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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.

The Tree of Good Semigroups in $${\mathbb {N}}^2$$ and a Generalization of the Wilf Conjecture