Definition:Well-Founded Ordered Set

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

Then the ordering $\preceq$ is a well-founded ordering iff every non-empty subset of $S$ has a minimal element.

An Ordered Set whose ordering is well-founded may also be described as well-founded:

Remark
The term well-founded is also commonly used for foundational relations. By the corollary to Foundational Relation is Antireflexive, a well-founded ordering is generally not a foundational relation.

Also see

 * Definition:Foundational Relation
 * Definition:Well-Ordering
 * Definition:Well-Ordered Set
 * Reflexive Reduction of Well-Founded Ordering is Foundational Relation