partially ordered set

A partial order is a relationship between (some) pairs of objects of a set. We often think of them as a type of “size” or “magnitude” leaning on intuition from and , though that may not always be what the ordering is meaning.

Definition

Non-strict partial orders

A reflexive, weak, or non-strict partial order, (often referred to simply as a partial order or a POSET), is a homogeneous relation  on a set  that is reflexive, antisymmetric, and transitive. That is, for all , the ordering must satisfy:

1. Reflexivity:  (every element is related to itself)
2. Antisymmetry: if  and , then  (no two distinct elements can precede each other)
3. Transitivity: if  and  then .

Strict partial orders

An irreflexive, strong, or strict partial order is a homogeneous relation on a set  that is irreflexive, asymmetric, and transitive; In other words, each element is not related to itself ( can’t happen), and if then cannot be true. Below more formally:

1. Irreflexivity: 
2. Asymmetry: if  then not .
3. Transitivity: if  and  then .