The covariety of saturated numerical semigroups with fixed Frobenius number

In this work we will show that if $F$ is a positive integer, then $\mathrm{S}\mathrm{a}\mathrm{t}(F)=\{S\mid SisasaturatednumericalsemigroupwithFrobeniusnumberF\}$ is a covariety. As a consequence, we present two algorithms: one that computes $\mathrm{S}\mathrm{a}\mathrm{t}(F),$ and the other which computes all the elements of $\mathrm{S}\mathrm{a}\mathrm{t}(F)$ with a fixed genus. If $X\subseteq S\mathrm{\backslash }\mathrm{\Delta }(F)$ for some $S\in \mathrm{S}\mathrm{a}\mathrm{t}(F),$ then we will see that there is the least element of $\mathrm{S}\mathrm{a}\mathrm{t}(F)$ containing a $X$. This element will denote by $\mathrm{S}\mathrm{a}\mathrm{t}(F)[X]\mathrm{.}$ If $S\in \mathrm{S}\mathrm{a}\mathrm{t}(F),$ then we define the $\mathrm{S}\mathrm{a}\mathrm{t}(F)$-rank of $S$ as the minimum of $\{cardinality(X)\mid S=\mathrm{S}\mathrm{a}\mathrm{t}(F)[X]\}\mathrm{.}$ In this paper, also we present an algorithm to compute all the element of $\mathrm{S}\mathrm{a}\mathrm{t}(F)$ with a given $\mathrm{S}\mathrm{a}\mathrm{t}(F)$-rank.