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   multiplicity m }. [ABSTRACT FROM AUTHOR]