Definition:Well-Founded Ordered Set

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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


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