $a_ito a$ and $a_i R b$ for all i, then $aRb$. See Deshpande (1968).

## Interval

For ab, the interval [a,b] is the set of points x satisfying ax and xb, also written axb. It contains at least the points a and b. One may choose to extend the definition to all pairs (a,b). The extra intervals are all empty.

Using the corresponding strict relation "<", one can also define the interval (a,b) as the set of points x satisfying a < x and x < b, also written a < x < b. An open interval may be empty even if a < b.

Also [a,b) and (a,b] can be defined similarly.

A poset is locally finite if every interval is finite. For example, the integers are locally finite under their natural ordering.

## References

• Jayant V. Deshpande, On Continuity of a Partial Order, Proceedings of the American Mathematical Society, Vol. 19, No. 2, 1968, pp. 383-386
• Bernd S. W. Schröder, Ordered Sets: An Introduction (Boston: Birkhäuser, 2003)
• Richard P. Stanley, Enumerative Combinatorics, vol.1, Cambridge Studies in Advanced Mathematics 49, Cambridge University Press, ISBN 0-521-66351-2