Definition:Well-Ordering/Definition 2

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

Then the ordering $\preceq$ is a well-ordering on $S$ iff $\preceq$ is well-founded.

Also see

 * Definition:Strict Well-Ordering


 * Well-Ordering is Total Ordering, which shows that every well-ordering is in fact a total ordering.