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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 generatedby $\mathcal C$ and by $S$ are equal. The semigroup $S$ is called $\mathcalC$-semigroup if $\mathcal C\setminus S$ is a finite set. In this paper, wecharacterize the $\mathcal C$-semigroups from their minimal generating sets,and we give an algorithm to check if $S$ is a $\mathcal C$-semigroup and tocompute its set of gaps. We also study the embedding dimension of $\mathcalC$-semigroups obtaining a lower bound for it, and introduce some families of$\mathcal C$-semigroups whose embedding dimension reaches our bound. In thelast section, we present a method to obtain a decomposition of a $\mathcalC$-semigroup into irreducible $\mathcal C$-semigroups.

Characterizing affine $\mathcal{C}$-semigroups