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:
“It is hard to be finite upon an infinite subject, and all subjects are infinite.”
—Herman Melville (1819–1891)
“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)
“The years seemed to stretch before her like the land: spring, summer, autumn, winter, spring; always the same patient fields, the patient little trees, the patient lives; always the same yearning; the same pulling at the chain—until the instinct to live had torn itself and bled and weakened for the last time, until the chain secured a dead woman, who might cautiously be released.”
—Willa Cather (1873–1947)