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Home > Partially ordered set

1 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:

aRa (reflexivity);

if aRb and bRa then a = b (antisymmetry); and

if aRb and bRc then aRc (transitivity).

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).

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.

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.

2 See also

• Order theory
• Preorder
• Total order

Compare: Equivalence class, Directed set.

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Topics: Partially Ordered Set Order Partial Strict Arb Theory Sets Weak...

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