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 $\mathrm{A}\mathrm{r}\mathrm{f}(F)=\{S\mid SisanArfnumericalsemigroupwithFrobeniusnumberF\}$ verifies the following conditions: 1) $\mathrm{\Delta }(F)=\{0,F+1,\to \}$ is the minimum of $\mathrm{A}\mathrm{r}\mathrm{f}(F),$ 2) if $\{S,T\}\subseteq \mathrm{A}\mathrm{r}\mathrm{f}(F)$, then $S\cap T\in \mathrm{A}\mathrm{r}\mathrm{f}(F),$ 3) if $S\in \mathrm{A}\mathrm{r}\mathrm{f}(F),$ $S\mathrm{\ne }\mathrm{\Delta }(F)$ and $\mathrm{m}(S)=\min (S\mathrm{\backslash }\{0\})$, then $S\mathrm{\backslash }\{\mathrm{m}(S)\}\in \mathrm{A}\mathrm{r}\mathrm{f}(F)$. The previous results will be used to give an algorithm which calculates the set $\mathrm{A}\mathrm{r}\mathrm{f}(F)\mathrm{.}$ Also we will see that if $X\subseteq S\mathrm{\backslash }\mathrm{\Delta }(F)$ for some $S\in \mathrm{A}\mathrm{r}\mathrm{f}(F),$ then there is the smallest element of $\mathrm{A}\mathrm{r}\mathrm{f}(F)$ containing $X\mathrm{.}$