Definition:Partial Ordering

Definition
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:


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

Weak vs. Strict Orderings
Compare strict partial ordering.

If it is necessary to emphasise that a partial ordering $$\preceq$$ is not strict, then the term weak partial ordering may be used.