Definition:Well-Founded Ordered Set

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

Then $\left({S, \preceq}\right)$ is well-founded 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 ordered set as a whole.

Remark
The term well-founded is also commonly used for foundational relations, which are closely related to, but different from, well-founded orderings.

Also see

 * Definition:Foundational Relation
 * Definition:Converse Well-Founded Ordered Set
 * Definition:Well-Ordering
 * Definition:Well-Ordered Set