Characterizing affine $\mathcal{C}$-semigroups

Let $\mathcal{C}\subset {\mathbb{N}}^p$ be a finitely generated integer cone and $S\subset \mathcal{C}$ be an affine semigroup such that the real cones generated by $\mathcal{C}$ and by $S$ are equal. The semigroup $S$ is called $\mathcal C$-semigroupif$\mathcal C\setminus S$ is a finite set. In this paper, we characterize the $\mathcal{C}$-semigroups from their minimal generating sets, and we give an algorithm to check if $S$ is a $\mathcal{C}$-semigroup and to compute its set of gaps. We also study the embedding dimension of $\mathcal C$-semigroupsobtainingalowerboundforit,andintroducesomefamiliesof$\mathcal C$-semigroups whose embedding dimension reaches our bound. In the last section, we present a method to obtain a decomposition of a $\mathcal C$-semigroupintoirreducible$\mathcal C$-semigroups.