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