Numerical semigroups with multiplicity three and four.
A numerical semigroup \(S\) is an additive submonoid of the set \(\mathbb{N}\) of nonnegative integers such that \(\mathbb{N}-S\) is finite. The multiplicity (resp., Frobenius number) of \(S\) is the least positive integer in \(S\) (resp., largest integer not in \(S\)), and the gender of \(S\) is the cardinality of \(\mathbb{N}-S\). Numerical semigroups with multiplicity two are uniquely determined by their Frobenius number. The author first shows that a numerical semigroup of multiplicity three is uniquely determined by its Frobenius number and gender, and then gives a formula for the number of numerical semigroups with multiplicity three and given Frobenius number or gender. The author next introduces a new invariant, called the ratio of \(S\), as \(n_2\), where \(\{n_1,n_2,\dots,n_k\}\) is the unique minimal set of generators for \(S\). He shows that a numerical semigroup of multiplicity four is uniquely determined by its Frobenius number, gender, and ratio, and then investigates the set of numerical semigroups of multiplicity four with given Frobenius number or gender.
scientific article; zbMATH DE number 5014183