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

A proportionally modular affine semigroup is the set of nonnegative integer solutions of a modular Diophantine inequalitywhere g 1 , . . . , g n , f 1 , . . . , f n ∈ Z and b ∈ N. In this work, a geometrical characterization of these semigroups is given. Moreover, some algorithms to check if a semigroup S in N n , with N n \ S a finite set, is a proportionally modular affine semigroup are provided by means of that geometrical approach.