Definition:Partial Ordering

Let $$\left({S; \preceq}\right)$$ be a poset.

Then the ordering $$\preceq$$ is a partial ordering on $$S$$ iff $$\preceq$$ is not connected.

That is, iff $$\left({S; \preceq}\right)$$ has at least one pair which is non-comparable:


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