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 (or subclass) 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. More formally, a relation is well-founded if: $${\displaystyle (\forall S\subseteq X)\;[S\neq \varnothing \implies (\exists m\in S)(\forall s\in S)\lnot (s\mathrel {R} m)].}$$ 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, meaning there is no infinite sequence x₀, x₁, x₂, ... 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.

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

it suffices to show that:

That is, $${\displaystyle (\forall x\in X)\;[(\forall y\in X)\;[y\mathrel {R} x\implies P(y)]\implies P(x)]\quad {\text{implies}}\quad (\forall x\in X)\,P(x).}$$