Definition:Well-Founded Ordered Set

Definition
Let $\struct {S, \preceq}$ be an ordered set.

Then $\struct {S, \preceq}$ is well-founded it satisfies the minimal condition:


 * 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 $\struct {S, \preceq}$ as a whole.

Also see

 * Definition:Descending Chain Condition
 * Definition:Converse Well-Founded Ordered Set

Stronger properties

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

Generalization

 * Definition:Strictly Well-Founded Relation