Definition:Ordering

Definition
Let $S$ be a set.

Partial vs. Total Orderings
Note that this definition of ordering does not demand that every pair of elements of $S$ is related by $\preceq$. The way we have defined an ordering, they may be, or they may not be, depending on the context.

If it is the case that $\preceq$ is a connected relation, i.e. that every pair of distinct elements is related by $\preceq$, then $\preceq$ is called a total ordering.

If it is not the case that $\preceq$ is connected, then $\preceq$ is called a partial ordering.

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 emphasise that it is not a strict ordering.

Also see

 * Equivalence of Definitions of Ordering


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


 * Definition:Strict Ordering


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