Dense, positioned and almost-positioned numerical semigroups - Faria, Manuel António Silva Ramos Caldas (2022)
The main purpose for this work is to solve the Frobenius problem for some classes of numerical semigroups and prove the Wilf’s conjecture for some other classes of numerical semigroups. First we study dense numerical semigroups, presenting algorithms that allow us to obtain the whole set of these semigroups, and we solve the Frobenius problem for embedding dimension three. Then we treat the positioned numerical semigroups, presenting algorithms that allow us to obtain the whole set of these semigroups and we prove Wilf’s conjecture. Finally we study the almost-positioned numerical semigroups, again proving Wilf’s conjecture.