The ratio-covariety of numerical semigroups having maximal embedding dimension with fixed multiplicity and Frobenius number

In this paper we will show that $\MED(F,m)=\{S\mid S \mbox{ is a numeri-}\\ \mbox{cal semigroup with maximal embedding dimension, Frobenius number $F$ and }\\ \mbox{multiplicity }m\}$ is a ratio-covariety. As a consequence, we present two algorithms: one that computes $\MED(F,m)$ and another one that calculates the elements of $\MED(F,m)$ with a given genus. If $X\subseteq S\backslash (\langle m \rangle \cup \{F+1,\rightarrow\})$ for some $S\in \MED(F,m)$, then there exists the smallest element of $\MED(F,m)$ containing $X$. This element will be denoted by $\MED(F,m)[X]$ and we will say that $X$ one of its $\MED(F,m)$-system of generators. We will prove that every element $S$ of $\MED(F,m)$ has a unique minimal $\MED(F,m)$-system of generators and it will be denoted by $\MED(F,m)\msg(S).$ The cardinality of $\MED(F,m)\msg(S)$, will be called $\MED(F,m)$-$\rank$ of $S.$ We will also see in this work, how all the elements of $\MED(F,m)$ with a fixed $\MED(F,m)$-$\rank$ are.