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:
“Nothing could his enemies do but it rebounded to his infinite advantage,—that is, to the advantage of his cause.... No theatrical manager could have arranged things so wisely to give effect to his behavior and words. And who, think you, was the manager? Who placed the slave-woman and her child, whom he stooped to kiss for a symbol, between his prison and the gallows?”
—Henry David Thoreau (1817–1862)
“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)
“To avoid tripping on the chain of the past, you have to pick it up and wind it about you.”
—Mason Cooley (b. 1927)