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:
“Hath not the morning dawned with added light?
And shall not evening call another star
Out of the infinite regions of the night,
To mark this day in Heaven? At last, we are
A nation among nations; and the world
Shall soon behold in many a distant port
Another flag unfurled!”
—Henry Timrod (1828–1867)
“The sun of her [Great Britain] glory is fast descending to the horizon. Her philosophy has crossed the Channel, her freedom the Atlantic, and herself seems passing to that awful dissolution, whose issue is not given human foresight to scan.”
—Thomas Jefferson (1743–1826)
“It could not have come down to us so far,
Through the interstices of things ajar
On the long bead chain of repeated birth,
To be a bird while we are men on earth,”
—Robert Frost (1874–1963)