From this section we begin to deal with Arf good semigroups of $\mathbb{N}^r$. First of all we recall some important definitions and results A good semigroup $S$ of $\mathbb{N}^r$ is a submonoid of $\left( \mathbb{N}^r,+\right)$ such that (cf. [@BDF])

-   For all $a,b \in S$, $    \min(a,b) \in S$;

-   If $a,b\in S$ and $a[i]=b[i]$ for some $i \in \left\{ 1, \ldots,n\right\}$, then there exists $c \in S$ such that $c[i] > a[i]=b[i]$, $c[j]\geq \min(a[j],b[j])$ for $j \in  \left\{ 1, \ldots,n\right\} \setminus \left\{ i\right\}$ and $c[j]=\min(a[j],b[j])$ if $a[j]\neq b[j]$;

-   There exists $\delta \in S$ such that $\delta+\mathbb{N}^r \subseteq S$

(where we are considering the usual partial ordering in $\mathbb{N}^r$: $a \leq b$ if $a[i] \leq b[i]$ for each $i=1,\ldots,r$). The previous properties guarantee that for a good semigroup $S$ there exists a minimal vector $c(S)$ such that $c(S)+ \mathbb{N}^r \subseteq S$. Such a vector is said to be the conductor of $S$.\
In this paper we will always deal with local good semigroups. A good semigroup $S$ is local if the zero vector is the only vector of $S$ with some component equal to zero. However, it can be shown that every good semigroup is the direct product of 