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:
“So that if you would form a just judgment of what is of infinite importance to you not to be misled in,—namely, in what degree of real merit you stand ... call in religion and morality.—Look,—What is written in the law of God?—How readest thou?—Consult calm reason and the unchangeable obligations of justice and truth;Mwhat say they?”
—Laurence Sterne (1713–1768)
“There is, however, this consolation to the most way-worn traveler, upon the dustiest road, that the path his feet describe is so perfectly symbolical of human life,—now climbing the hills, now descending into the vales. From the summits he beholds the heavens and the horizon, from the vales he looks up to the heights again. He is treading his old lessons still, and though he may be very weary and travel-worn, it is yet sincere experience.”
—Henry David Thoreau (1817–1862)
“How have I been able to live so long outside Nature without identifying myself with it? Everything lives, moves, everything corresponds; the magnetic rays, emanating either from myself or from others, cross the limitless chain of created things unimpeded; it is a transparent network that covers the world, and its slender threads communicate themselves by degrees to the planets and stars. Captive now upon earth, I commune with the chorus of the stars who share in my joys and sorrows.”
—Gérard De Nerval (1808–1855)