# Well-founded relation

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Type of binary relation

"Noetherian induction" redirects here. For the use in topology, see [Noetherian topological space](/source/Noetherian_topological_space).

Transitive binary relations v t e Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Total, Semiconnex Anti- reflexive Equivalence relation Y ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Preorder (Quasiorder) ✗ ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Partial order ✗ Y ✗ ✗ ✗ ✗ Y ✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict weak order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict total order ✗ Y Y ✗ ✗ ✗ ✗ Y Y Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Definitions, for all a , b {\displaystyle a,b} and S ≠ ∅ : {\displaystyle S\neq \varnothing :} a R b ⇒ b R a {\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}} a R b and b R a ⇒ a = b {\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}} a ≠ b ⇒ a R b or b R a {\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}} min S exists {\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}} a ∨ b exists {\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}} a ∧ b exists {\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}} a R a {\displaystyle aRa} not a R a {\displaystyle {\text{not }}aRa} a R b ⇒ not b R a {\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}} Y indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive: for all a , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle aRb} and b R c {\displaystyle bRc} then a R c . {\displaystyle aRc.} A term's definition may require additional properties that are not listed in this table.

In [mathematics](/source/Mathematics), a [binary relation](/source/Binary_relation) R is called **well-founded** (or **wellfounded** or **foundational**)[1] on a [set](/source/Set_(mathematics)) or, more generally, a [class](/source/Class_(set_theory)) X if every [non-empty](/source/Non-empty) [subset](/source/Subset) (or subclass) *S* ⊆ *X* has a [minimal element](/source/Minimal_element) with respect to R; that is, there exists an *m* ∈ *S* such that for every *s* ∈ *S*, one does not have *s* *R* *m*. More formally, a relation is well-founded if: ( ∀ S ⊆ X ) [ S ≠ ∅ ⟹ ( ∃ m ∈ S ) ( ∀ s ∈ S ) ¬ ( s R m ) ] . {\displaystyle (\forall S\subseteq X)\;[S\neq \varnothing \implies (\exists m\in S)(\forall s\in S)\lnot (s\mathrel {R} m)].} Some authors include an extra condition that R is [set-like](/source/Set-like_relation), i.e., that the elements less than any given element form a set.

Equivalently, assuming the [axiom of dependent choice](/source/Axiom_of_dependent_choice), a relation is well-founded when it contains no [infinite descending chains](/source/Infinite_descending_chain), meaning there is no infinite sequence *x*0, *x*1, *x*2, ... of elements of X such that *x**n*+1 *R* *x**n* for every natural number n.[2][3]

In [order theory](/source/Order_theory), a [partial order](/source/Partial_order) is called well-founded if the corresponding [strict order](/source/Strict_order) is a well-founded relation. If the order is a [total order](/source/Total_order), then it is called a [well-order](/source/Well-order).

In [set theory](/source/Set_theory), a set x is called a **well-founded set** if the [set membership](/source/Element_(mathematics)) relation is well-founded on the [transitive closure](/source/Transitive_closure_(set)) of x. The [axiom of regularity](/source/Axiom_of_regularity), which is one of the axioms of [Zermelo–Fraenkel set theory](/source/Zermelo%E2%80%93Fraenkel_set_theory), asserts that all sets are well-founded.

A relation R is **converse well-founded**, **upwards well-founded**, or **Noetherian** on X, if the [converse relation](/source/Converse_relation) *R*−1 is well-founded on X. In this case R is also said to satisfy the [ascending chain condition](/source/Ascending_chain_condition). In the context of [rewriting](/source/Rewriting) systems, a Noetherian relation is also called **terminating**.

## Induction and recursion

An important reason that well-founded relations are interesting is because a version of [transfinite induction](/source/Transfinite_induction) can be used on them: if (*X*, *R*) is a well-founded relation, *P*(*x*) is some property of elements of X, and we want to show that

- *P*(*x*) holds for all elements x of X,

it suffices to show that:

- If x is an element of X and *P*(*y*) is true for all y such that *y* *R* *x*, then *P*(*x*) must also be true.

That is, ( ∀ x ∈ X ) [ ( ∀ y ∈ X ) [ y R x ⟹ P ( y ) ] ⟹ P ( x ) ] implies ( ∀ x ∈ X ) P ( x ) . {\displaystyle (\forall x\in X)\;[(\forall y\in X)\;[y\mathrel {R} x\implies P(y)]\implies P(x)]\quad {\text{implies}}\quad (\forall x\in X)\,P(x).}

Well-founded induction is sometimes called Noetherian induction,[4] after [Emmy Noether](/source/Emmy_Noether).

On par with induction, well-founded relations also support construction of objects by [transfinite recursion](/source/Transfinite_recursion). Let (*X*, *R*) be a [set-like](/source/Binary_relation#Relations_over_a_set) well-founded relation and F a function that assigns an object *F*(*x*, *g*) to each pair of an element *x* ∈ *X* and a function g on the set {*y*: *y* *R* *x*} of predecessors of x. Then there is a unique function G such that for every *x* ∈ *X*, G ( x ) = F ( x , G | { y : y R x } ) . {\displaystyle G(x)=F\left(x,G\vert _{\left\{y:\,y\mathrel {R} x\right\}}\right).}

That is, if we want to construct a function G on X, we may define *G*(*x*) using the values of *G*(*y*) for *y* *R* *x*.

As an example, consider the well-founded relation (**N**, *S*), where **N** is the set of all [natural numbers](/source/Natural_numbers), and S is the graph of the successor function *x* ↦ *x*+1. Then induction on S is the usual [mathematical induction](/source/Mathematical_induction), and recursion on S gives [primitive recursion](/source/Primitive_recursive_functions). If we consider the order relation (**N**, <), we obtain [complete induction](/source/Complete_induction), and [course-of-values recursion](/source/Course-of-values_recursion). The statement that (**N**, <) is well-founded is also known as the [well-ordering principle](/source/Well-ordering_principle).

There are other interesting special cases of well-founded induction. When the well-founded relation is the usual ordering on the class of all [ordinal numbers](/source/Ordinal_numbers), the technique is called [transfinite induction](/source/Transfinite_induction). When the well-founded set is a set of recursively defined data structures, the technique is called [structural induction](/source/Structural_induction). When the well-founded relation is set membership on the universal class, the technique is known as [∈-induction](/source/%E2%88%88-induction). See those articles for more details.

## Examples

Well-founded relations that are not totally ordered include:

- The positive [integers](/source/Integer) {1, 2, 3, ...}, with the order defined by *a* < *b* [if and only if](/source/If_and_only_if) a [divides](/source/Divisor) b and *a* ≠ *b*.

- The set of all finite [strings](/source/String_(computer_science)) over a fixed alphabet, with the order defined by *s* < *t* if and only if s is a proper substring of t.

- The set **N** × **N** of [pairs](/source/Cartesian_product) of [natural numbers](/source/Natural_number), ordered by (*n*1, *n*2) < (*m*1, *m*2) if and only if *n*1 < *m*1 and *n*2 < *m*2.

- Every class whose elements are sets, with the relation ∈ ("is an element of"). This is the [axiom of regularity](/source/Axiom_of_regularity).

- The nodes of any finite [directed acyclic graph](/source/Directed_acyclic_graph), with the relation R defined such that *a* *R* *b* if and only if there is an edge from a to b.

Examples of relations that are not well-founded include:

- The negative integers {−1, −2, −3, ...}, with the usual order, since any unbounded subset has no least element.

- The set of strings over a finite alphabet with more than one element, under the usual ([lexicographic](/source/Lexicographic_ordering)) order, since the sequence "B" > "AB" > "AAB" > "AAAB" > ... is an infinite descending chain. This relation fails to be well-founded even though the entire set has a minimum element, namely the empty string.

- The set of non-negative [rational numbers](/source/Rational_number) (or [reals](/source/Real_numbers)) under the standard ordering, since, for example, the subset of positive rationals (or reals) lacks a minimum.

## Other properties

If (*X*, <) is a well-founded relation and x is an element of X, then the descending chains starting at x are all finite, but this does not mean that their lengths are necessarily bounded. Consider the following example: Let X be the union of the positive integers with a new element ω that is bigger than any integer. Then X is a well-founded set, but there are descending chains starting at ω of arbitrary great (finite) length; the chain ω, *n* − 1, *n* − 2, ..., 2, 1 has length n for any n.

The [Mostowski collapse lemma](/source/Mostowski_collapse) implies that set membership is a universal among the [extensional](/source/Extensionality#In_mathematics) well-founded relations: for any set-like well-founded relation R on a class X that is extensional, there exists a class C such that (*X*, *R*) is isomorphic to (*C*, ∈).

## Reflexivity

A relation R is said to be [reflexive](/source/Reflexive_relation) if *a* *R* *a* holds for every a in the domain of the relation. Every reflexive relation on a nonempty domain has infinite descending chains, because any constant sequence is a descending chain. For example, in the natural numbers with their usual order ≤, we have 1 ≥ 1 ≥ 1 ≥ .... To avoid these trivial descending sequences, when working with a partial order ≤, it is common to apply the definition of well foundedness (perhaps implicitly) to the alternate relation < defined such that *a* < *b* if and only if *a* ≤ *b* and *a* ≠ *b*. More generally, when working with a [preorder](/source/Preorder) ≤, it is common to use the relation < defined such that *a* < *b* if and only if *a* ≤ *b* and *b* ≰ *a*. In the context of the natural numbers, this means that the relation <, which is well-founded, is used instead of the relation ≤, which is not. In some texts, the definition of a well-founded relation is changed from the definition above to include these conventions.

## References

1. **[^](#cite_ref-1)** See Definition 6.21 in Zaring W.M., G. Takeuti (1971). *Introduction to axiomatic set theory* (2nd, rev. ed.). New York: Springer-Verlag. [ISBN](/source/ISBN_(identifier)) [0387900241](https://en.wikipedia.org/wiki/Special:BookSources/0387900241).

1. **[^](#cite_ref-2)** ["Infinite Sequence Property of Strictly Well-Founded Relation"](https://proofwiki.org/wiki/Infinite_Sequence_Property_of_Strictly_Well-Founded_Relation). *ProofWiki*. Retrieved 10 May 2021.

1. **[^](#cite_ref-3)** [Fraisse, R.](/source/Roland_Fra%C3%AFss%C3%A9) (15 December 2000). [*Theory of Relations, Volume 145 - 1st Edition*](https://www.elsevier.com/books/theory-of-relations/fraisse/978-0-444-50542-2) (1st ed.). Elsevier. p. 46. [ISBN](/source/ISBN_(identifier)) [9780444505422](https://en.wikipedia.org/wiki/Special:BookSources/9780444505422). Retrieved 20 February 2019.

1. **[^](#cite_ref-4)** [Bourbaki, N.](/source/Nicolas_Bourbaki) (1972) *Elements of mathematics. Commutative algebra*, Addison-Wesley.

- Just, Winfried and Weese, Martin (1998) *Discovering Modern Set Theory. I*, [American Mathematical Society](/source/American_Mathematical_Society) [ISBN](/source/ISBN_(identifier)) [0-8218-0266-6](https://en.wikipedia.org/wiki/Special:BookSources/0-8218-0266-6).

- [Karel Hrbáček](/source/Karel_Hrb%C3%A1%C4%8Dek) & [Thomas Jech](/source/Thomas_Jech) (1999) *Introduction to Set Theory*, 3rd edition, "Well-founded relations", pages 251–5, [Marcel Dekker](/source/Marcel_Dekker) [ISBN](/source/ISBN_(identifier)) [0-8247-7915-0](https://en.wikipedia.org/wiki/Special:BookSources/0-8247-7915-0)

## Further reading

- [https://ncatlab.org/nlab/show/well-founded+coalgebra](https://ncatlab.org/nlab/show/well-founded+coalgebra)

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Adapted from the Wikipedia article [Well-founded relation](https://en.wikipedia.org/wiki/Well-founded_relation) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Well-founded_relation?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.