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

## TL;DR

This paper introduces new concepts of length and genus for good semigroups in multi-dimensional natural numbers, explores their enumeration, and examines their relationships with existing concepts, extending the Wilf conjecture to this context.

## Contribution

It generalizes the notion of length and genus for good semigroups in ^n and provides methods to count local good semigroups with a fixed genus, extending previous work in two-branch cases.

## Key findings

- Counted local good semigroups with fixed genus
- Established relationships with existing concepts in two-branch cases
- Extended the Wilf conjecture to higher dimensions

## 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.