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, but it can be. (In mathematical usage, a total order is a kind of partial order.)

Contents 1 Formal definition 2 Examples 3 Strict and weak partial orders 4 Category theory 5 See also

Formal definition

A partial order is a binary relation R over a set P which is reflexive, antisymmetric and transitive. A set with a partial order is called a partially ordered set (poset). If (x, y) ∈ R then the notation x ≤ y is typically used instead of xRy (and x < y when x ≠ y). A partial order R on X is called a (total) order on X if for any x, y ∈ X one of the conditions is satisfied: x < y, x = y or x > y.

Alternatively, 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

Image:Hasse diagram of powerset of 3.png

• The set of natural numbers equipped with the lesser than or equal to relation.

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

• The set of subsets of a given set (power set) ordered by inclusion.

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 that is irreflexive and transitive, and therefore antisymmetric. In other words, for all a, b, and c in P, we have that:

• ¬(aRa) (irreflexivity);
• if a ≠ b and aRb then ¬(bRa) (antisymmetry); 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 definition of each is 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) : x ≤ y} 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

• order theory
• preorder
• total order
• directed set
• equivalence relation
• Hasse diagram
• graded posetde:Halbordnung

he:סדר חלקי hu:Rendezési reláció it:Relazione d'ordine pl:Poset ru:Частично упорядоченное множество es:Conjunto parcialmente ordenado sl:Relacija urejenosti zh:偏序关系