Infinite descending chain

Given a set S with a partial order ≤, an infinite descending chain is a chain V that is a subset of S upon which ≤ defines a total order such that V has no least element, that is, an element m such that for all elements n in V it holds that m ≤ n.

As an example, in the set of integers, the chain −1, −2, −3, ... is an infinite descending chain, but there exists no infinite descending chain on the natural numbers, as every chain of natural numbers has a minimal element.

If a partially ordered set does not contain any infinite descending chains, it is called well-founded or, in some case, Artinian; it is then said to satisfy the descending chain condition. A stronger condition, that there be no infinite descending chains and no infinite antichains, defines the well-quasi-orderings. A totally ordered set without infinite descending chains is called well-ordered.

References

• Yiannis N. Moschovakis (2006) Notes on set theory, Undergraduate texts in mathematics (Birkhäuser) ISBN 0-387-28723-X, p.116