Definition:Well-Founded Ordered Set

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

Then $\left({S, \preceq}\right)$ is well-founded iff every non-empty subset of $S$ has a smallest 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
A well-founded ordering is not a foundational relation, but a well-founded strict ordering is.

Also see

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