# Well-founded relation

In mathematics, a binary relation R is called well-founded (or wellfounded) on a class X if every non-empty subset SX has a minimal element with respect to R, that is, an element m not related by sRm (for instance, "s is not smaller than m") for any sS. In other words, a relation is well founded if

$(\forall S\subseteq X)\;[S\neq \emptyset \implies (\exists m\in S)(\forall s\in S)\lnot (sRm)].$ 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 if it contains no countable infinite descending chains: that is, 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.