Partially ordered set Guide, Meaning , Facts, Information and Description
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.
A binary relation R over a set P is a weak partial order if it is reflexive, antisymmetric, and transitive, i.e., for all a, b and c in P, we have that:
Formal definition
Alternatively, a strict partial order is a binary relation which is irreflexive, asymmetric, and transitive. 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).
In mathematics, partial order usually means weak partial order. However, 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.
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
Compare: Equivalence class, Directed set.
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