Degree of singularity

Definition

Let \(S\) be a numerical semigroup, \(t\) be an indeterminant and \(\mathbb{K}\) be a field. As in the semigroup ring case, it is defined \(\mathbb{K}[[t^s ~ | ~ s \in S]]\), denoted by \(\mathbb{K}[[S]]\), and is a subring of the ring power series \(\mathbb{K}[[t]]\). It is defined the degree of singularity of \(\mathbb{K}[[S]]\) as the length of the \(\mathbb{K}[[S]]-\)module \(\mathbb{K}[[t]]/\mathbb{K}[[S]]\). Some authors use the terminology degree of singularity to refer to the genus of a numerical semigroup.

References

Assi, Abdallah, Marco D’Anna, and Pedro A. Garcı́a-Sánchez. 2020. Numerical Semigroups and Applications. Vol. 3. RSME Springer Series. Springer, [Cham]. https://doi.org/10.1007/978-3-030-54943-5.

Delgado, M., P. A. Garcia-Sanchez, and J. Morais. 2024. “NumericalSgps, a Package for Numerical Semigroups, Version 1.4.0.” https://gap-packages.github.io/ numericalsgps.