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:
“Vanity is as advantageous to a government as pride is dangerous. To be convinced of this we need only represent, on the one hand, the numberless benefits which result from vanity, as industry, the arts, fashions, politeness, and taste; and on the other, the infinite evils which spring from the pride of certain nations, a laziness, poverty, a total neglect of everything.”
—Charles Louis de Secondat Montesquieu (1689–1755)
“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 name of the town isn’t important. It’s the one that’s just twenty-eight minutes from the big city. Twenty-three if you catch the morning express. It’s on a river and it’s got houses and stores and churches. And a main street. Nothing fancy like Broadway or Market, just plain Broadway. Drug, dry good, shoes. Those horrible little chain stores that breed like rabbits.”
—Joseph L. Mankiewicz (1909–1993)