Definition:Well-Founded Ordered Set

Not to be confused with Definition:Well-Founded Set.

Definition

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

Then $\struct {S, \preceq}$ is well-founded if and only if 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:Foundational Relation

Sources

• 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.5$: Relations