Gordan's lemma

Gordan's lemma is a lemma in convex geometry and algebraic geometry. It can be stated in several ways.

• Let ${\displaystyle A}$ be a matrix of integers. Let ${\displaystyle M}$ be the set of non-negative integer solutions of ${\displaystyle A\cdot x=0}$. Then there exists a finite subset of vectors in ${\displaystyle M}$, such that every element of ${\displaystyle M}$ is a linear combination of these vectors with non-negative integer coefficients.^[1]
• The semigroup of integral points in a rational convex polyhedral cone is finitely generated.^[2]
• An affine toric variety is an algebraic variety (this follows from the fact that the prime spectrum of the semigroup algebra of such a semigroup is, by definition, an affine toric variety).

The lemma is named after the mathematician Paul Gordan (1837–1912). Some authors have misspelled it as "Gordon's lemma".