Definition:Partial Preordering

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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:

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