Well-founded Relation
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Well-founded Relation

In mathematics, a binary relation R is called well-founded (or wellfounded) on a class X if every non-empty subset S ? X has a minimal element with respect to R, that is, an element m not related by sRm (for instance, "s is not smaller than m") for any s ? S. In other words, a relation is well founded if

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 the axiom of dependent choice, a relation is well-founded 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+1R xn for every natural number n.[1][2]

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,

Well-founded induction is sometimes called Noetherian induction,[3] 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,

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 ("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 ?, which is bigger than any integer. Then X is a well-founded set, but there are descending chains starting at ? of arbitrary great (finite) length; the chain ?, 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 aRa 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 To avoid these trivial descending sequences, when working with a partial order a < b if and only if a b and a ? b. More generally, when working with a preorder 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

References

  1. ^ "Condition for Well-Foundedness". ProofWiki. Retrieved 2019.
  2. ^ Fraisse, R. (15 December 2000). Theory of Relations, Volume 145 - 1st Edition (1st ed.). Elsevier. p. 46. ISBN 9780444505422. Retrieved 2019.
  3. ^ Bourbaki, N. (1972) Elements of mathematics. Commutative algebra, Addison-Wesley.

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