Definition:Well-Founded Ordered Set

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

Then $$\left({S; \preceq}\right)$$ is well-founded iff every non-empty subset of $$S$$ has a minimal element.

The term well-founded can equivalently be said to apply to the ordering $$\preceq$$ itself rather than to the poset as a whole.

Also see

 * Well-Ordering
 * Well-Ordered Set