Characterizing affine C-semigroups

Let C ⊂ N p be a finitely generated integer cone and S ⊂ C be an affine semigroup such that the real cones generated by C and by S are equal. The semigroup S is called C -semigroup if C \ S is a finite set. In this paper, we characterize the C -semigroups from their minimal generating sets, and we give an algorithm to check if S is a C -semigroup and to compute its set of gaps. We also study the embedding dimension of C -semigroups obtaining a lower bound for it, and introduce some families of C -semigroups whose embedding dimension reaches our bound. In the last section, we present a method to obtain a decomposition of a C -semigroup into irreducible C -semigroups.