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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.
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