In mathematics, a partially ordered set (or poset for short) is a set equipped with a partial order relation, formalizing the intuitive concept of a (not necessarily total) ordering.
Example
Unlike a total ordering, a partial ordering need not guarantee the mutual comparability of all objects in the set. For example, we could define an ordering ⊆ on the set of all political organizations such that a⊆b if every member of a is also a member of b. This would be only a partial ordering: if a is the Sierra Club and b is the Democratic Party, then neither a⊆b nor b⊆a holds. An example of a total ordering would be to define a≤b if the name of organization a precedes that of b in alphabetical order.
Partially ordered sets are studied in order theory.
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. However, most articles should not cause confusion as long as all formal definitions employ exact terminology.
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 essentially equivalent.
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