Dickson's lemma


In mathematics, Dickson's lemma states that every set of -tuples of natural numbers has finitely many minimal elements. This simple fact from combinatorics has become attributed to the American algebraist L. E. Dickson, who used it in order to prove a result in number theory about perfect numbers. However, the lemma was certainly known earlier, for example to Paul Gordan in his research on invariant theory.

Example

Let be a fixed number, and let be the set of pairs of numbers whose product is at least. When defined over the positive real numbers, has infinitely many minimal elements of the form, one for each positive number ; this set of points forms one of the branches of a hyperbola. The pairs on this hyperbola are minimal, because it is not possible for a different pair that belongs to to be less than or equal to in both of its coordinates. However, Dickson's lemma concerns only tuples of natural numbers, and over the natural numbers there are only finitely many minimal pairs. Every minimal pair of natural numbers has and, for if x were greater than K then would also belong to S, contradicting the minimality of, and symmetrically if y were greater than K then would also belong to S. Therefore, over the natural numbers, has at most minimal elements, a finite number.

Formal statement

Let be the set of non-negative integers, let n be any fixed constant, and let be the set of -tuples of natural numbers. These tuples may be given a pointwise partial order, the product order, in which if and only if, for every,.
The set of tuples that are greater than or equal to some particular tuple forms a positive orthant with its apex at the given tuple.
With this notation, Dickson's lemma may be stated in several equivalent forms:
Dickson used his lemma to prove that, for any given number, there can exist only a finite number of odd perfect numbers that have at most prime factors. However, it remains open whether there exist any odd perfect numbers at all.
The divisibility relation among the P-smooth numbers, natural numbers whose prime factors all belong to the finite set P, gives these numbers the structure of a partially ordered set isomorphic to. Thus, for any set S of P-smooth numbers, there is a finite subset of S such that every element of S is divisible by one of the numbers in this subset. This fact has been used, for instance, to show that there exists an algorithm for classifying the winning and losing moves from the initial position in the game of Sylver coinage, even though the algorithm itself remains unknown.
The tuples in correspond one-for-one with the monomials over a set of variables. Under this correspondence, Dickson's lemma may be seen as a special case of Hilbert's basis theorem stating that every polynomial ideal has a finite basis, for the ideals generated by monomials. Indeed, Paul Gordan used this restatement of Dickson's lemma in 1899 as part of a proof of Hilbert's basis theorem.