In this work we will show that if $F$ is a positive integer, then the set ${ \mathrm{ Arf}}(F)=\{ S\mid S \mbox{ is an Arf numerical semigroup with Frobenius number } F\}$ verifies the following conditions: 1) $\Delta(F)=\{ 0,F+1,\rightarrow\}$ is the minimum of ${ \mathrm{ Arf}}(F),$ 2) if $\{ S, T\} \subseteq { \mathrm{ Arf}}(F)$, then $S \cap T \in { \mathrm{ Arf}}(F),$ 3) if $S \in { \mathrm{ Arf}}(F),$ $S\neq \Delta(F)$ and ${ \mathrm m}(S)=\min (S \backslash \{ 0\})$, then $S\backslash \{ { \mathrm m}(S)\} \in { \mathrm{ Arf}}(F)$. The previous results will be used to give an algorithm which calculates the set ${ \mathrm{ Arf}}(F).$ Also we will see that if $X\subseteq S\backslash \Delta(F)$ for some $S\in { \mathrm{ Arf}}(F),$ then there is the smallest element of ${ \mathrm{ Arf}}(F)$ containing $X.$
Categories: math.AC