The ratio-covariety of numerical semigroups with fixed multiplicity and Frobenius number

In this work we will introduce the concept of ratio-covariety, as a nonempty family $\mathcal{R}$ of numerical semigroups verifying certain properties. This concept will allow us to: \begin{enumerate} \item Describe an algorithmic process to compute $\mathcal{R}\mathrm{.}$ \item Prove the existence of the smallest element of $\text{sR}$ that contains a set of positive integers. \item Talk about the smallest ratio-covariety that contains a finite set of numerical semigroups. \end{enumerate} In addition, in this paper we will apply the previous results to the study of the ratio-covariety $\mathcal{R}(F,m)=\{S\mid SisanumericalsemigroupwithFro-\text{mbox}beniusnumberFandmultiplicitym\}\mathrm{.}$