Definition:Well-Founded Relation/Definition 1
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Definition
Let $\struct {S, \RR}$ be a relational structure.
$\RR$ is a well-founded relation on $S$ if and only if:
- $\forall T \subseteq S: T \ne \O: \exists z \in T: \forall y \in T \setminus \set z: \tuple {y, z} \notin \RR$
where $\O$ is the empty set.
That is, $\RR$ is a strictly well-founded relation on $S$ if and only if:
- for every non-empty subset $T$ of $S$, there exists an element $z$ in $T$ such that for all $y \in T \setminus \set z$, it is not the case that $y \mathrel \RR z$.
Also known as
The term well-founded relation is often used the literature for what on $\mathsf{Pr} \infty \mathsf{fWiki}$ we call a strictly well-founded relation.
In order to emphasise the differences between the two, at some point a $\mathsf{Pr} \infty \mathsf{fWiki}$ editor coined the term strongly well-founded relation.
However, $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the less unwieldy term well-founded relation in preference to others.
Some sources do not hyphenate, and present the name as wellfounded relation.
Also see
- Results about well-founded relations can be found here.
Sources
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Definition $1$