Partial order
In mathematics, partially ordered sets, or posets for short, are special binary relations which formalize the intuitive concept of an ordering. Partially ordered sets are studied in order theory and a much more detailed introduction to the field can be found within the corresponding article. In contrast, this article serves as a quick lookup for the formal definition.
Formal definition
A binary relation R over a set P is a partial order if it 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)
- if aRb and bRc then aRc (transitivity)
A set with a partial order is called a partially ordered set, or poset for short. 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.
See also
