Partially ordered set

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.

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:

Related Topics:
Irreflexive - Transitive - Asymmetric

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• ¬(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.

Related Topics:
Directed acyclic graph - Transitive closure

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

See also: strict weak ordering

~ ~ ~ ~ ~ ~ ~ ~ ~ ~