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:
“The process of writing has something infinite about it. Even though it is interrupted each night, it is one single notation.”
—Elias Canetti (b. 1905)
“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)
“Nae living man I’ll love again,
Since that my lovely knight is slain.
Wi ae lock of his yellow hair
I’ll chain my heart for evermair.”
—Unknown. The Lament of the Border Widow (l. 25–28)