\begin{definition}[Definition:Partial Ordering]
Let $\struct {S, \preceq}$ be an ordered set.
Then the ordering $\preceq$ is a '''partial ordering''' on $S$ {{iff}} $\preceq$ is not connected.
That is, {{iff}} $\struct {S, \preceq}$ has at least one pair which is non-comparable:
:$\exists x, y \in S: x \npreceq y \land y \npreceq x$
\end{definition}