The minimal system of generators of an affine, plane and normal semigroup

Abstract

If X is a nonempty subset of Qk , the cone generated by X is C(X) = {q1x1 + · · ·+qnxn | n ∈ N\{0},{q1, . . . ,qn} ⊆ Q+0 and {x1, . . . ,xn} ⊆ X}. In this work we present an algorithm which calculates from {(a1,b1), (a2,b2)} ⊆ N2 , the minimal system of generators of the affine semigroup C({(a1,b1), (a2,b2)}) ∩N2. This algorithm is based on the study of proportionally modular Diophantine inequalities carried out in [1]. Also, we present an upper bound for the embedding dimension of this semigroup.