Definition:Partial Ordering

Definition
Let $\left({S, \preceq}\right)$ be an ordered set.

Then the ordering $\preceq$ is a partial ordering on $S$ $\preceq$ is not connected.

That is, $\left({S, \preceq}\right)$ has at least one pair which is non-comparable:


 * $\exists x, y \in S: x \npreceq y \land y \npreceq x$

Also defined as
Some sources define a partial ordering to be the structure known on as an ordering, that is, whose nature (partial or total) is unspecified.

Also known as
A partial ordering as defined here is sometimes referred to as a weak partial ordering, to distinguish it from a strict partial ordering

Also see

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