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:
 * $\exists x, y \in S: x \not \prec y \land y \not \prec x$