Definitions

Well-founded_relation

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 of X has a minimal element with respect to R; that is, for every non-empty subset S of X, there is an element m of S such that for every element s of S, the pair (s,m) is not in R:
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 xn+1 R xn 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 or upwards well-founded 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.

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 and P(x) is some property of elements of X and you want to show that P(x) holds for all elements 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. I.e., forall x in X ((forall yin X (y,R,x to P(y))) to 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 F a function, which 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,Gvert_{{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 xx + 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 ab.
  • 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 R defined such that a R b if and only if a is an element of b (assuming 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.

Counterexamples include:

  • the rational numbers under the standard ordering, since, for example, the set of positive rationals 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 well-founded relation: for any set-like well-founded relation R on a class X, 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 leq 1 leq 1 leq 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 ab. 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-ordered relation is changed from the definition above to include this convention.

References

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

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