Definition:Ordering

Definition
Let $S$ be a set.

Also known as
An ordering is also referred to as an order relation or an order, although the latter term is also used for several other concepts so bears the risk of ambiguity.

Some sources use the word partial for an ordering which may or may not be connected, while others insist on reserving the word partial for one which is specifically not connected. It is wise to be certain of what is meant.

An ordering as defined here is sometimes referred to as a weak ordering if it is necessary to place emphasis on the fact that it is not a strict ordering.

Also defined as
defines an ordering as a transitive relation.

He also allows the synonyms partial ordering (which this is), and quasi-ordering (which is generally used as a synonym for preordering).

This approach glosses over the antisymmetric nature of an ordering, and in fact what is ended up with appears to be what on is defined as a strict ordering.

This approach is not used on.

Also see

 * Equivalence of Definitions of Ordering


 * Definition:Partial Ordering
 * Definition:Total Ordering
 * Definition:Well-Ordering


 * Definition:Strict Ordering


 * Definition:Ordered Set
 * Definition:Partially Ordered Set
 * Definition:Totally Ordered Set
 * Definition:Well-Ordered Set