There are several categorizations of binary relations based on which axioms they hold. These are called orders.

• preorder
• weak order
• strict weak order
• strict order
• strict partial order
• equivalence

From the orderings a framework is built up.

• strict order
• weak order

Examples:

• subset on all sets is a preorder
• => on R is a strict order
• "=>" on R^2 is a antisymmetric preorder
• || || (magnitude) on R^2 is a weak order