Definition:Well-Ordered Set

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

Then $\left({S, \preceq}\right)$ is a well-ordered set (or woset) if the ordering $\preceq$ is well-founded.

That is, if every $T \subseteq S: T \ne \varnothing$ has a minimal or first element.

That is, $\exists a \in T: \forall x \in T: a \preceq x$.

Note the every in the above.

Also see

 * Partially ordered set (poset)
 * Totally ordered set (toset)


 * Well-Ordering


 * Well-Ordering is Total Ordering, which shows that every woset is in fact a toset.