Gordan's lemma
Introduction '''Gordan's lemma''' is a lemma in convex geometry and algebraic geometry. It can be stated in several ways. * Let  be a matrix of integers. Let  be the set of non-negative integer solutions of . Then there exists a finite subset of vectors in , such that every element of  is a linear combination of these vectors with non-negative integer coefficients. * The semigroup of integral points in a rational convex polyhedral cone is finitely generated. * 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".
Proofs There are topological and algebraic proofs.=== Topological proof ===Let  be the dual cone of the given rational polyhedral cone. Let  be integral vectors so that  Then the 's generate the dual cone ; indeed, writing ''C'' for the cone generated by 's, we have: , which must be the equality. Now, if ''x'' is in the semigroup:then it can be written as:where  are nonnegative integers and . But since ''x'' and the first sum on the right-hand side are integral, the second sum is a lattice point in a bounded region, and so there are only finitely many possibilities for the second sum (the topological reason). Hence,  is finitely generated.=== Algebraic proof ===The proof is based on a fact that a semigroup ''S'' is finitely generated if and only if its semigroup algebra  is a finitely generated algebra over . To prove Gordan's lemma, by induction (cf. the proof above), it is enough to prove the following statement: for any unital subsemigroup ''S'' of ,: If ''S'' is finitely generated, then , ''v'' an integral vector, is finitely generated.Put , which has a basis . It has -grading given by:.By assumption, ''A'' is finitely generated and thus is Noetherian. It follows from the algebraic lemma below that  is a finitely generated algebra over . Now, the semigroup  is the image of ''S'' under a linear projection, thus finitely generated and so  is finitely generated. Hence,  is finitely generated then. '''Lemma''': Let ''A'' be a -graded ring. If ''A'' is a Noetherian ring, then  is a finitely generated -algebra.Proof: Let ''I'' be the ideal of ''A'' generated by all homogeneous elements of ''A'' of positive degree. Since ''A'' is Noetherian, ''I'' is actually generated by finitely many , homogeneous of positive degree. If ''f'' is homogeneous of positive degree, then we can write  with  homogeneous. If ''f'' has sufficiently large degree, then each  has degree positive and strictly less than that of ''f''. Also, each degree piece  is a finitely generated -module. (Proof: Let  be an increasing chain of finitely generated submodules of  with union . Then the chain of the ideals  stabilizes in finite steps; so does the chain ) Thus, by induction on degree, we see  is a finitely generated -algebra.
Applications A ''multi-hypergraph'' over a certain set  is a multiset of subsets of '''' (it is called "multi-hypergraph" since each hyperedge may appear more than once). A multi-hypergraph is called ''regular'' if all vertices have the same degree. It is called ''decomposable'' if it has a proper nonempty subset that is regular too. For any integer ''n'', let  be the maximum degree of an indecomposable multi-hypergraph on ''n'' vertices. Gordan's lemma implies that  is finite. ''Proof'': for each subset ''S'' of vertices, define a variable ''xS'' (a non-negative integer). Define another variable ''d'' (a non-negative integer). Consider the following set of ''n'' equations (one equation per vertex):Every solution ('''x''',''d'') denotes a regular multi-hypergraphs on , where '''x''' defines the hyperedges and ''d'' is the degree. By Gordan's lemma, the set of solutions is generated by a finite set of solutions, i.e., there is a finite set  of multi-hypergraphs, such that each regular multi-hypergraph is a linear combination of some elements of . Every non-decomposable multi-hypergraph must be in  (since by definition, it cannot be generated by other multi-hypergraph). Hence, the set of non-decomposable multi-hypergraphs is finite.
See also * Birkhoff algorithm is an algorithm that, given a bistochastic matrix (a matrix which solves a particular set of equations), finds a decomposition of it into integral matrices. It is related to Gordan's lemma in that it shows that the set of these matrices is generated by a finite set of integral matrices.
References
See also *Dickson's lemma