Definition:Total Ordering

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

Then the ordering $$\preceq$$ is a total ordering on $$S$$ iff $$\left({S; \preceq}\right)$$ has no non-comparable pairs:


 * $$\forall x, y \in S: x \preceq y \lor y \preceq x$$

If this is the case, then $$\left({S; \preceq}\right)$$ is referred to as a totally ordered set.