or any s ∈ S. In other words, a relation is well founded if
(∀ S ⊆ X)[S ≠ Φ → (∃ m ∈ S)(∀ s ∈ S) ¬ (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.[1][2]

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

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