Definition:Well-Ordering

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

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

That is, iff every non-empty subset of $S$ has a smallest element under $\preceq$.

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

Also known as
Some sources write this without the hyphen: well ordering.

Also see

 * Strict Well-Ordering


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