The set of Arf numerical semigroups with given Frobenius number

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 $X.$