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)
“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)
“The conclusion suggested by these arguments might be called the paradox of theorizing. It asserts that if the terms and the general principles of a scientific theory serve their purpose, i. e., if they establish the definite connections among observable phenomena, then they can be dispensed with since any chain of laws and interpretive statements establishing such a connection should then be replaceable by a law which directly links observational antecedents to observational consequents.”
—C.G. (Carl Gustav)