In mathematics, a binary relation is called well-founded (or wellfounded or foundational) on a set or, more generally, a class if every non-empty subset (or subclass) has a minimal element with respect to ; that is, there exists an such that for every , one does not have . More formally, a relation is well-founded if:
<math display=block>(\forall S \subseteq X)\; [S \neq \varnothing \implies (\exists m \in S) (\forall s \in S) \lnot(s \mathrel{R} m)].</math>
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, meaning there is no infinite sequence of elements of such that for every natural number .
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.
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 () 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,
<math display=block>(\forall x \in X)\;[(\forall y \in X)\;[y\mathrel{R}x \implies P(y)] \implies P(x)]\quad\text{implies}\quad(\forall x \in X)\,P(x).</math>
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 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 set of predecessors of . Then there is a unique function such that for every ,
<math display=block>G(x) = F\left(x, G\vert_{\left\{y:\, y\mathrel{R}x\right\\right).</math>
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.
Examples
Well-founded relations that are not totally ordered include:
- The positive integers , with the order defined by if and only if divides and .
- The set of all finite strings over a fixed alphabet, with the order defined by if and only if is a proper substring of .
- The set of pairs of natural numbers, ordered by if and only if and .
- 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 defined such that if and only if there is an edge from to .
Examples of relations that are not well-founded include:
- The negative integers , 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 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 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, but
there 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 .
Reflexivity
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.
References
- Just, Winfried and Weese, Martin (1998) Discovering Modern Set Theory. I, American Mathematical Society .
- Karel Hrbáček & Thomas Jech (1999) Introduction to Set Theory, 3rd edition, "Well-founded relations", pages 251–5, Marcel Dekker
Further reading
- https://ncatlab.org/nlab/show/well-founded+coalgebra