Well-founded relation

In mathematics, a binary relation, R, is well-founded (or wellfounded) on a class X if and only if every non-empty subset S ⊆ X has a minimal element; that is, some element m of any S is not related by sRm (for instance, "m is not smaller than") for any s ∈ S.

$\forall S \subseteq X\ (S \neq \varnothing \to \exists m \in S\;\; \forall s \in S\;\, ( s, m) \notin R).$

(Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form a set.)

Equivalently, assuming some choice, a relation is well-founded if and only if it contains no countable infinite descending chains: that is, there is no infinite sequence x₀, x₁, x₂, ... of elements of X such that x_n+1 R x_n for every natural number n.

In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. If the order is a total order then it is called a well-order.

In set theory, a set x is called a well-founded set if the set membership relation is well-founded on the transitive closure of x. The axiom of regularity, which is one of the axioms of Zermelo–Fraenkel 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 R⁻¹ is well-founded on X. In this case R is also said to satisfy the ascending chain condition. In the context of 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 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,

$\forall x\in X\,[(\forall y\in X\,(y\,R\,x \to P(y))) \to P(x)]\to\forall x \in X\,P(x).$

Well-founded induction is sometimes called Noetherian induction,^[1] after Emmy Noether.

On par with induction, well-founded relations also support construction of objects by transfinite recursion. Let (X, R) be a set-like 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 initial segment {y: y R x} of X. Then there is a unique function G such that for every x ∈ X,

$G(x)=F(x,G\vert_{\{y: y\,R\,x\}})$

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, and S is the graph of the successor function x → x + 1. Then induction on S is the usual mathematical induction, and recursion on S gives primitive recursion. If we consider the order relation (N, <), we obtain complete induction, and course-of-values recursion. The statement that (N, <) is well-founded is also known as the 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, the technique is called transfinite induction. When the well-founded set is a set of recursively-defined data structures, the technique is called structural induction. When the well-founded relation is set membership on the universal class, the technique is known as ∈-induction. See those articles for more details.

Examples

Well-founded relations which are not totally ordered include:

• the positive integers {1, 2, 3, ...}, with the order defined by a < b if and only if a divides b and a ≠ b.
• the set of all finite strings 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 of natural numbers, ordered by (n₁, n₂) < (m₁, m₂) if and only if n₁ < m₁ and n₂ < m₂.
• the set of all regular expressions over a fixed alphabet, with the order defined by s < t if and only if s is a proper subexpression of t.
• any class whose elements are sets, with the relation $\in$ ("is an element of"). This is the axiom of regularity.
• the nodes of any finite 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) 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 rational numbers (or reals) under the standard ordering, since, for example, the set 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 and a new element ω, which 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 implies that set membership is a universal among the extensional well-founded relations: for any set-like well-founded relation R on a class X which is extensional, there exists a class C such that (X,R) is isomorphic to (C,∈).

Reflexivity

A relation R is said to be reflexive 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 $1 \geq 1 \geq 1 \geq \cdots$. To avoid these trivial descending sequences, when working with a reflexive relation R it is common to use (perhaps implicitly) the alternate relation R′ defined such that a R′ b if and only if a R b and a ≠ b. 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 this convention.

References

1. Bourbaki, N. (1972) Elements of mathematics. Commutative algebra, Addison-Wesley.

• Just, Winfried and Weese, Martin, Discovering Modern Set theory. I, American Mathematical Society (1998) ISBN 0-8218-0266-6.