Every positive integer is the Frobenius number of an irreducible numerical semigroup with at most four generators.
The Frobenius number \(g(S)\) of a numerical semigroup \(S\) is the maximum of the set \(\mathbb{N}\setminus S\). A (numerical) semigroup is irreducible if it cannot be expressed as the intersection of two numerical semigroups properly containing it. Irreducible semigroups are closely related to symmetric and pseudo-symmetric ones. The interest in these types of semigroups comes from ring theory, since they provide Gorenstein and Kunz semigroup rings. For any positive integer \(g\), the authors give an easy method to get irreducible semigroups of Frobenius number \(g\). These semigroups have at most four generators, being that the least number of generators needed for the general case.
scientific article; zbMATH DE number 2103850