Partially Ordered Set - Strict and Non-strict Partial Orders

Strict and Non-strict Partial Orders

In some contexts, the partial order defined above is called a non-strict (or reflexive, or weak) partial order. In these contexts a strict (or irreflexive) partial order "<" is a binary relation that is irreflexive and transitive, and therefore asymmetric. In other words, asymmetric (hence irreflexive) and transitive.

Thus, for all a, b, and c in P, we have that:

• ¬(a < a) (irreflexivity);
• if a < b then ¬(b < a) (asymmetry); and
• if a < b and b < c then a < c (transitivity).

There is a 1-to-1 correspondence between all non-strict and strict partial orders.

If "≤" is a non-strict partial order, then the corresponding strict partial order "<" is the reflexive reduction given by:

a < b if and only if (a ≤ b and a ≠ b)

Conversely, if "<" is a strict partial order, then the corresponding non-strict partial order "≤" is the reflexive closure given by:

a ≤ b if and only if a < b or a = b.

This is the reason for using the notation "≤".

Strict partial orders are 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.