# Category:Well-Founded Relations

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This category contains results about **Well-Founded Relations**.

Definitions specific to this category can be found in Definitions/Well-Founded Relations.

### Definition 1

$\RR$ 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.

### Definition 2

$\RR$ is a **well-founded relation on $S$** if and only if:

- for every non-empty subset $T$ of $S$, $T$ has a minimal element.

## Subcategories

This category has the following 5 subcategories, out of 5 total.

## Pages in category "Well-Founded Relations"

The following 18 pages are in this category, out of 18 total.

### E

### I

### R

- Rank Function Property of Well-Founded Relation
- Reflexive Reduction of Well-Founded Relation is Strictly Well-Founded Relation
- Restriction of Strictly Well-Founded Relation is Strictly Well-Founded
- Restriction of Well-Founded Ordering is Well-Founded
- Restriction of Well-Founded Relation is Well-Founded