Characterizing affine C-semigroups

Let C subset of N-p be a finitely generated integer cone and S subset of 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.