In mathematics, a binary relation is called well-founded (or wellfounded or foundational[1]) on a set or, more generally, a class if every non-empty subset has a minimal element with respect to ; that is, there exists an such that, for every, one does not have . In other words, a relation is well founded if:Some authors include an extra condition that 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, which can be proved when there is no infinite sequence of elements of such that for every natural number .[2] [3]
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 is called a well-founded set if the set membership relation is well-founded on the transitive closure of . The axiom of regularity, which is one of the axioms of Zermelo–Fraenkel set theory, asserts that all sets are well-founded.
A relation is converse well-founded, upwards well-founded or Noetherian on, if the converse relation is well-founded on . In this case is also said to satisfy the ascending chain condition. In the context of rewriting systems, a Noetherian relation is also called terminating.
An important reason that well-founded relations are interesting is because a version of transfinite induction can be used on them: if is a well-founded relation, is some property of elements of, and we want to show that
holds for all elements of,
it suffices to show that:
If is an element of and is true for all such that, then must also be true.
That is,
Well-founded induction is sometimes called Noetherian induction,[4] after Emmy Noether.
On par with induction, well-founded relations also support construction of objects by transfinite recursion. Let be a set-like well-founded relation and a function that assigns an object to each pair of an element and a function on the initial segment of . Then there is a unique function such that for every,
That is, if we want to construct a function on, we may define using the values of for .
As an example, consider the well-founded relation, where is the set of all natural numbers, and is the graph of the successor function . Then induction on is the usual mathematical induction, and recursion on gives primitive recursion. If we consider the order relation, we obtain complete induction, and course-of-values recursion. The statement that 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.
Well-founded relations that are not totally ordered include:
Examples of relations that are not well-founded include:
If is a well-founded relation and is an element of, then the descending chains starting at are all finite, but this does not mean that their lengths are necessarily bounded. Consider the following example: Let be the union of the positive integers with a new element ω that is bigger than any integer. Then is a well-founded set, butthere are descending chains starting at ω of arbitrary great (finite) length; the chain has length for any .
The Mostowski collapse lemma implies that set membership is a universal among the extensional well-founded relations: for any set-like well-founded relation on a class that is extensional, there exists a class such that is isomorphic to .
A relation is said to be reflexive if holds for every 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 . 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 if and only if and . More generally, when working with a preorder ≤, it is common to use the relation < defined such that if and only if and . 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.