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Concept
Well-founded relation
Summary
In mathematics, a binary relation R is called well-founded (or wellfounded or foundational) on a class X if every non-empty subset S ⊆ X has a minimal element with respect to R, that is, an element m ∈ S not related by s R m (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 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.
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, after Emmy Noether.
On par with induction, well-founded relations also support construction of objects by transfinite recursion.
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