The tree of good semigroups in $\mathbb{N}^2$ and a generalization of the Wilf conjecture

Abstract

In this work, we study good semigroups of $\mathbb{N}^n$ introducing the definition of length and genus for these objects. We show how to count the local good semigroup with a fixed genus. Furthermore, we study the relationships of these concepts with other ones previously defined in the case of good semigroups with two branches.