Definition:Partial Ordering

Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

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:

$\exists x, y \in S: x \not \preceq y \land y \not \preceq x$

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.