Definition:Well-Founded Relation/Definition 2
Jump to navigation
Jump to search
Definition
Let $\struct {S, \RR}$ be a relational structure.
$\RR$ is a well-founded relation on $S$ if and only if:
- for every non-empty subset $T$ of $S$, $T$ has a minimal element.
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
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.): $\S 10.1$