Given a set S with a partial order ≤, an infinite descending chain is an infinite, strictly decreasing sequence of elements x1 > x2 > ... > xn > ...
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 possess any infinite descending chains, it is said then, that it satisfies the descending chain condition. Assuming the axiom of choice, the descending chain condition on a partially ordered set is equivalent to requiring that the corresponding strict order is well-founded. 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.
Famous quotes containing the words infinite, descending and/or chain:
“Philosophy, certainly, is some account of truths the fragments and very insignificant parts of which man will practice in this workshop; truths infinite and in harmony with infinity, in respect to which the very objects and ends of the so-called practical philosopher will be mere propositions, like the rest.”
—Henry David Thoreau (1817–1862)
“Man is a stream whose source is hidden. Our being is descending into us from we know not whence. The most exact calculator has no prescience that somewhat incalculable may not balk the very next moment. I am constrained every moment to acknowledge a higher origin for events than the will I call mine.”
—Ralph Waldo Emerson (1803–1882)
“We are all bound to the throne of the Supreme Being by a flexible chain which restrains without enslaving us. The most wonderful aspect of the universal scheme of things is the action of free beings under divine guidance.”
—Joseph De Maistre (1753–1821)