Definition:Ordering

Definition
Let $S$ be a set.

Smaller and Larger
An ordering is often considered to be a comparison of the size of objects, in some perhaps intuitive sense. Depending on the nature of the sets being ordered, and depending on the nature of the ordering relation, this aspect may or may not be intellectually sustainable.

Be that as it may, one frequently encounters terminology such as:


 * $A$ is smaller than $B$ to mean $A \preceq B$
 * $A$ is larger than $B$ to mean $B \preceq A$.

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.

Weak vs. Strict Orderings
Compare strict ordering.

If it is necessary to emphasise that an ordering $\preceq$ is not strict, then the term weak ordering may be used.

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.

Also see

 * Equivalence of Definitions of Ordering


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


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