Partially ordered set

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In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. This relation formalizes the intuitive concept of an ordering, sequencing, or arrangement of that set's elements. Such an ordering does not necessarily need to be total, that is, it need not guarantee the mutual comparability of all objects in the set.

Formal definition

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

  • aRa (reflexivity);
  • if aRb and bRa then a = b (antisymmetry); and
  • if aRb and bRc then aRc (transitivity).

A set with a partial order is called a partially ordered set. 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

  • The set of natural numbers equipped with the divides relation.

Strict and weak partial orders

In some contexts, the partial order defined above is called a weak (or reflexive) partial order. In these contexts a strict (or irreflexive) partial order is a binary relation which is irreflexive and transitive, and therefore asymmetric. In other words, for all a, b, and c in P, we have that:

  • ¬(aRa) (irreflexivity);
  • if aRb then ¬(bRa) (asymmetry); and
  • if aRb and bRc then aRc (transitivity).

If R is a weak partial order, then R − {(a, a) | a in P} is the corresponding strict partial order. Similarly, every strict partial order has a corresponding weak partial order, and so the two definitions are both readily expressed in terms of the other.

Strict partial orders are also 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.

See also: strict weak ordering

Category theory

When considered as a category where hom(x, y) = {(x, y) : xy} 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.

See also

de:Halbordnung he:סדר חלקי hu:Rendezési reláció it:Relazione d'ordine es:Conjunto parcialmente ordenado sl:Relacija urejenosti zh:偏序关系