In mathematics, Dickson's lemma is a finiteness statement applying to n-tuples of natural numbers. It is a simple fact from combinatorics, which has become attributed to the American algebraist Dickson . It was certainly known earlier, for example to Gordan in his researches on invariant theory.

Stating it first for clarity for N², for any pair (m,n) of natural numbers we can introduce R_m,n, the 'rectangle' of numbers (r, s) with r at least m and s at least n. This is semi-infinite in the north and east directions, in the usual plane representation. The lemma then states that any union of the R_m,n is a finite union.

The generalization to N^k is the natural one, with k-tuples in place of pairs.

The statement says something about N^k as the topological space with the product topology arising from N, where the latter has the (semi-continuity) topology in which the open sets are all sets R_m defined as all n with n at least m. The 'rectangles' are by definition a base for the topology; it says finite unions give all open sets.

As for the proof of the lemma, it can be derived directly, but a slick way is to show that it is a special case of Hilbert's basis theorem - in fact is essentially the case of ideals generated by monomials.