The set of Arf numerical semigroups with given Frobenius number

Abstract

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) $\Delta (F)=\{0,F+1,\to \}$ 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 $S\in \mathrm{Arf}(F),$ $S\ne \Delta (F)$ 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 $X.$

15 pages