Let \(\mathcal {C}\subseteq \mathbb {N}^p\) be an integer cone. A \(\mathcal {C}\) -semigroup \(S\subseteq \mathcal {C}\) is an affine semigroup such that the set \(\mathcal {C}{\setminus }S\) is finite. Such \(\mathcal {C}\) -semigroups are central to our study. We develop new algorithms for computing \(\mathcal {C}\) -semigroups with specified invariants, including genus, Frobenius element, and their combinations, among other invariants. To achieve this, we introduce a new class of \(\mathcal {C}\) -semigroups, termed \(\mathcal {B}\) -semigroups. By fixing the degree lexicographic order, we also research the embedding dimension for both ordinary and mult-embedded \(\mathbb {N}^2\) -semigroups. These results are applied to test some generalizations of Wilf’s conjecture.
