A strict weak ordering is a binary relation that defines an equivalence relation and has the properties stated below.

Two elements x and y are considered equivalent under the equivalence relation of the ordering R if x R y and y R x are both false.

A strict weak ordering has these properties:

• irreflexivity: x R x must be false.
• asymmetric: x R y implies not y R x (i.e. antisymmetric, in the context of irreflexivity)
• transitivity: (x R y and y R z) implies x R z
• transitivity of equivalence: If x is equivalent to y under the equivalence relation stated above and y is equivalent to z, then x is equivalent to z.

A strict weak ordering is similar to a weak partial order, but stricter.

Example: a<b<d, a<c<d, no other elements or relationships. Then b and c are equivalent.

A weak but not strict weak order: a<b<c<e. a<d<e: the derived relation mentioned above holds for b,d and d,c but not for b,c, so is not transitive.

A common example of a strict weak ordering is the less than relationship over real numbers. However, less than also satisfies the requirements for a total ordering, which are stronger than those for a strict weak ordering.