Definition:Well-Founded

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 smallest 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

Sources

• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.5$