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: $$(\forall S \subseteq X)\; [S \neq \varnothing \implies (\exists m \in S) (\forall s \in S) \lnot(s \mathrel{R} m)].$$ 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 $x_{0}, x_{1}, x_{2}, ...$ of elements of $X$ such that $x_{n+1} R x_{n}$ 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, $$(\forall x \in X)\;[(\forall y \in X)\;[y\mathrel{R}x \implies P(y)] \implies P(x)]\quad\text{implies}\quad(\forall x \in X)\,P(x).$$

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 $x ∈ X$, $$G(x) = F\left(x, G\vert_{\left\{y:\, y\mathrel{R}x\right\}}\right).$$

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: Examples of relations that are not well-founded 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 $(n_{1}, n_{2}) < (m_{1}, m_{2})$ if and only if $n_{1} < m_{1}$ and $n_{2} < m_{2}$.
 * 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 $b$.
 * 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.