• Source: Well-founded relation

In mathematics, a binary relation R is called well-founded (or wellfounded or foundational) on a set or, more generally, a class X if every non-empty subset S ⊆ X has a minimal element with respect to R; that is, there exists an m ∈ S such that, for every s ∈ S, one does not have s R m. In other words, a relation is well-founded if:




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∀
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S
≠
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⟹

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∃
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{\displaystyle (\forall S\subseteq X)\;[S\neq \varnothing \implies (\exists m\in S)(\forall s\in S)\lnot (s\mathrel {R} m)].}


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 when it contains no infinite descending chains, which can be proved when 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, 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,




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∀
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∈
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(
∀
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∈
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[
y

R

x

⟹

P
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implies


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{\displaystyle (\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).}


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 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,




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{\displaystyle G(x)=F\left(x,G\vert _{\left\{y:\,y\mathrel {R} x\right\}}\right).}


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 that 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.
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 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 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 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 with a new element ω that 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 among the extensional well-founded relations: for any set-like well-founded relation R on a class X that 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 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 ≥ 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 ISBN 0-8218-0266-6.
Karel Hrbáček & Thomas Jech (1999) Introduction to Set Theory, 3rd edition, "Well-founded relations", pages 251–5, Marcel Dekker ISBN 0-8247-7915-0

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