Ratio-Covarieties of Numerical Semigroups

In this work, we will introduce the concept of ratio-covariety, as a family R of numerical semigroups that has a minimum, denoted by min (R), is closed under intersection, and if S∈R and S≠min (R), then S\r (S) ∈R, where r (S) denotes the ratio of S. The notion of ratio-covariety will allow us to: (1) describe an algorithmic procedure to compute R; (2) prove the existence of the smallest element of R that contains a set of positive integers; and (3) talk about the smallest ratio-covariety that contains a finite set of numerical semigroups. In addition, in this paper we will apply the previous results to the study of the ratio-covariety R (F, m) =S∣S is a numerical semigroup with Frobenius number F and multiplicitym.