\begin{definition}[Definition:Partial Preordering]
Let $S$ be a set.
Let $\precsim$ be a preordering on $S$.
Then $\precsim$ is a '''partial preordering''' on $S$ {{iff}} $\precsim$ is not connected.
That is, {{iff}} there is at least one pair of elements of $S$ which is non-comparable:
:$\exists x, y \in S: x \not \precsim y \land y \not \precsim x$
Category:Definitions/Preorder Theory
Category:Definitions/Partial Orderings
\end{definition}