Definition:Well-Ordered Set

Definition 1
Let $\left({S, \preceq}\right)$ be an ordered set.

Then $\left({S, \preceq}\right)$ is a well-ordered set (or woset) iff:

For each non-empty subset $T$ of $S$: $T$ has a smallest (or first, or least) element.

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

Definition 2
Let $\left({S, \preceq}\right)$ be a totally ordered set.

Then $\left({S, \preceq}\right)$ is a well-ordered set iff:

For each non-empty subset $T$ of $S$: $T$ has a minimal element.

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.