# 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:
$\exists x, y \in S: x \not \precsim y \land y \not \precsim x$