Definition:Well-Founded Ordered Set
- Not to be confused with Definition:Well-Founded Set.
Definition
Let $\left({S, \preceq}\right)$ be an ordered set.
Then $\left({S, \preceq}\right)$ 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 as a whole.
Remark
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The term well-founded is also commonly used for foundational relations, which are closely related to, but different from, well-founded orderings.
Also see
Stronger properties
Generalization
Sources
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1.5$: Relations
