Definition:Partial Ordering

Definition

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

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

That is, if and only if $\left({S, \preceq}\right)$ 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 $\mathsf{Pr} \infty \mathsf{fWiki}$ 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