Definition:Strict Partial Ordering

Definition
Let $$\left({S, \prec}\right)$$ be a relational structure.

Let $$\prec$$ be a strict ordering.

Then $$\prec$$ is a strict partial ordering on $$S$$ iff $$\prec$$ is not connected.

That is, iff $$\left({S, \prec}\right)$$ has at least one pair which is non-comparable:
 * $$\exists x, y \in S: x \not \prec y \and y \not \prec x$$