Well-founded relation

Template:Binary relations

In mathematics, a binary relation Template:Mvar is called well-founded (or wellfounded or foundational) on a set or, more generally, a class Template:Mvar if every non-empty subset (or subclass) S ⊆ X has a minimal element with respect to Template:Mvar; 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: $$(\mathrm{\forall }S\subseteq X)[S\ne \varnothing ⟹(\mathrm{\exists }m\in S)(\mathrm{\forall }s\in S)\mathrm{\neg }(sRm)].$$ Some authors include an extra condition that Template:Mvar is set-like, i.e., that the elements less than any given element form a set.

Equivalently, assuming the axiom of dependent choice, a relation is well-founded when it contains no infinite descending chains, meaning there is no infinite sequence x₀, x₁, x₂, ... of elements of Template:Mvar such that x_n+1 R x_n for every natural number Template:Mvar.

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 Template:Mvar is called a well-founded set if the set membership relation is well-founded on the transitive closure of Template:Mvar. The axiom of regularity, which is one of the axioms of Zermelo–Fraenkel set theory, asserts that all sets are well-founded.

A relation Template:Mvar is converse well-founded, upwards well-founded, or Noetherian on Template:Mvar, if the converse relation R⁻¹ is well-founded on Template:Mvar. In this case Template:Mvar 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 Template:Mvar, and we want to show that

P(x) holds for all elements Template:Mvar of Template:Mvar,

it suffices to show that:

If Template:Mvar is an element of Template:Mvar and P(y) is true for all Template:Mvar such that y R x, then P(x) must also be true.

That is, $$(\mathrm{\forall }x\in X)[(\mathrm{\forall }y\in X)[yRx⟹P(y)]⟹P(x)]\text{implies}(\mathrm{\forall }x\in X)P(x).$$

Well-founded induction is sometimes called Noetherian induction, 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 Template:Mvar a function that assigns an object F(x, g) to each pair of an element x ∈ X and a function Template:Mvar on the set {y: y R x} of predecessors of Template:Mvar. Then there is a unique function Template:Mvar such that for every x ∈ X, $$G(x)=F(x,G{|}_{\{y:yRx\}}).$$

That is, if we want to construct a function Template:Mvar on Template:Mvar, 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 Template:Mvar is the graph of the successor function x ↦ x+1. Then induction on Template:Mvar is the usual mathematical induction, and recursion on Template:Mvar 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 that are not totally ordered include:

• The positive integers {1, 2, 3, ...}, with the order defined by a < b if and only if Template:Mvar divides Template:Mvar and a ≠ b.
• The set of all finite strings over a fixed alphabet, with the order defined by s < t if and only if Template:Mvar is a proper substring of Template:Mvar.
• The set N × N of pairs of natural numbers, ordered by (n₁, n₂) < (m₁, m₂) if and only if n₁ < m₁ and n₂ < m₂.
• Every class whose elements are sets, with the relation ∈ ("is an element of"). This is the axiom of regularity.
• The nodes of any finite directed acyclic graph, with the relation Template:Mvar defined such that a R b if and only if there is an edge from Template:Mvar to Template:Mvar.

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 set of non-negative rational numbers (or reals) 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 Template:Mvar is an element of Template:Mvar, then the descending chains starting at Template:Mvar are all finite, but this does not mean that their lengths are necessarily bounded. Consider the following example: Let Template:Mvar be the union of the positive integers with a new element ω that is bigger than any integer. Then Template:Mvar 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 Template:Mvar for any Template:Mvar.

The Mostowski collapse lemma implies that set membership is a universal among the extensional well-founded relations: for any set-like well-founded relation Template:Mvar on a class Template:Mvar that is extensional, there exists a class Template:Mvar such that (X, R) is isomorphic to (C, ∈).

Reflexivity

A relation Template:Mvar is said to be reflexive if a R a holds for every Template:Mvar 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 ≤, 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

• Just, Winfried and Weese, Martin (1998) Discovering Modern Set Theory. I, American Mathematical Society Template:Isbn.
• Karel Hrbáček & Thomas Jech (1999) Introduction to Set Theory, 3rd edition, "Well-founded relations", pages 251–5, Marcel Dekker Template:ISBN