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:


 * $$\exists x, y \in S: x \not \precsim y \and y \not \precsim x$$