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

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