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abstract: 'In this paper we give an algorithm for the computation of all the Arf numerical semigroups with a given genus. Moreover, we generalize the concept of genus of a numerical semigroup to good semigroups of $\mathbb{N}^r$ and we give a procedure to calculate all the Arf semigroups of $\mathbb{N}^r$ with a given genus.'
author:
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    Giuseppe Zito\
    \
    \
title: Arf good semigroups with fixed genus
---

Introduction {#introduction .unnumbered}
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In this paper we present procedures concerning Arf subsemigroups of $\mathbb{N}^r$ for $r \geq 1$. In particular, for $r=1$, we deal with the concept of Arf numerical semigroup whose importance is motivated by its role in the study of equivalence between algebroid branches. Specifically, two algebroid branches are said to be equivalent if they have the same multiplicity sequence (cf. [@Campi Definition 1.5.11]), that is the sequence of the multiplicities of their succesive blow ups. In [@Arf], Arf gave an algebraic interpretation of this equivalence, introducing the concept of Arf ring and showing that for each ring $R$ there exists a smallest Arf overring $R'$, called the Arf closure of $R$. A ring $R$ and its Arf closure have the same multiplicity sequence, and from this 